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x - ray emission is a major contributor to the observed spectrum of seyfert galaxies , and yet the physics of the emitting region is still not well understood . the most common and ( thus far ) successful approach to this problem , to which we shall refer as the ` spectral approach ' ( sa ) , makes very simple assumptions about the geometry and/or the particle heating mechanisms , but uses a detailed microphysical approach to account for the particle - photon interactions and to derive the spectrum . this spectrum is then compared with the observations in order to place constraints on the parameters of the emitting regions . the early models assumed a non - thermal pair dominated plasma . ( for a comprehensive review of non - thermal models see svensson 1994 . ) however , with the more recent substantial progress made in the x - ray observation of seyfert galaxies ( e.g. , jourdain et al . 1992 ; johnson et al . 1993 ) , it is now evident that thermal models are strongly favored by the data . accordingly , much of the current attention is focused on thermal models ( svensson 1996a ) . aside from the question concerning the nature of the particle distribution , there is also the issue regarding the emitter s geometry . haardt & maraschi ( 1991 , 1993 ) argued that if most of the energy is dissipated in a hot corona overlying a cold accretion disk , then the resulting spectrum naturally explains many of the observed features in these sources . in particular , roughly half of the coronal x - ray emission is directed towards the cold disk , where it gets absorbed and re - emitted as uv radiation , which then re - enters the corona and contributes to the cooling of the electrons . thus , the lepton cooling rate becomes proportional to the heating rate . in this case , the inverse compton up - scattering of the uv radiation leads to an almost universal x - ray spectral index , consistent with the observations ( e.g. , according to nandra & pounds 1994 , @xmath2 for a sample of seyfert galaxies ) . the hardening of the spectrum above about 10 kev ( nandra & pounds 1994 ) and a broad hump at @xmath3 50 kev ( e.g. , zdziarski et al 1995 ) are accounted for by reflection of the hard x - rays in the cold disk . however , observationally the hard x - ray luminosity , @xmath4 , can be a few times smaller than the luminosity , @xmath5 , in the soft uv - component . this is inconsistent with the uniform two - phase disk coronal model , because the latter predicts about the same luminosity in both x - rays and uv ( due to the fact that all the uv radiation arises as a consequence of reprocessing of the hard x - ray flux , which is about equal in the upward and downward directions ) . to overcome this apparent difficulty , haardt , maraschi & ghisellini ( 1994 ) introduced a patchy disk - coronal model , which assumes that the x - ray emitting region consists of separate active regions(ar ) independent of each other . in this case , a portion of the reprocessed as well as intrinsic radiation from the cold disk escapes to the observer directly , rather than entering ars , thus allowing for a greater ratio of @xmath6 . recently , stern et al . ( 1995 ) and poutanen & svensson ( 1996a ) carried out state of the art calculations of the radiative transport of the anisotropic polarized radiation , for a range of ar geometries . they showed that this type of model indeed reproduces the observed x - ray spectral slope , the compactness , and the high - energy cutoff . the model has very few parameters , namely , the compactness and the temperature of the intrinsic / reprocessed radiation from the cold disk . therefore , it appears that the model is very robust in its predictions . on the other hand , another somewhat less common approach to explaining the x - rays from galactic black hole candidates ( gbhc ) and seyfert galaxies , which we shall call the ` magnetic flare ' ( mf ) approach , is being developed by analogy with the strong , energetic ( x - ray emitting ) flares observed on the sun . a pioneering paper in this field was that of galeev , rosner & vaiana ( 1979 ) , who showed that the physical conditions in an accretion disk surrounding a black hole are such that magnetic fields are likely to grow to equipartition values . this magnetic field is then transported to the surface of the disk by buoyancy forces where its energy is released in a flare - like event . the magnetic flare approach is , in a sense , complementary to the spectral approach , in that it attempts to include all the relevant physics self - consistently ( e.g. , de vries & kuijpers 1992 ; van oss , van den oord & kuperus 1993 ; volwerk , van oss & kuijpers 1993 ) . unfortunately , the physics involved is quite complex and still somewhat open to debate . the resulting spectrum is a combination of time - averaged components from many different flares , and is subject to many uncertainties clearly the mf model must invoke many more parameters , or assumptions about the magnetic field reconnection , than does the sa approach . therefore , no detailed spectra from these events ( in the case of seyfert galaxies ) have yet been computed . one may argue that to make substantial progress , these two approaches need to find an overlap of self - consistency . in particular , the sa model does not specify the mechanism by which the gravitational energy dissipated within the cold disk is transported out to the optically thin corona . it is _ assumed _ that some process can provide the needed electron heating , and often a reference is made to magnetic fields . moreover , the particle dynamics is ignored , imposing instead the artificial constraint that the particles are confined to a closed box . thus , even though the sa model can reproduce the observed spectrum quite well , the situation is unsatisfactory from a broader theoretical perspective . correspondingly , it appears that the most important results obtained within the framework of the sa model have not been fully incorporated into the magnetic flare scenarios . for example , it is well known that the universal x - ray spectral index in seyfert galaxies is best explained by the inverse comptonization of soft uv photons . this requires a relatively high value of the compactness parameter ( see discussion below ) in the emitting region . as far as we can tell , no work has yet been done to show ( based on the physics of reconnection or some other mechanism for the transfer of energy from the magnetic field to the particles ) that a specific mf model can indeed provide the needed high compactness during the active phase , though haardt , maraschi & ghisellini ( 1994 , hereafter hmg94 ) did use the physics of magnetic flares to account for the heating rates and the required confinement of the ars . they showed that the compactness can be high enough during the active phase if one assumes that the entire magnetic field energy is transfered to the particles during a few light - crossing time scales . they did not , however , explicitly consider the question of how the spectrum from these highly transient phenomena is formed . more recently , nayakshin & melia ( 1997b ) considered the issue of pressure balance within the plasma trapped inside the flare during the active phase . they found that under certain conditions , a pressure equilibrium can be maintained in the source if its thomson optical depth is @xmath7 . they also showed that the current data can not distinguish between a spectrum comprised of a single flare component and one formed from many different flares with a range @xmath8 in @xmath9 . in other words , one can always find a @xmath9 for the spectrum assuming a single flare that represents the composite spectrum quite well out to about 100 kev , where the quality of the data deteriorates . in addition , nayakshin & melia ( 1997c ) have considered the implications of a time - dependent x - ray reflection and reprocessing by the cold disk underneath the flare . they find that due to the short lived , but very intense x - ray flux from the ar , the upper layer of the disk is compressed to a density in excess of that found in the disk s mid - plane . under these conditions , the x - ray reprocessing leads to a temperature of the emitted uv radiation that is roughly independent of the x - ray luminosity and the overall bolometric luminosity of the source , as suggested by the euv - soft x - ray observations ( walter & fink 1993 ; fink et al . 1994 ; zhou et al . 1997 ) . due to the increased gas density in the compressed layer , the ionization parameter is smaller than that arising in time - independent x - ray reflection ( i.e. , when the x - ray source is assumed to be stationary a condition that is clearly violated in magnetic flares ) . this may explain those observations of seyfert galaxies that suggest the presence of a nearly neutral reflector ( zdziarski et al . 1996 ) . these results strengthen the mf model and motivate us here to attempt to assemble the various components of this picture . we first discuss the different physical constraints imposed on the ars by both the spectral observations and the physics of the corresponding processes , without necessarily confining our discussion to the mf model . we will then show that magnetic flares above the cold disk are probably the best candidates for producing these ars , and we discuss the physics of the mf model in greater depth . we conclude by listing some of the unresolved issues . our first task here will be to assemble the various constraints imposed on the ars in seyfert galaxies from observations and theoretical considerations . in so doing , we shall first summarize the better known results , and then discuss the additional constraints that follow from various attempts to construct realistic ars based on the idea that these may be magnetic structures , characterized by a well - defined confinement and energy supply . the most important parameter of the ar is the compactness @xmath10 , where @xmath11 is the radiation energy flux at the top of the ar and @xmath12 is its typical size . note that this definition is for the local compactness , i.e. , the one that characterizes the local properties of the plasma , unlike the global compactness @xmath13 , where @xmath14 is the total luminosity of the object and @xmath15 is the typical size of the region that emits this luminosity . it is the latter that should be compared to the observed compactness rather than the former . consider the following example . assume that the emitting region is a full disk - like corona . in this case the local and global compactnesses are related in this way : @xmath16 where @xmath17 is the coronal scale height , which is unlikely to be larger than the accretion disk scale height , @xmath18 , and so the local compactness @xmath19 can be much smaller than the global one . at the same time , as suggested by the frequently observed large ratio @xmath6 ( e.g. , hmg94 ; svensson 1996a ) , the emitting region can consist of a large number of small localized areas . since the total x - ray luminosity from these ars should be the same as that in the model with a full corona , it is clear that the local compactness of each region must be larger than that of the full corona . in particular , depending on the ratio of the total active area @xmath20 covered by the ars to @xmath21 , the local compactness can be either larger or smaller than the global one . therefore , even though the observed values of global compactness for seyfert galaxies lie in the range @xmath22 ( done & fabian 1989 ; but see also fabian 1994 ) , one can not argue that the local compactness @xmath19 should be larger than these values based on observations alone . however , a large local compactness is strongly preferred in current pair - dominated two - phase models ( e.g. , svensson 1996a ; zdziarski et al . 1996 ) . to produce the correct spectrum , @xmath23 should be relatively large ( @xmath1 ) . in the context of the pair - dominated two - phase model , the only mechanism for fixing the optical depth is by pair equilibrium , and thus one needs @xmath24 in order to create them , in which case the optical depth of the ars becomes a function of compactness . however , as we show in 3.3 below , the optical depth of the x - ray emitting regions may be dominated by electrons rather than pairs . for the purposes of setting theoretical limits on the compactness parameter , this nevertheless implies the same result since both cases require @xmath0 . in addition , radiation mechanisms put their own limitations on the local compactness . the fact that the x - ray spectral index for seyfert galaxies lies in a rather narrow range ( nandra & pounds 1994 ) is most naturally explained by the approximately constant compton @xmath25-parameter ( defined in , e.g. , rybicki & lightman 1979 ) . fabian ( 1994 ) shows that in order for the compton emissivity to dominate over the bremsstrahlung one , the compactness of the plasma should be larger than @xmath26 where @xmath27 is the electron temperature in the units of @xmath28 . for the typical value @xmath29 , this requires that @xmath30 . note that the gas does not necessarily need to be maxwellian , as long as the optical depth is sufficiently large ( e.g. , ghisellini et al . 1994 ; nayakshin & melia 1997a ) , since then the comptonized spectrum looks very much the same for different electron distributions having the same @xmath25-parameter . moreover , in the presence of a strong magnetic field , the synchrotron self - absorption is an efficient mechanism for thermalizing the electrons , to the extent that it becomes a more important thermalization mechanism than coulomb collisions ( svensson 1996b ; nayakshin & melia 1997a ) . thus , constraints imposed on the compactness by the coulomb thermalization process ( fabian 1994 ) can be violated . to summarize this section , we note that all current explanations for the x - ray emission from seyfert galaxies require a large local compactness parameter @xmath0 . as already discussed in the introduction , observational evidence very strongly favors the geometry of localized x - ray sources above the accretion disk . we note that this immediately requires the active regions to be transient with a lifetime comparable to ( or less than ) the disk thermal time scale , @xmath31 , where @xmath32 in terms of the sound speed @xmath33 . an integral assumption of the two - phase model is that the internal disk emission is negligible compared with the x - ray flux of the ar , at least during the active phase ( poutanen & svensson 1995 ) . assuming that a fraction ( @xmath1 ) of the total energy content in the surface area of the disk immediately below the ar is transferred into the ar , the time scale for the release of this energy must then be much shorter than @xmath34 , during which time the disk s internal energy is radiated . our calculations show that if this condition is not satisfied , then the localized ars actually produce a _ steeper _ spectrum than that of a full corona , due to the enhanced internal emission from regions of the disk that surround the ar . this is an effect that is neglected in the two - phase corona - disk model . physically , the internal disk emission provides too much cooling in this case , unless the x - ray emitting region somehow snatches heating power even from disk regions that are not directly below it , which appear to be unrealistic . the plasma in the ars should be confined during the active phase , otherwise the energy will be lost to the expansion of the plasma rather than producing the x - rays . not confined , the source would expand at the sound speed ( which turns out to be a fraction of @xmath35 for these conditions ) . the lifetime of the ar would then be limited to a few light crossing times . it is not clear that the spectrum from such an expanding and short lived source can resemble anything studied thus far in the literature . the familiar gravitational confinement , operating in the main part of the accretion disk , does not work here due to several reasons . first of all , the _ locally _ limited eddington compactness @xmath19 is at most @xmath36 for ars with a roughly semi - spherical shape , where @xmath37 is the positron number density @xmath38 divided by that of the protons @xmath39 , while the relatively large thomson optical depth @xmath40 obtained by zdziarski et al . ( 1996 ) requires a compactness of a few hundred ( if no magnetic field is involved and the particles are confined to a rigid box ) . second , there is no mechanism for counter balancing a side - ways expansion of the plasma . therefore , since there seems to be no other reasonable possibility for confinement of the ar plasma , it may be argued that a magnetic field is required to provide the bounding pressure . any confinement mechanism will fail to confine the plasma for a time longer than about one dynamical time scale for the disk , since adjacent points with slightly different radii are torn apart on this time scale due to the disk s differential rotation . in addition , if the pairs are important for the model , then the lifetime of the ar should be large enough to allow establishment of the pair equilibrium . to put it another way , there should be enough time to create enough pairs if the plasma is initially optically thin and proton - dominated . we experimented with time - dependent codes in which radiation transfer is treated in the frequency - dependent eddington approximation , and found that this condition leads to the requirement that the lifetime of the region should be roughly an order of magnitude longer than the light crossing time for the ar . in the thin disk approximation one can always satisfy both requirements as long as the size @xmath12 of the ar is of the order of the disk scale height @xmath41 , since @xmath42 , where @xmath33 is the local sound speed . we should also note that there can be other than pair creation mechanisms for the plasma to adjust its optical depth ( see 3.3 ) , so this constraint is only important when the optical depth is dominated by pairs . to be consistent with the observations and the physics of the two - phase accretion disk - corona model , one needs very short lived phenomena to occur above the disk s atmosphere . in fact , the whole evolution of the ar should happen faster than the disk s hydrostatic time scale . to confine the plasma with a high compactness parameter @xmath0 , one needs mechanisms other than gravitational confinement . we suggest that this points to magnetic flares as the most likely mechanism for the ar formation . galeev , rosner & vaiana ( 1979 ) showed that magnetic flares are likely to occur on the surface of an accretion disk , since the internal dissipative processes are ineffective in limiting the growth of magnetic field fluctuations . as a consequence of buoyancy , magnetic flux should be expelled from the disk into a corona , consisting of many magnetic loops , where the energy is stored . it has also been speculated that just as in the solar case , the magnetically confined , loop - like structures ( which we shall collectively call magnetic flares ; see , e.g. , priest 1982 ) produce the bulk of the x - ray luminosity . the x - rays are assumed to be created by upscattering of the intrinsic disk emission . since then , several solar magnetic flare workers have elaborated on this subject ( e.g. , kuperus & ionson 1985 ; burm 1986 ; burm & kuperus 1988 ; stepinski 1991 ; de vries & kuijpers 1992 ; volwerk , van oss & kuijpers 1993 ; van oss , van oord & kuperus 1993 ) . unfortunately , these models are very much more complicated than simpler plasma models that take into account the detailed interaction of particles and radiation but leave out the question of how the plasma is confined and energy is supplied . thus , although the models invoking magnetic flares above the cold accretion disk have been viable , the detailed spectrum from such a flare could not be computed , and the model has remained somewhat of an abstraction . an important step forward was that by haardt , maraschi & ghisellini ( 1994 ) , who for the first time attempted to connect the physics of magnetic flares with the observational need for localized active regions above the disk . however , the actual consideration of the magnetic field structure that confines the plasma to the ar was still missing . furthermore , the amount of energy stored in the magnetic field has been treated as just a parameter , depending on how long and at what rate the energy is supplied to the ar . in reality , the field value is limited by the equipartition field in the disk ( galeev , rosner & vaiana 1979 ) . the question of how the pressure equilibrium in the ar ( important when discussing @xmath9 of the source ) is set up has not been discussed . one of the purposes of this paper is to pay more attention to the magnetic flare model for the x - ray emission from accretion disks in black hole systems in general , and in seyfert galaxies in particular . in the rest of the paper , we point out that the mf model can account for many , if not all , of the observed x - ray and uv spectral features of seyfert galaxies . very importantly , we shall also demonstrate that these flares are physically consistent with the constraints imposed on the ars discussed above . in the standard accretion disk theory , the gas density has an approximately gaussian vertical profile , and thus it decreases very fast with increasing height . let us also assume that the magnetic flux tube is rooted in the midplane of the disk . the `` flare region '' , i.e. , that part of the flux tube above the accretion disk surface , is then dominated by magnetic field pressure . it is well known that a magnetic field , left to its own devices , tends to fill all the available space ( e.g. , parker 1979 , 8.4 ) . for the magnetic flux tube rooted in the midplane of the disk , this means that the tube cross section expands ; the tube is thick in the sense that the cross sectional radius is of the order of the tube length . the whole structure has a roughly semi - spherical shape ( fig . 1 ) . we note that the observations actually require the magnetic flux tubes to be thick if they are to explain the x - ray emission from seyferts . indeed , if the tubes are slim , then most of the photons reflected from the disk will not re - enter the ar , but leave system . the amount of cooling of the ar due to these photons is then not enough to explain the x - ray indexes of seyfert galaxies from spectral modeling , it is known that the fraction of photons re - entering the ar should be relatively large , @xmath43 ( e.g. , svensson 1996a ) . 0.2 in we will now assume that by some process ( e.g. , by magnetic reconnection or dissipation of magnetic waves ) the magnetic field energy is being transferred to the particles . we can estimate the maximum compactness of the ar by the following considerations . the magnetic field is limited by the equipartition value in the midplane of the disk . the size of the ar , @xmath12 , is of the order of one turbulent cell , which is at best of the order of the disk scale height @xmath41 . let us assume that the field annihilation ( which provides the energy transfer to the particles ) occurs on a time scale @xmath44 equal to the light crossing time @xmath45 times some number @xmath46 a few . we will also assume that the flare occurs at 6 gravitational radii , where most of the bolometric luminosity is produced . using the results of sz94 , we obtain : @xmath47 here , the ` critical ' column energy density is @xmath48 , where @xmath49 is the thomson cross section and @xmath50 is the midplane energy density . using the results of sz94 , we obtain @xmath51 where @xmath52 is the radius in units of 3 gravitational radii , @xmath53 ( @xmath54 is the mass of the black hole ) , @xmath55 is the standard viscosity parameter , and @xmath56 is the dimensionless luminosity , @xmath57 , where @xmath58 is the eddington luminosity . finally , @xmath59 accounts for the assumed stress - free boundary condition at the disk s inner edge . the constant @xmath60 has the value @xmath61 if the disk is gas pressure dominated , and it is @xmath62 if the dominant pressure is due to radiation . taking @xmath63 as an example , we get @xmath64 where @xmath65 is the magnetic energy density @xmath66 . hmg94 suggested that plausible values for @xmath67 and @xmath55 are 10 and 0.1 , respectively . we can also assume that @xmath68 . it is then seen that @xmath0 , but it is not likely to be as high as a few hundred . the two - phase model is often criticized for a lack of self - consistency : one of the most important quantities determining the spectrum the thomson optical depth of the ar is either fixed in an ad hoc manner , or is said to be given by pair equilibrium . the latter may be viable if the pairs are strongly confined inside the ar and if the compactness of the region is @xmath3 several hundred . however , a physical description of how this happen is needed in order to validate the basic assumptions of the model . haardt , maraschi & ghisellini ( 1994 ) have made an attempt in this direction , but their description of magnetic flares was rather simplistic and did not provide an explanation for the observed optical depth . to address this issue in greater depth , nayakshin & melia ( 1997b ) considered the role played by pressure balance in establishing an equilibrium optical depth during the active phase of a magnetic flare . the main difference with the solar case is that here the compactness of the flare is much larger than unity , and thus radiation pressure dominates over particle pressure ( if the proton temperature is the same as that of the electrons ) . the conditions providing a pressure balance are therefore drastically different from those in the sun , where the particles dictate the nature of the equilibrium . nayakshin & melia ( 1997b ) assumed that the energy is supplied to the gas by magnetohydrodynamic waves . under the conditions typical for seyfert galaxies , the group velocity of these waves ( @xmath69 ) is expected to be close to the speed of light @xmath35 . because momentum is transferred to the gas , as well as energy , a compressional force is imposed on the plasma . the radiation pressure within the active region is approximately @xmath70 , where @xmath71 is the x - ray flux leaving the source . in quasi - equilibrium the energy influx is equal to the energy outflux , and radiation pressure is equal to the momentum influx due to the magnetic waves . this then requires that the thomson optical depth @xmath9 be in the range @xmath72 , depending on the actual geometry of the flare . the alfvn velocity can be used as an estimate for the group velocity of the magnetohydrodynamic waves . taking the disk structure to be that of a standard shakura - sunyaev model in its radiation pressure dominated region , one can show that the alfvn velocity @xmath69 ( at a distance of @xmath73 gravitational radii from the black hole ) is @xmath74^{1/2}\;.\ ] ] it is evident that for @xmath75 and @xmath56 not too small , @xmath69 can be quite close to @xmath35 ( if it exceeds @xmath35 , the relativistic corrections will permit it to saturate at @xmath35 only ) . in this estimate , we assume that the thomson optical depth @xmath9 of the plasma within the flare region is entirely due to the accreting electrons . if in addition pairs are produced , then equation ( 6 ) should be used with @xmath76 instead , where @xmath76 is the thomson optical depth of the ar due to the electrons accreting with the protons , which further increases @xmath77 . we conclude from this that @xmath69 must be close to @xmath35 for quite a broad range of the @xmath55-parameter , @xmath78 , and it is _ completely independent _ of the black hole mass @xmath54 . as already noted by haardt , maraschi & ghisellini ( 1994 ) , the spectrum of a magnetic flare should be similar to that of a static active region of the same size and compactness , as long as the lifetime of the flare exceeds several light - crossing time scales . this is certainly true if pairs are not important , since the time scales for other processes that may influence the spectrum ( e.g. , poutanen & svensson 1996a ) are of the order of a light crossing time . however , the life time of one single flare is short compared with the typical integration time of current x - ray instruments . moreover , it is very likely that there are many magnetic flares present at any given moment of time . therefore , it becomes clear that if magnetic flares are responsible for the x - ray emission from seyfert galaxies , the spectrum must be a composite of the contributions from many different flares . nayakshin & melia ( 1997b ) tested this possibility , assuming that the energy balance is fixed by requiring the compton @xmath25-parameter to be constant for all the flares ( which is reasonable , given that @xmath25 is fixed by the geometry of the two - phase model ) , and they summed over the spectra from flares with different @xmath9 . for illustrative purposes , the distribution of flares was taken to be a gaussian over @xmath9 , centered on @xmath79 with a dispersion of @xmath80 . the resulting spectrum is practically indistinguishable from that of a single flare with @xmath81 up to a photon energy of about a hundred kev . the osse error bars are much larger than the deviations of the composite and single flare spectra , and so the current observations can not distinguish between these two possibilities . thus , magnetic flares can conceivably account for the observed x - ray/@xmath82-ray spectra of seyfert galaxies . nayakshin & melia ( 1997c ) considered the x - ray reflection / reprocessing due to a transient , energetic flare above the accretion disk to compare with other studies reported in the literature that assume a stationary state . the main difference between the two is the structure of the emitting ( i.e. , reprocessing ) layer . in particular , since the flare lifetime is shorter than the disk thermal time scale , a pressure and energy equilibrium between the incident x - ray flux and the underlying disk is not established . a typical photon does not have sufficient time to diffuse to the mid plane of the disk during one lifetime of the flare . however , the x - ray skin , i.e. , the layer that absorbs and reprocesses the x - rays , is only a tiny fraction of the whole disk , and thus a quasi - equilibrium is established within it . as a result of the incident flux , the x - ray skin is compressed to much higher densities than the density of the undisturbed accretion disk . it turns out that the pressure and energy equilibrium of this x - ray skin yields a unique temperature @xmath83 few @xmath84k _ independently _ of the mass of the central engine . this seems to account well for the observed independence of the big blue bump temperature on the luminosity of the source ( walter 1994 ; zhou et al . 1997 ) . by comparison , a stationary , time independent reflection can not easily explain these observations . an additional attractive feature of the mf model is that due to a much larger gas density in the reflecting layer , the ionization parameter ( @xmath85 ) remains relatively small , in which case the reflected / reprocessed spectrum is indistinguishable from that of a neutral reflector , which appears to be favored by current observations ( zdziarski et al . static x - ray reflection / reprocessing , on the other hand , may have difficulties complying with the observed low ionization parameter of the reflecting matter , since in this case the x - ray skin density is much lower . summarizing , many of the attractive features of reflection / reprocessing in a static layer below the ar are preserved in the case of a time - dependent , short - lived magnetic flare , but the latter has the additional advantage of being able to account for the approximate universality of the bbb temperature and the low ionization fraction in the reflector . one of the central questions in the modeling of seyfert galaxies has always been whether a pair equilibrium is established within the source , since this has some serious observational consequences . however , pairs have successfully eluded detection in seyfert galaxies . with the discovery of a high - energy break above @xmath3 100 kev and the non - detection of a predicted annihilation line , it has become apparent that the non - thermal power in seyfert galaxies , if at all present , is quite small ( e.g. , svensson 1996a ; zdziarski et al . 1996 , and references therein ) . thus , it was concluded that the plasma is mostly thermal ( e.g. , haardt & maraschi 1991 ; fabian 1994 ) . this inference was supported by the finding that an annihilation line would not be observed from a thermal plasma because it is always hidden in the broad comptonized spectrum ( zdziarski & coppi 1995 ) . recent work by ( zdziarski et al . 1996 ) suggests that in the context of a thermal pair equilibrium , an optical depth of roughly unity is then the consequence of a large compactness ( @xmath3 several hundred ) . we , however , suggest that this situation is achieved by pressure equilibrium , as discussed in 3.3 . in this case , the plasma consists primarily of the electrons and protons stripped from the disk , at least at the beginning of the flare , since during the magnetic energy storage phase the plasma is not sufficiently hot to provide enough hard photons that would create electron - positron pairs . thus , in this framework , the pairs are not important in determining the spectrum from the flare , and this is again consistent with the lack of any observed pair signature . of course , a detailed modeling of a magnetic flare event must take into account the pair creation process which continuously produces new pairs when @xmath86 . it is the total optical depth ( i.e. , the sum of the thomson optical depths of electrons and pairs ) that matters for the pressure equilibrium . if this pressure balance fixes the optical depth to some particular value @xmath1 , then clearly , compared to the no - pair case , the plasma must expand to accommodate the new particles . let us assume that the total energy supplied to the plasma is a constant , which means that the luminosity @xmath14 remains constant . then , as the plasma expands , its compactness decreases as @xmath87 since @xmath88 . since the pair creation rate is proportional to @xmath89 , an equilibrium is reached at some @xmath12 such that the pairs are now responsible for a fraction of the total optical depth @xmath9 . this fraction turns out to be quite small unless the initial compactness is as high as several hundred . it is interesting to note that even flares with an initial value of @xmath19 that would lead to a pair runaway ( e.g. , svensson 1982 ) find an equilibrium configuration with a source compactness below this critical value . we intend to quantify the character of the pair equilibrium in this situation in a future publication , but we may already anticipate that a compactness as high as several hundred is only barely permitted by equation ( 5 ) , and that therefore pairs should be of relatively low importance to the dynamics and energetics of magnetic flares . several authors have suggested that magnetic flares above the accretion disk are responsible for the observed variations in the agn and gbhc luminosity ( e.g. , galeev , rosner & vaiana 1979 ; de vries & kuijpers 1992 ; volwerk , van oss & kuijpers 1993 ; van oss , van den oord & kuperus 1993 , and others ) . the power density spectrum ( pds ) from these sources is typically a power - law ( lawrence et al . 1987 ; mchardy & czerny 1987 ; krolik et al . 1991 ) . in the case of the sun , dmitruk & gomez ( 1997 ) have shown that magnetic flares can naturally account for a power - law shape in the pds with an index @xmath90 . since in principle the flares in black hole systems may have different spatial sizes , and thus different durations and overall power , one can reasonably expect that a similar pds may be produced by these transient events above the accretion disks in agns and gbhcs . we note here that the power - law pds should be explained by local variations of the magnetic flare properties , rather than variations occurring systematically with a changing location of the flare ( compare with the rotating bright - spots model , e.g. , abramovicz et al . the observed x - ray pds spans a wide range in frequencies , typically @xmath91-@xmath92 hz . this range corresponds to the range in radius @xmath93 , since @xmath94 , where @xmath95 is the rotational frequency of the keplerian disk . but the local contribution to the overall luminosity goes as @xmath96 , and thus the smallest frequencies contribute less than the largest ones , in contradiction to the observed power spectrum . only if one assumes that the luminosity of the flare is independent of its location does one obtain the right power spectrum . however , such an assumption is unphysical , since we know that the x - ray luminosity is a major component of the bolometric luminosity , and thus it should scale in the same way as the local gravitational dissipation in the disk . therefore , since the emission comes from a relatively narrow range in radii , it should be the flare size that varies and produces the observed pds . alternatively , since disturbances propagate along magnetic field lines in a strong magnetic field , and since the magnetic flux tube is thick , there can be a wide range in characteristic scales @xmath97 even in one source ( @xmath97 is essentially the length of the given magnetic field line [ see fig . 1 ] ) . moreover , the energy density of the magnetic field will scale roughly as @xmath98 ( that would be so for a potential field that has no currents even at the boundary , i.e. in the footpoints ) . thus , one might expect to see a power - law pds even from a _ single _ event in this case . we intend to investigate this question in future work , but we caution that the analysis of the pds is unlikely to provide any valuable information about one single magnetic flare , since at any given instant of time there should be a number of such events . these flares occur roughly at random , and thus information about a single flare is washed out . the complete annihilation of the magnetic field energy @xmath65 ( @xmath99 ) within a volume @xmath100 during a time @xmath101 provides an estimate of the single flare luminosity : @xmath102 where we have used the sz94 accretion disk parameters with their @xmath103 set equal to @xmath104 . based on similar considerations , hmg94 estimated the required number of magnetic flares to be about 10 . we are therefore in agreement with this estimate , although in principle the number of less energetic or smaller flares may still be larger , since equation ( 7 ) is only an upper limit on @xmath105 . an implicit assumption thus far has been that the magnetic flare can indeed sustain a sufficient number of protons roughly one disk height @xmath18 above the disk . for this to be viable , we need to demonstrate that the gravitational energy of the protons trapped inside the flux tube is very much smaller than the magnetic field energy . the latter is at most the total thermal energy of the disk immediately below the flare , while the former may be estimated as @xmath106 . using expressions from sz94 for @xmath107 , we see that @xmath108 which satisfies the constraint . as before , @xmath109 is the thomson optical depth of the material trapped inside the flux tube . the nature of accretion disk instabilities has received a great deal of attention ( for recent references , see chen 1995 ) . while not attempting to consider this question in detail here , we can make several comments on the stability of the mf model . magnetic flares may be viewed as an additional channel by which energy can be transported out of the disk . of course , in the standard disk model , the dissipated gravitational energy is lost directly to radiation . since the time taken by a photon to diffuse outward from the midplane to the disk s surface is a very strongly increasing function of the optical depth , it is conceivable that under some conditions the energy transported by the magnetic field is greater than that due to the radiation . the total energy content of the disk plus corona system is then expected to be lower than that of the standard theory , though with the same luminosity , and such a situation leads to greater stability ( e.g. , sz94 ) . although it is not clear what role advection would have in such a model , it is expected that magnetic flares may help to quench some of the disk instabilities encountered in standard models . we have seen that magnetic flares are physically consistent with the multi - wavelength spectra of seyfert galaxies . very importantly , the mf model seems to account for several observed characteristics that can not be easily reconciled with a picture in which the ars are static . however , a host of unanswered questions and problems remain . first , accretion disk flares have been considered only in a highly schematic fashion thus far . unfortunately , the physics of magnetic energy release in a non - static and turbulent gas is not well known , other than the fact that it must happen , as is seen in the sun . in addition , a detailed model of the magnetic flare should also include a consideration of all the relevant aspects of magnetic flux tube formation in the underlying turbulent disk , a problem that also has not been solved . this , however , does not mean we can ignore the magnetic flare model for the x - ray emission in seyferts . instead , additional studies are called for , especially in view of the fact that very recent observations of solar flares seem to support much of the current theoretical thinking in this area and are generating enthusiasm among solar theorists ( e.g. , innes et al . 1997 , klimchuk 1997 ) . another major unresolved issue is how the disk viscosity is connected to the magnetic field . if we knew this relationship , we would be able to eliminate @xmath55 or @xmath65 from equation ( 5 ) , and thus get better constraints on the maximum compactness of a magnetic flare . this follows from the fact that the structure of a cold disk is quite sensitive to the viscosity law . in addition , viscosity figures very prominently in the physics of magnetic flux tubes ( e.g. , vishniac 1995 ) . in this paper we have attempted to address the problems that arise when physical constraints are imposed on the active regions thought to exist in the two - phase corona - accretion disks in seyfert galaxies . we showed that these regions should necessarily be highly transient , i.e. , evolve faster than one thermal disk time scale due to spectrum formation constraints . a consideration of the plasma confinement lead us to require an overall magnetic field with a stress much larger than the x - ray radiation pressure . furthermore , putting these constraints together , we concluded that the magnetic flare model appears to be consistent with the type of transient active regions required by the observations . we then proceeded to show that the model is probably capable of explaining the observed optical depth , x - ray reflection and uv reprocessing implied by the data , and the observed power - law power density spectra . finally , we discussed the unresolved issues that need to be investigated in future work . this work was partially supported by nasa grant nag 5 - 3075 . we have benefitted from many discussions with randy jokipii and eugene levy . burm , h. 1986 , a&a , * 165 * , 120 burm , h. & kuperus , m. 1988 , a&a , * 192,165 dmitruk , p. & gomez , d.o . 1997 , apjl , in press ( also astro - ph/9705050 ) johnson et al . 1993 , bull . american astron . , 183 , # 64.03 jourdain et al . 1992256l38 nayakshin & melia 1997a , submitted to apj ( available at astro - ph/9705011 ) nayakshin & melia 1997b , submitted to apj letters nayakshin & melia 1997c , apj letters , in press ( also astro - ph/9705010 ) . van oss , r.f . , van den oord , g.h.j . , & kuperus , m. 1993 , a&a , 270 * , 275 parker , e.n . 1979 , cosmical magnetic fields , clarendon press , oxford . priest , e.r . , solar magneto - hydrodynamics , kluwer academic publishers , 1984 . rybicki , g. b. , & lightman , a.p . , 1979 , radiative processes in astrophysics , john wiley and sons : new york . shakura & sunyaev 1973 , a&a , 24 , 337 svensson , r. 1996b , invited review at relativistic astrophysics , available at astro - ph/9612081 volwerk , m. , van oss , r.f . , & kuijpers , j. , 1993 , a&a , * 270 * , 265 de vries , m. , & kuijpers , j. , 1992 , a&a , * 266 * , 77 walter , r. , & fink , h.h . 1993 , a&a , * 274 * , 105 walter , r. , et al . 1994 , a&a , * 285 * , 119	we discuss several physical constraints on the nature of the active regions ( ar ) in seyfert 1 galaxies , and show that a plausible model consistent with these constraints is one in which the ars are magnetically confined and `` fed '' . the unique x - ray index of these sources points to a large compactness parameter ( @xmath0 ) . this , together with the conditions required to account for the observed optical depth being close to unity , suggests that the magnetic energy density in the ar should be comparable to the equipartition value in the accretion disk , and that it should be released in a flare - like event above the surface of the cold accretion disk . we consider the various issues pertaining to magnetic flares and attempt to construct a coherent picture , including a reason for the optical depth in the ar being @xmath1 , and an understanding of the characteristics of the x - ray reflection component and the power density spectra associated with this high - energy emission . #1#2#3#1 , a&a , # 2 , # 3 # 1#2#3#1 , a&as , # 2 , # 3 # 1#2#3#1 , aj , # 2 , # 3 # 1#2#3#1 , _ apj _ , * # 2 * , # 3 . # 1#2#3#1 , _ apj ( letters ) _ , * # 2 * , # 3 . # 1#2#3#1 , apjs , # 2 , # 3 # 1#2#3#1 , ara&a , # 2 , # 3 # 1#2#3#1 , baas , # 2 , # 3 # 1#2#3#1 , icarus , # 2 , # 3 # 1#2#3#1 , _ mnras _ , * # 2 * , # 3 . # 1#2#3#1 , _ nature _ , * # 2 * , # 3 . # 1#2#3#1 , pasj , # 2 , # 3 # 1#2#3#1 , pasp , # 2 , # 3 # 1#2#3#1 , qjras , # 2 , # 3 # 1#2#3#1 , science , # 2 , # 3 # 1#2#3#1 , soviet astr . , # 2 , # 3 # 1#2#3#1 , soviet astr . lett . , # 2 , # 3 # 1#2#3#4#1 , # 2 , # 3 , # 4 # 1#1 submitted to the astrophysical journal	12,460	685
recent years have seen a great expansion of topological quantum materials beyond time - reversal - invariant topological insulators @xcite , driven by the search for symmetry - protected topological ( spt ) states of matter that are distinct from trivial states only in the presence of certain symmetry . this underlying symmetry can be associated with conservation of internal quantum numbers such as charge and spin @xcite , or with spatial operations such as rotation and reflection @xcite . since spatial symmetry is a common property of all crystals , a wide array of topological band insulators protected by various crystal symmetries , commonly referred to as topological crystalline insulators ( tcis ) @xcite , has been theorized . the hallmark of a tci is the existence of topologically protected gapless excitations on surfaces that preserve the relevant crystal symmetry . a notable class of tcis protected by reflection symmetry was predicted and observed in the iv - vi semiconductors sn@xmath3pb@xmath4(te , se ) @xcite , and the symmetry protection of the topological surface states has been demonstrated @xcite . more recently , tcis have been generalized to band insulators with magnetic point group symmetries @xcite , nonsymmorphic symmetries @xcite , and with both glide reflection and time - reversal symmetry @xcite . in addition , topological insulators protected by translation @xcite and magnetic translation symmetry @xcite were studied in early works . the interplay between topology and crystallography is continuing to knit together abstract mathematics and real materials . recently , a new type of electronic tcis protected by reflection symmetry has been theoretically constructed @xcite , which is enabled by electron interactions and do not exist in free fermion systems . in a broader context , interaction - enabled topological crystalline phases were also been found in fermion superconductors @xcite and boson insulators @xcite . such phases are now attracting wide attention , and it is of great interest to find their material realizations and experimental signatures . in this work , we find a new class of interaction - enabled topological crystalline insulators in two and three dimensions , which are protected by time - reversal ( @xmath0 ) and reflection / rotation symmetry ( @xmath1 ) , or simply the combined symmetry @xmath5 . this phase exists in systems of spin-@xmath6 electrons with spin - orbit interaction , and can not be adiabatically connected to any slater insulator in the presence of @xmath5 symmetry . instead , this phase admits a natural description in terms of a magnetic system of interacting spins , hence is termed `` topological crystalline magnets '' ( tcms ) . a distinctive feature of tcms is the presence of gapless spin excitations on the edge parallel to the axis of reflection . these edge states exhibit strongly anisotropic response to magnetic fields in directions parallel and perpendicular to edge . our model for two- and three - dimensional tcms is adiabatically connected to an array of _ decoupled _ one - dimensional symmetry - protected topological ( spt ) states , on which the @xmath5 symmetry acts as an internal anti - unitary @xmath7 symmetry . this stacking approach provides a unifying description of all previously known topological crystalline insulators @xcite , both with @xcite and without @xcite interactions . the one - dimensional spt state serving as the building block of our higher dimensional tcms apparently looks similar to , but , in fact , is remarkably different from the affleck , kennedy , lieb , and tasaki ( aklt ) state @xcite . the aklt state belongs to the haldane phase , which is a _ bosonic _ spt phase protected , for example , by the dihedral ( @xmath8 ) symmetry or the time - reversal symmetry @xcite . however , the haldane phase is not a _ fermionic _ spt phase and is hence trivial as an electronic phase @xcite . namely , when we decompose the @xmath9 spins of the aklt model into _ mobile _ electrons with spin-@xmath10 , the ground state is adiabatically deformable into a trivial band insulator @xcite while keeping the dihedral and the time - reversal symmetry . in contrast , our 1d tcm state is a robust fermionic spt phase protected by @xmath5 as we shall see now . s and orange dots illustrate the edge degrees of freedom . a gapless edge state appears on each edge of a finite - size system . the edge degrees of freedom satisfy @xmath11 , which is distinct from physical electrons or edge states of noninteracting topological insulators . our 1d model is composed of a four - dimensional hilbert space @xmath12 on each site arising from the spin and orbital degrees of freedom of an _ even _ number of spin-@xmath6 electrons . the time - reversal operator @xmath13 thus satisfies @xmath14 on @xmath12 . as the simplest realization of such anti - unitary symmetry we take the complex conjugation @xmath15 . we also assume that states in @xmath12 are all even or all odd under a spatial symmetry @xmath1 , which is either the reflection about @xmath16 plane @xmath17 or the @xmath18-rotation about @xmath19-axis @xmath20 . the operator @xmath21 is hence represented by the identity operator @xmath22 on @xmath12 . in one dimension @xmath1 is essentially an internal symmetry , but will become a true spatial symmetry in higher dimensional cases to be studied later . as an explicit example , @xmath12 can be identified as a subset of the states of two spin-@xmath6 electrons occupying two orbitals . assuming each orbital is invariant under reflection or rotation , the operator @xmath21 only acts on the spin part of the two - electron wavefunction . there are in total six two - electron states , consisting of spin - singlet states formed by two electrons on the same orbital , as well as spin - singlet and spin - triplet states formed by two electrons on different orbitals . we denote the electron operators associated with these two orbitals by @xmath23 and @xmath24 respectively , where @xmath25 is the spin projection along the @xmath26 axis . then , out of the six two - electron states , the following four satisfy @xmath27 and @xmath28 ( @xmath29 ) and span the desired hilbert space @xmath12 : @xmath30 reads @xmath31 where both @xmath32 and @xmath33 are a set of three hermitian operators that generate the @xmath34 algebra and mutually commute , i.e. , @xmath35&=&i \epsilon^{abc}\hat{\gamma}^{\mu c } , \;\ ; [ \hat{\gamma}^{1 a } , \hat{\gamma}^{2 b } ] = 0\end{aligned}\ ] ] with @xmath36 and @xmath37 . the components of these @xmath38 operators are explicitly given by the following @xmath39 matrices in the basis of @xmath40 @xmath41 note that @xmath42 are pure imaginary and are hence odd under time - reversal symmetry @xmath0 . the hamiltonian consists of bilinears of @xmath38 s and is therefore time - reversal invariant . it is also invariant under @xmath1 since @xmath1 does not transform @xmath38 at all . to analyze the topological nature of the ground state of @xmath43 , it is more convenient to switch the basis of @xmath12 from @xmath44 to @xmath45 by the local linear transformation @xmath46 : @xmath47 is nothing but the spin operator acting on @xmath48 , @xmath49 for example , the usual spin algebras such as @xmath50 and @xmath51 hold . therefore , @xmath43 in eq . is just an antiferromagnetic spin chain whose exchange coupling is nonzero every other bond . the ground state is the valence - bond solid ( vbs ) state : @xmath52 and @xmath53 ( @xmath54 ) . the nontrivial topology of the model is encoded in the symmetry property of the edge states . although the auxiliary field @xmath55 apparently behaves like an electronic spin , its transformation under @xmath56 is in fact quite distinct from the physical spin . in the @xmath57 basis , @xmath13 and @xmath21 are represented by @xmath58 and @xmath59 , respectively . namely , @xmath55 transforms under @xmath0 in the same way as the physical spins , while it does not change under @xmath1 ( @xmath60 ) unlike electrons . this peculiar transformation property of the auxiliary field @xmath55 can be summarized as @xmath61 on the two dimensional hilbert space spanned by @xmath48 . equation must be compared to @xmath62 and hence @xmath63 of a physical spin-@xmath6 electron . one may think one can redefine @xmath64 to get @xmath65 , but even after that @xmath66 remains unchanged since @xmath13 is anti - unitary . although the hamiltonian @xmath43 is invariant under @xmath13 and @xmath21 separately , we can add arbitrary symmetry - breaking perturbations keeping only the combined symmetry @xmath56 and the bulk gap . since @xmath56 is an anti - unitary symmetry that squares into @xmath67 , it protects the kramers degeneracy on each edge . the fact that the value of @xmath68 of our edge state is different from that of physical electrons has two important implications . ( i ) the edge state of any ( noninteracting ) topological insulator satisfies @xmath63 . therefore , the vbs state in eq . can not be adiabatically connected to electronic topological insulators . in other words , the vbs state is an interaction - enabled topological phase protected by @xmath56 . ( ii ) the edge state of the vbs state is robust against the perturbation of attaching physical spin-@xmath6 electrons to the edge . in the case of the standard aklt model , for example , the edge spin-@xmath6 can be gapped by attaching an electron , since both of them fall into the same class of projective representations @xmath69 . on the other hand , the edge state of our model can not be gapped this way , since even after attaching an electron , the anti - unitary symmetry @xmath56 remains @xmath11 . to summarize , we have presented a simple 1d model of interacting electrons that realizes an interaction - enabled topological phase protected by the combined symmetry @xmath56 . the edge degrees of freedom satisfies @xmath11 and are stable against attaching additional electrons to the edge . ] now we move onto 2d tcm models . this time the reflection / rotation symmetry is truly a spatial symmetry and the 2d tcm phases are hence protected purely by non - local symmetries . we will discuss two models . the first one is stacked 1d chains shown in fig . [ fig2 ] ( a ) . the hamiltonian is @xmath70 where @xmath71 and @xmath72 in the basis of @xmath73 . the second one is a square - lattice model depicted in fig . [ fig2 ] ( b ) . @xmath74 where @xmath75 , @xmath76 , @xmath77 , and @xmath78 in the basis of @xmath79 . for both models , each auxiliary field @xmath80 ( @xmath81 ) transforms as @xmath82 so that @xmath83 satisfies @xmath84 the first transformation in eq . is again distinct from that of spin-@xmath6 electrons . as a consequence , @xmath85 satisfies @xmath11 unlike electrons as before . although both @xmath86 and @xmath87 themselves are invariant under @xmath21 and @xmath13 separately , arbitrary perturbations can be added to these hamiltonians as long as the combined symmetry @xmath56 is respected and the bulk gap is not closed . note that the reflection / rotation symmetry @xmath1 here needs to be site - centered [ @xmath88 and can not be bond - centered [ @xmath89 . the bond - centered one does not protect gapless edge states as we discuss below . to break the body - centered symmetry without affecting the site - centered one , one can introduce a - b sublattice structure [ gray shadows in fig . [ fig](b ) ] by modifying the spin hamiltonian by weak perturbation . of the 2d models in fig . the color represents the nonuniform perturbation @xmath90 with @xmath91 , for example , which respects the @xmath56 symmetry . the edge state in the region @xmath92 and @xmath93 opens a gap proportional to @xmath94 , while there will be a residual gapless edge state protected by @xmath11 on the domain wall . [ fig3 ] ] the ground state of these 2d hamiltonians is the vbs state illustrated in fig . [ fig2 ] , analogous to eq . . there is a 1d edge state formed by @xmath95}$ ] along the line @xmath96 , and another 1d edge state formed by @xmath97}$ ] along @xmath98 . to see the gaplessness of the edge states , we add a @xmath56-symmetric perturbation @xmath90 along the line @xmath96 as shown in fig . [ fig3 ] , where @xmath99 is an odd function of @xmath100 that approaches to a constant @xmath101 for @xmath102 . note that @xmath99 must flip sign at @xmath103 to be consistent with the @xmath56 symmetry , forming a domain wall around @xmath103 . all @xmath104 s along the edge away from the domain wall open a gap proportional to @xmath94 . however , the edge state at the domain wall @xmath103 must remain gapless . this is protected , again , by the anti - unitary symmetry @xmath56 with @xmath11 . this unavoidable gaplessness of the edge state signals the topological nature of our 2d models . essentially , @xmath105 s on the @xmath106 line plays the role of the 1d spin chain discussed above . in contrast , when @xmath1 is bond - centered , there will be an even number of @xmath107 s at the domain wall and @xmath108 and the edge may be completely gapped . an experimental signature of tcms is the anisotropic response of the edge state to the external magnetic field @xmath109 . we start with the case where @xmath1 is the reflection @xmath110 about the @xmath16 plane . recall that the @xmath111 under @xmath110 , while @xmath104 does not react to @xmath110 . both @xmath109 and @xmath104 flips sign under @xmath0 . the familiar form of the coupling to the external field @xmath112 is thus not allowed by symmetry @xmath113 . instead , arbitrary linear coupling to @xmath114 , i.e. , @xmath115 , is allowed . when @xmath114 is set to a constant value , this term breaks the @xmath113 symmetry and the edge states will be gapped and the gap should be proportional to @xmath116 . on the other hand , @xmath117 and @xmath118 do not couple linearly to @xmath119 . we therefore expect anisotropic response of the edge state towards the external magnetic field . when @xmath1 is the @xmath18-rotation @xmath120 around @xmath19 axis , the magnetic field @xmath121 changes to @xmath122 under @xmath120 . thus , arbitrary linear coupling between @xmath123 and @xmath124 is allowed . thus a constant @xmath123 can induce a gap to the edge , while @xmath114 and @xmath125 can not . we thus expect similar anisotropic response in this case too . ] one can readily construct a 3d tcm model in the same way as we did for the 2d models . the 3d model is a 2d array of the 1d tcm chains , illustrated in fig . [ fig4 ] . for this 3d model , @xmath1 must be the site - centered @xmath18-rotation about the @xmath19-axis . namely , the rotation axis must coincide with one of the 1d chain . the gapless 2d surfaces at @xmath96 and @xmath98 are protected by the combined symmetry @xmath126 . to see this , let us again add a @xmath126-symmetric perturbation @xmath127 . to be consistent with the @xmath126 symmetry , @xmath128 should satisfy @xmath129 , meaning that @xmath130 . therefore , there will be a residual zero mode at the `` vortex core '' of the perturbed surface , protected by @xmath11 . in this paper we introduced tcm phases protected by non - local symmetry @xmath126 in two and three dimension . they are interaction - enabled and are robust against attaching physical electrons to the edge . they can be detected in experiment from their anisotropic response of the edge state towards external magnetic fields . xu , c. liu , n. alidoust , m. neupane , d. qian , i. belopolski , j.d . denlinger , y.j . wang , h. lin , l. a. wray , g. landolt , b. slomski , j.h . dil , a. marcinkova , e. morosan , q. gibson , r. sankar , f.c . chou , r.j . cava , a. bansil , and m. z. hasan , nat . commun . * 3 * , 1192 ( 2012 ) .	we introduce a novel class of interaction - enabled topological crystalline insulators in two- and three - dimensional electronic systems , which we call `` topological crystalline magnet . '' it is protected by the product of the time - reversal symmetry @xmath0 and a mirror symmetry or a rotation symmetry @xmath1 . a topological crystalline magnet exhibits two intriguing features : ( i ) it can not be adiabatically connected to any slater insulator and ( ii ) the edge state is robust against coupling electrons to the edge . these features are protected by the anomalous symmetry transformation property @xmath2 of the edge state . an anisotropic response to the external magnetic field can be an experimental signature .	4,966	186
the general solution for an asymptotically flat black hole in the einstein - maxwell theory is given by the kerr - newman solution . it describes an electrically charged rotating black hole with its three parameters : mass , charge and angular momentum . a systematic study of the fields dynamic in the vicinity of a black hole is essential for understanding black - hole evaporation , quasinormal modes and stability . the linear dynamics of a charged massive scalar field in the background of a charged black hole is characterized by the two dimensionless parameters @xmath0 and @xmath1 . a black hole is not believed to be formed possessing considerable electric charge , and , once it is formed , it undergoes a rather quick discharging @xcite . yet , even if a black hole has a very small charge of order @xmath7 , the parameter @xmath1 need not be small . in addition , a charge induced by an external magnetic field , may be formed at the surface of an initially neutral , but rotating black hole @xcite . thus , the complete analysis of a massive charged scalar field dynamics should include consideration of the whole range of values @xmath8 and @xmath0 . in this work we shall study the stability and evolution of perturbation of a massive charged scalar field in the kerr - newman background in terms of its _ quasinormal modes _ and _ asymptotic tails _ at late times . it is believed that if the quasinormal modes are damped , the system under consideration is stable , though a rigorous mathematical proof of stability is usually complicated and sometimes includes a nontrivial analysis of the initial value problem . by now , quite a few papers have been devoted to scalar field perturbations in the black - hole background , yet , while the behavior of the massless neutral scalar field is indeed very well studied , the quasinormal modes of charged fields was studied only in the regime @xmath9 , @xmath10 @xcite , except for @xcite , where the wkb estimation for quasinormal modes ( qnms ) of a massless charged scalar field around the reissner - nordstrm black hole was given in the regime @xmath11 , where @xmath12 is the multipole number . the complete analysis of quasinormal modes ( allowing also to judge about stability ) for a massive charged scalar field _ for arbitrary values _ @xmath8 and @xmath0 has been lacking so far not only for the kerr - newman , but even for reissner - nordstrm solutions . here we shall solve this problem by adopting the two numerical methods of computation ( the frobenius method and the time - domain integrations ) based on convergent procedures , which allow us to find quasinormal modes accurately and with no restriction on the parameters of the system . perturbation of a charged massive field in the background of a rotating charged black hole has rich physics , because there are a number of phenomena which must be taken into consideration : * _ superradiance _ , that is the amplification of waves with particular frequencies reflected by a black hole , if it is rotating @xcite or electrically charged @xcite . thus , there will be the two regimes of superradiance for kerr - newman black holes , owing to charge and rotation @xcite . * _ superradiant instability _ of bound states around black holes owing to the massive term , which creates a local minimum far from the black hole , so that the wave will be reflected repeatedly and can grow . it is essential that this instability occurs under the bound states boundary condition , which differ from the quasinormal modes ones @xcite . * _ quasiresonances . _ when a field is massive , quasinormal modes with arbitrarily long lifetimes , called quasiresonances , appear , once some critical value of mass of the field is achieved @xcite . when the damping rate goes to zero , the quasinormal asymptotically approach the bound state , but still remain quasinormal modes for whatever small but nonzero damping . @xcite . * _ instability of the extremal black holes _ that apparently occurs for fields of any spin and both for extremal reissner - nordstrm and kerr black holes @xcite , and therefore , must be expected for the extremal kerr - newman solution as well . however , in the linear approximation this instability develops only on the event horizon and can not be seen by an external observer . * _ mode branching_. quasinormal modes of kerr black holes were believed to be completely studied until a few months ago when an interesting observation has been made @xcite . it was shown that , for the near - extremal rotation there are two distinct sets of damped quasinormal modes , which merge to a single set in the exactly extremal state @xcite . here , through the numerical analysis of quasinormal modes and asymptotic tails we have shown that a massive charged scalar field is stable in the vicinity of the kerr - newman black hole , in spite of the instability of the corresponding bound states . we found that at some values of the field s charge @xmath3 quasinormal modes may behave qualitatively differently from those of the neutral field : the fundamental mode ( dominating at late times ) may have arbitrarily small real part ( real oscillation frequency ) which appears in the time domain as a very short period of quasinormal ringing consisting of damped oscillations and the quick onset of asymptotic power - law tails . in addition , we generalized earlier results on mode branching of massless fields around nearly extremal kerr black holes to the massive fields and kerr - newman solutions . an analytic formula has been obtained for large @xmath1 . the paper is organized as follows . in sec ii the basic formulas for a charged massive scalar field in the kerr - newman background is given . the wave equation is reduced to the schrdinger - like form with an effective potential . sec iii describes the numerical methods which we used : the frobenius methods , two schemes of time - domain integration ( for neutral and charged fields ) together with the method for extraction of frequencies from the time - domain profiles , called the prony method , and the wkb approach . we have related separately perturbations of nonextremal black holes ( sec iv ) and nearly and exactly extremal ones ( sec . v ) , as near extremal black holes shows new phenomena , such as mode branching . in sec . vi we discuss some technical difficulties which appear when one considers higher overtones of a charged scalar field or approach closely to the extremal state , keeping @xmath3 non zero . in sec vii we summarize the results obtained . in the boyer - lindquist coordinates the kerr - newman metric has the form @xmath13 ^ 2,\end{aligned}\ ] ] where @xmath14 @xmath15 and @xmath5 is the black - hole charge , @xmath4 is its mass . the electromagnetic background of the black hole is given by the four - vector potential , @xmath16 we shall parameterize the metric by the following three parameters : the event horizon @xmath17 , the inner horizon @xmath18 , and the rotation parameter @xmath19 , @xmath20 the black hole s mass and charge are then @xmath21 a massive charged scalar field satisfies the klein - gordon equation , @xmath22 where @xmath3 and @xmath2 are the field s charge and mass respectively . one separates variables by the following ansatz @xmath23 where @xmath24 obeys the following equation @xmath25 and @xmath26 is the separation constant . this equation can be solved numerically for any value of @xmath27 in the same way as the equation for a massive scalar field in the kerr black - hole background @xcite . let us note that , when @xmath28 , eq . ( [ angularpart ] ) can be reduced to the well - known equation for the spheroidal functions . in this case , for any fixed value of @xmath27 the separation constant @xmath26 can be found numerically using the continued fraction method @xcite . when the effective mass is not zero , the separation constant @xmath29 can be expressed , in terms of the eigenvalue for spheroidal functions @xmath30 @xcite , as @xmath31 when @xmath32 , one has @xmath33 . for nonzero values of @xmath19 , the separation constant can be enumerated by the integer multipole number @xmath34 . the radial function satisfies a schrdinger equation , @xmath35 where @xmath36 is the tortoise coordinate , @xmath37 and the effective potential is @xmath38 the asymptotics of the effective potential near the event horizon and at spatial infinity are @xmath39 where we fix the sign of @xmath40 such that @xmath41 is of the same sign as @xmath42 . note , that @xmath43 and @xmath42 can have different signs . this corresponds to the superradiant regime in which one has @xmath44 by definition , quasinormal modes ( qnms ) are proper oscillation frequencies which correspond to purely incoming wave at the event horizon and purely outgoing wave at infinity , so that no incoming waves from either of the `` infinities '' are allowed . thus , the boundary conditions for the qnms can be written as follows @xmath45 here , we shall briefly relate the three numerical methods used for finding quasinormal frequencies : * leaver method , which is based on a convergent procedure and , thereby , allowing one to find qn modes accurately , * wkb method ( accurate in the regime of high multipole numbers ) , * time - domain integration which includes contribution of all modes , and , together with the prony method , usually allows extracting a few lower dominant frequencies from a time - domain profile . we shall see that in some ranges of parameters , when one method becomes slowly convergent or inapplicable , the other can be used , so that the use of a few alternative methods is necessary here not only for an additional checking , but also for getting the complete picture of quasinormal modes and stability in the full range of parameters . equation ( [ radialpart ] ) has an irregular singularity at spatial infinity and four regular singularities at @xmath46 , @xmath47 and @xmath48 . the appropriate frobenius series is determined as @xmath49 where @xmath50 and @xmath51 is the hawking temperature @xmath52 the function @xmath53 must be regular at the horizon and spatial infinity , so that the series in the vicinity of the event horizon @xmath54 satisfies both these requirements . this series converges everywhere outside the event horizon ( @xmath55 ) . when boundary conditions ( [ boundary - conditions ] ) are satisfied , that is when @xmath27 is a quasinormal ( qn ) frequency , the series convergence also at the spatial infinity @xcite . the coefficients @xmath56 satisfy the three - term recurrence relation . @xmath57 where @xmath58 , @xmath59 , @xmath60 can be found in an analytic form . by comparing the ratio of the series coefficients @xmath61we obtain an equation with a convergent _ infinite continued fraction _ on its right side : @xmath62which can be solved numerically by minimizing the absolute value of the difference between its left- and right - hand sides . equation ( [ continued_fraction ] ) has an infinite number of roots , but the most stable root depends on @xmath63 . the larger number @xmath63 corresponds to the larger imaginary part of the root @xmath27 @xcite . as we study qnms with slower decay rate , we usually choose @xmath64 . in order to improve convergence of the infinite continued fraction for nonzero mass of a scalar field , we use the nollert procedure @xcite . as for the study of mode branching we are also interested in modes with high multipole numbers , the wkb approach is useful here , which is accurate in the eikonal regime @xmath65 and usually provides very good accuracy at moderate @xmath66 . the wkb formula for calculation of qnms has the following form : @xmath67 where @xmath68 and @xmath69 are the values of the effective potential ( [ effective - potential ] ) and its second derivative with respect to the tortoise coordinate @xmath36 at the potential s peak . the terms @xmath70 depend on higher derivatives of @xmath71 at its maximum , and @xmath63 labels the overtones . the wkb approach was developed by schutz and will @xcite and later extended to higher orders @xcite . since there is implicit dependence on @xmath27 ( either through the separation constant @xmath26 for nonvanishing rotation or due to the @xmath1 coupling in the effective potential ( [ effective - potential ] ) ) , one has to search for the roots of equation ( [ wkbformula ] ) by minimizing the absolute value of the difference between its left and right sides . for nonrotating black holes we are able to construct a time - dependent profile of the wave function at a fixed @xmath36 . the recently found instability of the exactly extremal reissner - nordstrm black hole @xcite makes it important to check stability of the nearly and exactly extremal black holes . even when the frobenius method finds no growing quasinormal modes , one could think that the growing modes were simply missed in the frequency domain at the stage of search for the roots of the equation with continued fractions eq . ( [ continued_fraction ] ) , or , that a different boundary condition should be imposed at the event horizon in the extremal case . in order to eliminate both these suspicions , we will use the numerical characteristic integration method in time domain . here , two different schemes of integration were used for _ neutral _ and _ charged _ fields . * a scheme for a neutral field . * the wave equation can be written in time - dependent form as follows : @xmath72 the technique of integration of the above wave equation in the time domain was developed in @xcite . the method uses the light - cone variables @xmath73 , @xmath74 , so that the wave equation reads @xmath75 the initial data are specified on the two null surfaces @xmath76 and @xmath77 . acting by the time evolution operator @xmath78 on @xmath79 and taking account of ( [ light - cone ] ) , one finds @xmath80 where one introduced letters to mark the points as follows : @xmath81 , @xmath82 , @xmath83 , and @xmath84 . equation ( [ integration - scheme ] ) allows us to calculate the values of @xmath79 inside the rhombus , which is built on the two null - surfaces @xmath85 and @xmath86 , starting from the initial data specified on them . as a result we can find the time profile data @xmath87 in each point of the rhombus . the time - domain integration includes a contribution from all overtones , and , thus , missing some mode is excluded . this method is based on the scattering of the gaussian wave on the potential barrier and therefore does not specify the boundary condition on the event horizon , so that the potentially missed instability due to possibly different boundary conditions must be also discarded . * a scheme for a charged field . * when the scalar field is charged , an extra term containing the first derivative in time appears . therefore , a different integration scheme is required . we shall use the finite difference scheme proposed in @xcite . first we rewrite the wave - like equation ( [ radialpart ] ) for the reissner - nordstrm black hole ( @xmath32 ) in the time - dependent form @xmath88 where @xmath89 following @xcite , one can derive the evolution of @xmath79 in an isosceles triangle with the base on the axis @xmath36 , where initial conditions are imposed . then , one has @xmath90 indexes @xmath91 and @xmath92 enumerate , respectively , the coordinates @xmath93 and @xmath36 of the grid : @xmath94 we choose the initial conditions again as a gaussian distribution whose maximum is near the maximum of the effective potential . since von neumann stability conditions require @xmath95 , in this scheme we chose @xmath96 . in order to achieve convergence we decrease @xmath97 . note , that as @xmath1 grows the convergence becomes seemingly slower , requiring smaller @xmath97 , what increases the computation time . when @xmath98 this scheme is reduced to the one above for the neutral field , yet , the codes for both schemes differ , so that letting @xmath98 in the mathematica code for the second scheme leads to a much longer computing than the first scheme . * prony method for mode extraction . * once a time domain profile is found , one can extract dominant frequencies from it with the help of the prony method . we fit the profile data by superposition of damped exponents @xmath99 and look for the convergence of the obtained frequencies at the increasing @xmath100 . we shall show that although the fit works well for the neutral scalar field , it can not be effectively used once modes with very small @xmath42 dominate in the spectrum at late times . in the regime of large @xmath1 and nonextremal @xmath5 , the frobenius method allows us to find an approximate analytic expression for the quasinormal frequencies . when @xmath101 one can observe that @xmath102 , @xmath103 , and @xmath104 . then , we can rewrite eq . ( [ continued_fraction ] ) as @xmath105 considering @xmath106 , from ( [ continued_fraction_qq ] ) we find that @xmath107 . we observe that @xmath108 when @xmath109 . then , we can write down @xmath26 as @xmath110 and find the asymptotic formula for the qnms @xmath111 here @xmath112 is the overtone number . from the data given in fig . [ figasymptot ] one can find that @xmath113 for @xmath114 . in order to find @xmath115 as a function of @xmath12 and @xmath116 we have analyzed numerically the asymptotic behavior of the eigenvalue of ( [ angularpart ] ) and found that @xmath117 + 2,\ ] ] where the brackets denote the integer part . when @xmath5 is not very close to its extremal value , the above analytical formula ( in the limit of vanishing rotation ) can be verified by the time - domain integration for a few lower modes , as it is shown in fig . [ figworotation ] . there one can see that the analytic formula ( [ asymptotical_frequency ] ) works very well already at moderate values of @xmath1 . nevertheless , even though we are able to obtain a stable time - domain profile for any values of @xmath3 and @xmath5 , the prony method does not converge for high overtones as well as near extremal black holes . we shall discuss this in detail in the last section . for @xmath32 ( [ asymptotical_frequency ] ) coincides with the asymptotic formula found in @xcite in the regime @xmath118 with the help of the wkb approximation . unlike @xcite in our calculations one does not need to be limited by the regime @xmath119 . it is well known that due to a symmetry of the kerr - newman metric ( @xmath120 , @xmath121 ) quasinormal modes of a neutral scalar field come in within a degenerate pair @xmath122 when the scalar field is charged , in addition to ( [ degeneration ] ) one needs to take @xmath123 for the symmetry to be restored . that is why we study only qnms with positive real parts . the degeneration ( [ degeneration ] ) does not exist as soon as the coupling @xmath1 is not zero ( figs . [ figworotation ] and [ figrotation ] ) . then , at negative values of @xmath1 , the mode with a positive real part has slower decay rate than the one with the negative one ( while for positive @xmath1 the situation is opposite ) . moreover , when the absolute value of @xmath1 increases ( figs . [ figworotation ] and [ figrotation ] ) , the real oscillation frequency ( given by @xmath42 ) approaches zero and then the mode `` disappears '' from the spectrum at @xmath1 larger than some critical value . such disappearing of a mode in some range of parameters is not unusual and happens , for example , for arbitrarily long living modes ( quasiresonances ) of a massive neutral scalar field @xcite . we observe that at asymptotically late times , a neutral massless scalar field decays as @xmath124 in concordance with @xcite , while for the charged massless field , the dominant asymptotical tail is @xmath125 as it was first found in @xcite . " note , that in the regime of large @xmath8 the correction formula to ( [ chargedtail ] ) was reported in @xcite , which might be correct , and , then , should appear at larger values of @xmath8 than those we considered in the numerical time - domain evolution here . the late - time tails in the background of the extremally charged black hole obey the same law ( see fig . [ figtailsextr ] ) . at intermediately late times , _ a massive charged _ scalar field decay as ( see fig . [ figasympttailsmassive ] ) @xmath126 while at asymptotically late times , the decay law is @xmath127 thus , the power laws of a massive charged scalar field s decay at intermediate and asymptotically late times do not depend on multipole @xmath12 or charges @xmath3 and @xmath5 . the same law was obtained in @xcite for the massive dirac field . we suppose thereby that the asymptotic behavior of the charged massive fields does not depend on spin . the decaying time - domain profiles for various values of the parameters @xmath3 , @xmath5 and @xmath2 show that no instability exist for nonextremal reissner - nordstrm black holes . the complementary frequency domain data gives no indication of growing modes when the rotation is not vanishing . therefore , we conclude that there are no signs of instability for a massive charged scalar field in the nonextremal kerr - newman background under the quasinormal modes boundary conditions . we shall measure all quantities in units of @xmath128 , that is we take @xmath129 . in order to keep staying near the extremal state , it is sufficient to keep @xmath130 close to @xmath128 . here we shall start from the generalization of level plots of @xcite , which show the branching of modes . in @xcite it was found that , in the near - extremal kerr limit , zero - damped mode ( zdm ) satisfies @xmath131 where @xmath132 , @xmath133 . therefore , in fig . [ fig1 ] the same multipole @xmath12 and azimuthal @xmath116 numbers were chosen as in @xcite . in fig . [ fig1 ] it is shown the logarithm of the absolute value of difference between the left and right hand sides of the continued fraction equation ( [ continued_fraction ] ) as a function of real and imaginary parts of @xmath27 . in fig . [ fig1 ] we can see that at some near - extremal @xmath130 , and @xmath19 , which starts from @xmath134 until its extremal value , the new branch of modes appears : the two branches of modes , which take place for high rotation ( @xmath135 ) , merge into a single one at slower rotation ( @xmath136 ) . as it is expected , for small values of black hole s charge @xmath5 , the obtained in fig . [ fig1 ] zero - damped mode is close to the one described by eq . ( [ eqhod2 ] ) . the mode branching is owing to extremal rotation and not owing to extremal charge . indeed , a neutral scalar field around the near - extremal reissner - nordstrm black hole shows no mode branching . we have checked that the time - domain profile of scalar and gravitational perturbations of the exactly extremal reissner - nordstrm black hole consists of damped quasinormal oscillations , what proves stability of the extremal reissner - nordstrm solution from the point of view of an external observer , because 1 ) the profile includes contribution from all the modes , so that none can be missed , 2 ) the method of calculations does not imply any specific boundary conditions at the event horizon . this does not contradict a special instability of the extremal reissner - nordstrm black holes @xcite as this instability develops only on the event horizon and can not be observed by an external observer @xcite . .fundamental quasinormal modes ( @xmath114 ) of the scalar field for the extremal kerr - newman black hole ( @xmath137 ) . [ cols="^,^",options="header " , ] we have also showed that the decay of a scalar field ( be it charged or neutral , massive or massless ) is dominated by the power - law tails ( see table ( [ tail - summary ] ) ) which are the same for the nonextremal black hole as for the extremal one . here we have considered an already formed black hole and the perturbation propagating outside of it , while the modeling of collapse of charged matter leads to different result for the late - time decay of the extremal reissner - nordstrm black hole @xcite . thus we complement earlier results on late - time tails of neutral massive @xcite and massless fields @xcite in the schwarzschild and reissner - nordstrm backgrounds . in addition , when @xmath138 the continued fraction equation ( [ continued_fraction ] ) indicates presence of some new roots in the frequency domain , which do not exist for the neutral field , and whose real parts are proportional to @xmath139 . these roots exist for any value of @xmath5 and @xmath19 ( and nonzero @xmath3 ) , but correspond to higher `` overtones '' for weakly charged black hole ( fig . [ figimaginarymode ] ) , yet , they become dominant for nearly extremal black holes . these `` modes '' , if they exist , can not be extracted from a time - domain profile , probably due to the quick onset of asymptotic tails , and , therefore , must be further verified with an alternative method of calculation . the obtained here conclusion on the stability of a massive charged scalar field around kerr - newman black hole allows us to go on the study of hawking radiation for this case . using numerical techniques makes it possible to find grey - body factors accurately for the full range of parameters @xcite . this work was supported by the european commission grant through the marie curie international incoming program . r. a. k. acknowledges support of his visit to universidade federal do abc by fapesp . a. z. was supported by conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq ) . 80 b. carter , phys . lett . * 33 * , 558 ( 1974 ) . g. w. gibbons , commun . phys . * 44 * , 245 ( 1975 ) . g. w. gibbons , monthly notices of the royal astronomical society , vol . 177 , oct . 1976 , p. 37p-41p . a. n. aliev and d. v. galtsov , sov . 32 * , 75 ( 1989 ) . s. hod and t. piran , phys . d * 58 * , 024017 ( 1998 ) [ gr - qc/9712041 ] . r. a. konoplya , phys . d * 66 * , 084007 ( 2002 ) [ gr - qc/0207028 ] . r. a. konoplya , phys . b * 550 * , 117 ( 2002 ) [ gr - qc/0210105 ] . j. jing , q. -y . pan and x. he , int . j. mod . d * 16 * , 81 ( 2007 ) . r. a. konoplya and a. zhidenko , phys . d * 76 * , 084018 ( 2007 ) [ arxiv:0707.1890 [ hep - th ] ] . k. d. kokkotas , r. a. konoplya and a. zhidenko , phys . d * 83 * , 024031 ( 2011 ) [ arxiv:1011.1843 [ gr - qc ] ] . a. flachi and j. s. lemos , phys . rev . d * 87 * , 024034 ( 2013 ) [ arxiv:1211.6212 [ gr - qc ] ] . s. hod , phys . b * 710 * , 349 ( 2012 ) [ arxiv:1205.5087 [ gr - qc ] ] . a. a. starobinsky , zh . fiz . , 64 , 48 , 1973 ( transl . in soviet phys . jetp , 37 , 28 ) ; 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distant powerful radio galaxies are important laboratories for studying the formation and evolution of massive galaxies , because they are among the most luminous and largest known galaxies in the early universe and likely progenitors of dominant cluster galaxies ( e.g. * ? ? ? they are generally embedded in giant ( cd - sized ) ionized gas halos ( e.g. * ? ? ? * ) surrounded by galaxy overdensities , whose structures have sizes of a few mpc @xcite . the radio galaxy hosts have clumpy optical morphologies @xcite , spectra indicative of extreme star formation ( e.g. * ? ? ? * ) , and large stellar masses @xcite . because the radio lifetimes ( few times 10@xmath4 yr ) are much smaller than cosmological timescales , the statistics are consistent with every dominant cluster galaxy having gone through a luminous radio phase during its evolution @xcite hence , distant radio galaxies may be typical progenitors of galaxies that dominate the cores of local clusters . the radio source mrc 1138262 , identified with a galaxy at @xmath5 , is one of the most intensively studied distant radio galaxies @xcite . several of its properties are those expected of the progenitor of a dominant cluster galaxy . the @xmath6-band luminosity corresponds to a stellar mass of @xmath7@xmath8 @xmath2 @xcite , implying that mrc 1138262 is one of the most massive galaxies known at @xmath9 . the host galaxy is surrounded by a giant ly@xmath10 halo @xcite and the faraday rotation of the radio source is among the largest known @xcite , indicating that the system is embedded in a dense hot ionized gas with an ordered magnetic field . the radio galaxy is associated with a 3 mpc - sized structure of galaxies , of estimated mass @xmath11 2 @xmath1@xmath12 @xmath2 , the presumed antecedent of a local cluster . the presence of this protocluster " has been deduced using three independent selection techniques . there are overdensities of ly@xmath10 and h@xmath10 emission lines objects and galaxies having 4000 break continuum features at the approximate redshift of the radio galaxy @xcite . previous observations of mrc 1138262 with the _ hubble space telescope ( hst ) _ indicated that its optical emission is clumpy @xcite , indicative of a merging structure . here we present a new _ hst _ image of the radio galaxy that reaches 2 mag fainter than previous images and shows the merging processes in unprecedented detail . throughout this letter we assume a standard cosmology with @xmath13 = 72 km s@xmath3 mpc@xmath3 , @xmath14 = 0.27 , and @xmath15 = 0.73 , implying that at the distance of mrc 1138262 , an angular scale of 1 corresponds to a projected linear scale of 8.3 kpc . we obtained deep images of mrc 1138262 with the advanced camera for surveys ( acs ; * ? ? ? * ) on the _ hst_. the centre of the 1138262 protocluster was observed with the acs during 2005 1722 may in two 34@xmath134 acs fields that overlapped by 1 in a region that includes the radio galaxy . the total exposure time in the overlapping region was nine orbits with the f475w ( @xmath16 ) filter and 10 orbits with the f814w ( @xmath17 ) filter . the filters were selected to sample the continuum radiation with maximum sensitivity and colour discrimination , while minimizing contamination from bright emission lines . although the @xmath181549 and @xmath181640 lines fall within the @xmath16 passband , their measured rest - frame equivalent widths are 6 and 10 , respectively @xcite , implying that their effect on the continuum image is negligible . the observations were processed through the acs gto pipeline @xcite to produce registered , cosmic - ray rejected images . a deep 19 orbit composite image of the continuum emission was then produced by adding the @xmath16 and @xmath17 images . the @xmath19 depth in the overlapping region is 29.3 and 29.0 mag in the respective @xmath16 and @xmath17 images measured in a square @xmath20 diameter aperture . the _ hst _ image of the host galaxy and its immediate surroundings is shown in figure 1 , with rest - frame velocities of the ly@xmath10 emission corresponding to several of the acs continuum clumps . figure 2 illustrates the relation of the continuum optical emission to the associated gaseous halo and relativistic plasma . the giant gaseous halo extends by at least 25 ( @xmath7200 kpc ) and is one of the largest ly@xmath10 structures known in the universe . there are several features in the figures that are of interest : \1 . the optical continuum emission of the galaxy consists of at least 10 distinct clumps . the clumps are presumably satellite galaxies that are still merging with mrc 1138262 . they have sizes of typically @xmath70105 , corresponding to @xmath715 kpc , i.e. comparable to the typical sizes of lyman break galaxies @xcite several of the satellites have elongated structures reminiscent of chain and tadpole galaxies recently found to dominate the resolved population of the hubble ultra deep field ( hudf , * ? ? ? * ; * ? ? ? examples are denoted by numbers 1 , 3 , and 4 in figure 1 . another linear , distorted galaxy is seen 3 north of the nucleus ( 2 ) , and several galaxies having double ( 6 , 8) or clumpy morphologies embedded in diffuse emission ( 5 , 7 ) lie at slightly larger distances from the main complex . these objects have @xmath21 colors of between 0.1 and 0.7 magnitudes and @xmath17 magnitudes of between 24.3 and 27.7 , consistent with star formation at rates between 0.5 and 26 @xmath22 yr@xmath3 @xcite . to determine whether there is a concentration of such objects around the radio galaxy , we analyzed the statistics of tadpole and chain galaxies having @xmath23 in the whole 3@xmath15 acs field around mrc 1138262 . to minimize systematic effects , the morphologies were classified manually by somebody not previously associated with the project ( e. h. ) . in the 10@xmath110 area around the radio galaxy , three such objects were found ( objects 1 , 3 , and 4 in fig . this should be compared with an expected number of 0.22 from our analysis of the whole field , a value consistent with the hudf analysis of @xcite . taking into account the clustering of these objects , the probability of finding three such galaxies within a 10@xmath110 region was estimated to be parts in 1000 . this implies that the chain and tadpole galaxies are concentrated at the position of the radio galaxy and connected with the forming massive galaxy at the protocluster centre . faint diffuse emission is visible between the obvious clumps . this extended emission is unlikely to be dominated by scattered light of an obscured nuclear quasar , because its morphology is not reminiscent of a scattering cone . furthermore , its mean color is comparable to that of the star - forming clumps , consistent with the occurrance of ongoing star formation over the whole central 50@xmath140 kpc region . the total extended luminosity ( comprising 45% of the total emission in @xmath16 ) implies a star formation rate of @xmath24100 @xmath22 yr@xmath3 . ly@xmath10 was detected from all of the satellite galaxies in the halo that have been studied spectroscopically . the width of the ly@xmath10 profile in the halo is consistent with the observed velocity dispersion of the associated protocluster , indicating that the galaxies are moving relative to each other with radial velocities of up to a few thousand kilometers per second . although the optical structure is extended approximately along the radio axis , most of the galaxy is located outside the narrow region occupied by the radio source and is therefore unlikely to be influenced directly by the radio source . the obvious interpretation of the new _ hst _ image is that it shows hierarchical merging processes occurring in a forming massive cd galaxy . the morphological complexity and clumpiness observed in the mrc 1138 - 262 system agrees qualitatively with predictions of hierarchical galaxy formation models ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . because of its striking appearance and its probable nature , we name the host galaxy of mrc 1138262 the `` spiderweb galaxy '' and refer to the small satellite galaxies located within and around it as `` flies . '' with relative velocities of several hundred kilometers per second ( fig . 1 ) , these flies will traverse the 100 kpc extent of the spiderweb many times in the interval between @xmath25 and 0 , consistent with the merger scenario . vigorous merging also provides a plausible mechanism for fueling the supermassive black hole , which may subsequently quench any ongoing star formation through radio feedback ( e.g. * ? ? ? * ) . using recent infrared spectroscopic observations of the gas in mrc 1138 - 262 , @xcite show that the pressure of the radio source is sufficiently large to expel @xmath26 of the gas during the radio source lifetime . after the gas available for star formation has been expelled , the growth of such a massive galaxy will proceed primarily through merging @xcite . assuming that the flies are undergoing single @xmath71 gyr starbursts , their star formation rates imply a final stellar mass for each satellite galaxy of several times @xmath27 @xmath2 and a total mass for all the flies of @xmath28 @xmath2 . this is less than a tenth of the mass of the whole galaxy , derived from its @xmath6-band luminosity @xcite , implying that a large fraction of the galaxy mass has already assembled by @xmath29 . such a relatively early growth disagrees with the predictions of some models for typical evolution of the dominant galaxies in the most massive clusters @xcite . the most intriguing feature in our _ hst _ image is the association of the linear flies ( chains and tadpole morphologies ) with the merging massive host galaxy . in the hudf the frequency of such objects increases dramatically at faint magnitudes ( * ? ? ? * @xmath30 @xmath24 24 ; ) . because such peculiar galaxies dominate the faint resolved galaxy population , they are likely to be an important source of star formation in the early universe . the nature of these objects is unclear . there are several possibilities . 1 . they may be spiral galaxies observed edge - on @xcite . however , it is difficult to account for the large numbers of faint linear galaxies observed in the hudf by such selection effects . their elongated appearance may be due to star formation associated with radio jets @xcite produced by primeval massive black holes @xcite . radio synchrotron jets are known to occur on varying scales , ranging from the most luminous galactic nuclei to x - ray binaries and jets are sometimes associated with star formation @xcite . they may be formed as the result of merging , either of galaxies that are formed along filamentary gravitational instabilities @xcite or in major events @xmath70.7 gyr after the merging process has commenced @xcite . the fact that the spiderweb linear flies are located in an environment where vigorous galaxy interactions are taking place is consistent with a merger hypothesis for their origin . the motions of the flies with velocities of several hundred kilometers per second through the dense gaseous halo , perturbed by superwinds from the nucleus @xcite and the radio jet , could result in shocks . the shocks would then lead to jeans - unstable clouds , enhanced star formation along the direction of motion , and chain and tadpole morphologies . the morphology of the spiderweb galaxy provides a unique new testbed for simulations of forming massive galaxies at the centers of galaxy clusters . the occurrence of tadpole and chain galaxies in the dense central environment of the protocluster places constraints on ( 1 ) evolution models for dominant cluster galaxies and ( 2 ) the nature of the chains and tadpoles , an important constituent of the early universe . our results are consistent with a merger scenario for the formation of these peculiar linear galaxies . deep observations with the _ hst _ of similar objects over a range of redshifts are needed to study whether linear flies are generally present in the vicinity of cd galaxy progenitors and how their luminosity functions in these special regions compare with those of other types of galaxies . the positions and morphological parameters of flies in spiderweb galaxies at @xmath9 together with hubble data on dominant cluster galaxies at @xmath31 will provide new constraints for models of massive galaxy formation . future spectroscopic observations will delineate the velocity field and color distributions of mrc 1128262 in more detail , thereby elucidating further how the flies are being captured by the spiderweb galaxy . g. k. m. acknowledges support from the royal netherlands academy for arts and sciences and the netherlands organization for scientific research ( nwo ) . j. k. was supported by dfg/ sfb 439 . the acs was developed under nasa contract nas 5 - 32864 . the research has been supported by nasa grant nag5 - 7697 and an equipment grant from sun microsystems , inc . this letter is based partially on observations made with ( 1 ) the acs on the nasa / esa hubble space telescope , obtained via the stsci , which is operated by the aura inc . , under nasa contract nas 5 - 26555 , ( 2 ) the vlt at eso , paranal , chile , program p63.o-0477(a ) , and ( 3 ) the vla operated by the nrao , which is operated by the aui . bicknell , g. v. , sutherland , r. s. , van breugel , w. j. m. , dopita , m. a. , dey , a. , & miley , g. k. 2000 , , 540 , 678 blakeslee , j. p. , anderson , k. r. , meurer , g. r. , bentez , n. , & magee , d. 2003 , in asp conf . 295 , astronomical data analysis software and systems xii , ed . e. payne , r. i. jedrzejewski , & r. n. hook ( san francisco ; asp ) , 257 bouwens , r. j. , illingworth , g. d. , blakeslee , j. p. , broadhurst , t. j. , & franx , m. 2004 , , 611 , l1 carilli , c. l. , rttgering , h. j. a. , van ojik , r. , miley , g. k. , & van breugel , w. j. m. 1997 , , 109 , 1 croton , d. j. , et al . 2006 , , 365 , 11 de lucia , g. , & blaizot , j. 2006 , , submitted ( astro - ph/0606519 ) dey , a. , van breugel , w. , vacca , w. d. , & antonucci , r. 1997 , , 490 , 698 dubinski , j. 1998 , , 502 , 141 di matteo , t. , springel , v. , & hernquist , l. 2005 , , 433 , 604 elmegreen , d. m. , elmegreen , b. g. , rubin , d. s. , & schaffer , m. a. 2005 , , 631 , 85 ferguson , h. c. , et al . 2004 , , 600 , l107 ford , h. c. , et al . 1998 , , 3356 , 234 gao , l. , loeb , a. , peebles , p. j. e. , white , s. d. m. , & jenkins , a. 2004 , , 614 , 17 kurk , j. d. , 2003 , ph.d . thesis , univ . leiden kurk , j. d. , pentericci , l. , overzier , r. a. , rttgering , h. j. a. , & miley , g. k. 2004a , , 428 , 817 kurk , j. d. , pentericci , l. , rttgering , h. j. a. , & miley , g. k. 2004b , , 428 , 793 larson , r. 1992 , in star formation in stellar systems , ed . g. tenorio - tagle , m. prieto , & f. sanchez ( cambridge : cambridge univ . press ) , 125 madau , p. , pozzetti , l. , & dickinson , m. 1998 , , 498 , 106 miley , g. 2000 , in from extrasolar planets to cosmology , ed . j. bergeron & a. renzini ( berlin : springer ) , 32 nesvadba , n. p. h. , et al . 2006 , , in press pentericci , l. , rttgering , h. j. a. , miley , g. k. , carilli , c. l. , & mccarthy , p. 1997 , , 326 , 580 pentericci , l. , rttgering , h. j. a. , miley , g. k. , mccarthy , p. , spinrad , h. , van breugel , w. j. m. , & macchetto , f. 1999 , , 341 , 329 pentericci , l. , rttgering , h. j. a. , miley , g. k. , spinrad , h. , mccarthy , p. j. , van breugel , w. j. m. , & macchetto , f. 1998 , , 504 , 139 pentericci , l. , et al . 2000 , , 361 , l25 rees , m. j. 1989 , , 239 , 1p rttgering , h. j. a. , van ojik , r. , miley , g. k. , chambers , k. c. , van breugel , w. j. m. , & de koff , s. 1997 , , 326 , 505 silk , j. , & rees , m. j. 1998 , , 331 , l1 springel , v. , di matteo , t. , & hernquist , l. 2005a , , 361 , 776 springel , v. , et al . 2005b , , 435 , 629 straughn , a. n. , cohen , s. h. , ryan , r. e. , hathi , n. p. , windhorst , r. a. , & jansen , r. a. 2006 , , 639 , 724 taniguchi , y. , & shioya , y. 2001 , , 547 , 146 van breugel , w. , filippenko , a. v. , heckman , t. , & miley , g. 1985 , , 293 , 83 van ojik , r. , rttgering , h. j. a. , miley , g. k. , & hunstead , r. w. 1997 , , 317 , 358 venemans , b. p. , et al . 2002 , , 569 , l11 venemans , b. p. , et al . 2005 , , 431 , 793 villar - martn , m. , et al . 2006 , , 366 , l1 zirm , a. w. , et al . 2005 , , 630 , 68	we present a deep image of the radio galaxy mrc 1138262 taken with the _ hubble space telescope ( hst ) _ at a redshift of @xmath0 . the galaxy is known to have properties of a cd galaxy progenitor and be surrounded by a 3 mpc - sized structure , identified with a protocluster . the morphology shown on the new deep _ hst _ acs image is reminiscent of a spider s web . more than 10 individual clumpy features are observed , apparently star - forming satellite galaxies in the process of merging with the progenitor of a dominant cluster galaxy 11 gyr ago . there is an extended emission component , implying that star formation was occurring over a 50@xmath140 kpc region at a rate of more than 100 @xmath2 yr@xmath3 . a striking feature of the newly named `` spiderweb galaxy '' is the presence of several faint linear galaxies within the merging structure . the dense environments and fast galaxy motions at the centres of protoclusters may stimulate the formation of these structures , which dominate the faint resolved galaxy populations in the hubble ultra deep field . the new image provides a unique testbed for simulations of forming dominant cluster galaxies .	5,872	330
_ real projective 3-space _ , @xmath2 , is defined to be the quotient @xmath3 , where @xmath4 is the antipodal relation @xmath5 and can be thought of as the disk , @xmath6 , with antipodal boundary points identified . projective space has a non - trivial first homology group , @xmath7 . the generator for the group , @xmath8 , is the cycle originating from the line in @xmath6 that runs between the north and south poles . mroczkowski @xcite has shown that every knot in @xmath2 can be transformed into either the trivial cycle or @xmath8 by crossing changes and generalized reidemeister moves on an @xmath9 projection of the knot . thus , there are two non - equivalent unknots in @xmath2 . cycles that can be unknotted " into a cycle homologous to @xmath8 will be referred to as _ 1-homologous cycles_. cycles that can be unknotted " into a trivial cycle will be referred to as _ 0-homologous cycles_. a _ link _ in @xmath2 is _ splittable _ if one of the components can be contained within a sphere , embedded in the space , while the other component remains in the complement of the sphere . otherwise , the link in @xmath2 is _ non - split_. a non - split link can be formed one of three ways in @xmath2 : two 0-homologous cycles , a 0-homologous cycle with a 1-homologous cycle , and two 1-homologous cycles . note : two disjoint 1-homologous cycles will always form a non - split link . similarly , a _ non - split triple - link _ is a non - split link of three components . in this paper we will refer to non - split linked cycles as _ linked cycles _ and an embedding of a graph as _ linked _ if it contains a non - split link . we will refer to a non - split triple - link as a _ triple - link _ and an embedding of a graph as _ triple - linked _ if it contains a non - split triple - link . a graph @xmath10 is a _ minor _ of a graph @xmath11 if @xmath10 can be obtained from @xmath11 through a series of vertex removals , edge removals , or edge contractions . a graph @xmath11 is _ minor - minimal _ with respect to a property @xmath12 if @xmath11 has property @xmath12 but no minor of @xmath11 has property @xmath12 . if @xmath11 is a graph , define an _ induced subgraph _ , @xmath13 $ ] , of @xmath11 to be the subgraph of @xmath11 on vertices @xmath14 and the set of edges in @xmath11 with both endpoints in the set @xmath15 . a graph @xmath11 is _ intrinsically linked in _ @xmath16 if and only if @xmath11 contains a non - split link in every spatial embedding . we define _ intrinsically linked in _ @xmath17 analogously . it has been shown that the complete set of minor - minimal intrinsically linked graphs in @xmath16 is the set of petersen family graphs @xcite ( including @xmath18 and graphs obtained from @xmath18 by @xmath19 and @xmath20 exchanges ) . however , all petersen family graphs except for @xmath21 embed linklessly in @xmath17 @xcite . while @xcite characterizes several families of graphs that are minor - minimally intrinsically linked in @xmath17 , the complete set of minor - minally intrinsically linked graphs in @xmath17 , which is finite due to the result in @xcite , remains to be found . a graph @xmath11 is _ intrinsically triple - linked in _ @xmath1 if and only if @xmath11 contains a non - split link of three components in every spatial embedding . we define _ intrinsically triple - linked in _ an embedding is said to be 3-_linkless _ if and only if it does not contain a triple - link . while conway , gordon @xcite , and sachs @xcite showed that @xmath18 is intrinsically linked in @xmath1 , @xmath18 can be linklessly embedded in @xmath2 ; it has been shown that 7 is the smallest @xmath22 for which @xmath23 is intrinsically linked in @xmath2 @xcite . in contrast , while 10 was shown to be the smallest @xmath22 for which @xmath23 is intrinsically triple - linked in @xmath1 @xcite , we have shown that 10 is also the smallest @xmath22 for which @xmath23 is intrinsically triple - linked in @xmath2 . it remains to be shown whether @xmath0 is minor - minimal with respect to triple - linking in @xmath2 . additionally , we have shown two other intrinsically triple - linked graphs in @xmath1 can be embedded without a triple - link in @xmath2 . a complete set of minor - minimal intrinsically triple - linked graphs remains to be found , in both @xmath1 and @xmath2 . such sets are finite due to the result in @xcite . we will need the following lemmas : [ k7 ] @xcite _ the graphs obtained by removing two edges from @xmath24 and removing one edge from @xmath25 are intrinsically linked in @xmath2 . _ [ ohk4 ] @xcite _ given a linkless embedding of @xmath18 in @xmath2 , no @xmath26 subgraph can have all 0-homologous cycles . _ we also use the following elementary observation . [ evenk4 ] _ for every embedding into @xmath2 , @xmath26 has an even number of 1-homologous cycles . _ the following lemma was shown true in @xmath1 by flapan , naimi , and pommersheim @xcite and the proof holds true analogously in @xmath2 . [ ll ] _ let @xmath11 be a graph embedded in @xmath2 that contains cycles @xmath27 , @xmath28 , @xmath29 and @xmath30 . suppose @xmath27 and @xmath30 are disjoint from each other and from @xmath28 and @xmath29 and suppose @xmath31 is a simple path . if @xmath32 and @xmath33 , then @xmath11 contains a non - split three - component link . _ the following proposition is not the main result of this paper . however , the proof is included because it is concise and since its method does not hold for proving @xmath0 is also triple - linked . _ the graph @xmath34 is intrinsically triple - linked in @xmath2 . _ let @xmath11 be a complete graph on the vertex set @xmath35 . embed @xmath11 in @xmath2 . consider @xmath36 $ ] . since @xmath24 is intrinsically linked in @xmath2 , this subgraph contains a pair of linked cycles that can be reduced to two linked @xmath37-cycles . without loss of generality , let @xmath38 and @xmath39 be the pair of linked cycles in @xmath36 $ ] . now consider @xmath40 $ ] . since @xmath24 is intrinsically linked in @xmath2 , this subgraph contains a pair of linked cycles that can be reduced to two linked @xmath37-cycles . in @xmath40 $ ] , one cycle must use @xmath41 and the other cycle must use @xmath42 , or lemma [ ll ] would apply immediately . without loss of generality , let @xmath43 and @xmath44 be the pair of linked cycles in @xmath40 $ ] . consider @xmath45 $ ] . by lemma [ ohk4 ] , @xmath46 $ ] must contain a @xmath47-homologous cycle or @xmath45 $ ] contains a pair of linked cycles and lemma [ ll ] applies with @xmath29 and @xmath30 . thus by lemma [ evenk4 ] , two cycles in @xmath48 must be @xmath47-homologous @xmath37-cycles . now consider @xmath49 $ ] . by lemma [ ohk4 ] , @xmath50 $ ] must contain a @xmath47-homologous cycle or @xmath49 $ ] contains a pair of linked cycles and lemma [ ll ] applies with @xmath27 and @xmath28 . thus by lemma [ evenk4 ] , two cycles in @xmath51 must be @xmath47-homologous @xmath37-cycles . since every cycle in @xmath52 is disjoint from every cycle in @xmath53 , and at least two cycles in each set are @xmath47-homologous , there exists a link using one cycle from @xmath52 and one cycle from @xmath53 . lemma [ ll ] then applies since every cycle in @xmath52 shares at least a simple path with @xmath27 , and @xmath28 and the cycle from @xmath53 are disjoint from each other , @xmath27 , and the cycle from @xmath52 . thus , @xmath11 contains a triple - link . -1.5 in in @xmath2.,title="fig : " ] -1.5 in _ if @xmath11 is @xmath18 embedded in @xmath2 and @xmath11 has two disjoint @xmath54-homologous cycles , then @xmath11 contains a non - split link . _ assume @xmath11 can be embedded so that it has two disjoint 0-homologous cycles and so that it does not have a non - split link . without loss of generality , let @xmath55 and @xmath56 be @xmath54-homologous cycles in @xmath11 . consider @xmath57 $ ] . since @xmath11 is not linked , by lemma [ ohk4 ] and lemma [ evenk4 ] , @xmath57 $ ] must have two @xmath47-homologous cycles . without loss of generality , let @xmath58 and @xmath59 be @xmath47-homologous cycles . similarly , @xmath60 $ ] must also have two @xmath47-homologous cycles . since @xmath56 is @xmath54-homologous by assumption and @xmath61 is disjoint from @xmath59 , @xmath62 and @xmath63 are @xmath47-homologous cycles . similarly , @xmath64 $ ] has two @xmath47-homologous cycles . since @xmath55 is @xmath54-homologous by assumption and @xmath65 is disjoint from @xmath62 , @xmath66 and @xmath67 are @xmath47-homologous cycles or @xmath11 would contain a pair of linked cycles . now consider @xmath68 $ ] , which must also have two @xmath47-homologous cycles by lemma 2 and lemma 3 . since @xmath69 is disjoint from @xmath63 , @xmath65 is disjoint from @xmath62 , and @xmath70 is disjoint from @xmath58 , @xmath69 , @xmath65 , and @xmath70 must be @xmath54-homologous . this forces @xmath71 $ ] to contain only 0-homologous cycles , and thus @xmath11 is linked by [ ohk4 ] . thus , @xmath11 can not have two disjoint @xmath54-homologous cycles and not be linked . _ up to ambient isotopy and crossing changes , figure [ linkless ] is the only way to linklessly embed @xmath18 in @xmath2 . _ let @xmath11 be a complete graph on the vertex set \{1 , 2 , 3 , 4 , 5 , 6}. embed @xmath11 in @xmath2 . the graph @xmath11 has a @xmath54-homologous 3-cycle , else @xmath11 has disjoint @xmath47-homologous cycles and is thus linked by proposition 6 . without loss of generality , let @xmath56 be a @xmath54-homologous 3-cycle . now consider vertices \{1 , 2 , 3}. if @xmath55 is @xmath54-homologous , @xmath11 is linked ; thus , we assume @xmath55 is a @xmath47-homologous cycle . mroczkowski @xcite showed that any cycle can be made into an unknotted @xmath54- or @xmath47-homologous cycle by crossing changes , so we can assume after crossing changes and ambient isotopy the embedding has a projection as drawn in figure [ linkless ] ( except the edges between the vertices @xmath72 and @xmath73 may be more complicated than in the figure ) with vertices @xmath72 on the boundary and the edges between them on the boundary . we may use ambient isotopy and crossing changes so that edges from @xmath72 to @xmath73 connect in the projection without crossing the boundary of @xmath74 . we now show that we may connect them , without loss of generality , as depicted in figure 1 . if vertex @xmath75 , @xmath76 must connect to at least one of \{4 , 5 , 6 } from @xmath77 and to at least one of \{4 , 5 , 6 } from @xmath78 , else there would be a @xmath54-homologous @xmath26 and @xmath11 would be linked by lemma 2 . without loss of generality , assume @xmath79 connects to @xmath80 and @xmath81 from @xmath82 and to @xmath83 from @xmath84 . if @xmath85 connects to @xmath80 and @xmath81 from @xmath86 , then @xmath87 $ ] is a @xmath54-homologous @xmath26 and @xmath11 is linked by lemma 2 . thus , @xmath85 connects to either @xmath80 or @xmath81 from @xmath86 and connects to the other from @xmath88 . without loss of generality , let @xmath86 connect to @xmath80 ; so , @xmath88 connects to @xmath81 . if @xmath86 connects to @xmath83 , then either @xmath89 or @xmath90 must connect to both @xmath83 and @xmath81 and the other to @xmath80 , else @xmath11 has a @xmath54-homologous @xmath26 . without loss of generality , let @xmath89 connect to @xmath83 and @xmath81 and @xmath90 connect to @xmath80 . then , @xmath91 and @xmath92 are disjoint @xmath47-homologous cycles so @xmath11 is linked . thus , @xmath88 connects to @xmath83 . now , if @xmath89 connects to either @xmath80 and @xmath81 or @xmath83 and @xmath81 , then @xmath11 has a @xmath54-homologous @xmath26 and is linked by lemma 2 . so , @xmath89 must connect to @xmath81 and @xmath90 must connect to @xmath80 and @xmath83 . -1.5 in in @xmath2.,title="fig : " ] -1.5 in signed graphs , that is , graphs with each edge assigned a @xmath93 or a @xmath94 sign , have been studied extensively and were first introduced by harary @xcite , see also @xcite . an embedding of a graph @xmath11 into @xmath2 induces a signed graph of @xmath11 as follows : deform the embedding so that no vertices touch the line at infinity and all intersections of edges with the line at infinity are transverse . assign @xmath93 edges to be edges that hit the boundary an even number of times and @xmath94 edges to be edges that hit the boundary an odd number of times . if a cycle has an odd number of @xmath94 edges , then the cycle is @xmath47-homologous . two embeddings , @xmath95 and @xmath96 , of a graph @xmath11 are _ crossing - change equivalent _ if and only if @xmath95 can be obtained from @xmath96 by crossing changes and ambient isotopy . thus , by proposition 7 , a linkless @xmath18 is crossing - change equivalent to the embedding in figure [ signed ] . that is , @xmath97 , @xmath98 , @xmath99 , @xmath100 , @xmath101 , and @xmath102 are @xmath94 edges , and the other nine edges are @xmath93 edges . _ the graph @xmath0 is intrinsically triple - linked in @xmath2 . _ let @xmath11 be a graph isomorphic to @xmath0 on the vertex set @xmath103 . embed @xmath11 in @xmath2 . assume the embedding is 3-linkless . if every subgraph of @xmath11 isomorphic to @xmath18 is linked , then flapan , naimi , and pommersheim s proof @xcite that @xmath0 is intrinsically linked in @xmath1 nearly works , except they do use the fact that @xmath104 is intrinsically linked at the very end and @xmath104 is not intrinsically linked in @xmath2 . bowlin - foisy @xcite , however , modify @xcite slightly so that only the fact that @xmath18 is intrinsically linked is needed . thus , in the case that every subgraph of @xmath11 isomorphic to @xmath18 is linked , then @xmath11 is triple - linked . so we may assume there exists a linkless @xmath18 subgraph in @xmath11 . without loss of generality , assume this linkless @xmath18 is on vertices @xmath105 . by proposition 7 , this @xmath18 has an embedding crossing - change equivalent to that in drawn in figure [ signed ] . * claim : * the embedded induced subgraph @xmath106 $ ] is @xmath54-homologous . assume @xmath50 $ ] has a @xmath47-homologous cycle . without loss of generality , let @xmath107 be a @xmath47-homologous cycle . now consider @xmath108 $ ] . if @xmath108 $ ] is not @xmath54-homologous , then two of @xmath109 , @xmath110 , and @xmath111 are @xmath47-homologous by lemma [ evenk4 ] . then @xmath55 , @xmath107 , and a cycle from @xmath108 $ ] comprise three disjoint @xmath47-homologous cycles , so @xmath11 is triple - linked . thus , @xmath108 $ ] is @xmath54-homologous and so @xmath112 $ ] has a pair of linked cycles by lemma [ ohk4 ] . since @xmath107 is @xmath47-homologous , and @xmath107 is disjoint from all the @xmath47-homologous cycles in the second column of table 1 , lemma [ ll ] applies and @xmath11 has a triple - link . thus , @xmath50 $ ] is @xmath54-homologous . [ cols="^,^ " , ] table @xmath113 . this list exhausts the possible embeddings with @xmath114 and @xmath115 . the same argument holds for the embedding has @xmath116 and @xmath117 . thus , in every embedding of @xmath11 in @xmath2 , @xmath11 has a triple - link . flapan , naimi , and pommersheim @xcite showed that @xmath118 can be embedded 3-linklessly in @xmath1 , and so @xmath118 can be embedded 3-linklessly in @xmath2 . thus , 10 is the smallest @xmath22 for which @xmath23 is intrinsically triple - linked in @xmath2 . _ a graph composed of @xmath22 disjoint copies of an intrinsically @xmath22-linked graph in @xmath1 is intrinsically @xmath22-linked in @xmath2 . in particular , three disjoint copies of intrinsically triple - linked graphs in @xmath1 are intrinsically triple - linked in @xmath2 _ if any of the three copies of the graph has all 0-homologous cycles , then it is crossing - change equivalent to a spatial embedding , and thus triple - linked , as its disjoint cycle pairs would have the same linking numbers as a spatial embedding . else , all three copies have at least one 1-homologous cycle . then we have three disjoint 1-homologous cycles , and thus have a triple - link . as shown above , @xmath0 is an example of a one - component graph that is intrinsically triple - linked in @xmath1 . in the following section , we will exhibit two examples of minor - minimal intrinsically triple - linked graphs , each comprised of two components , that are intrinsically triple - linked in @xmath1 . the question remains whether there exists a minor - minimal intrinsically triple - linked graph of three components in @xmath2 . we will use the following theorem : @xcite _ let @xmath11 be a graph containing two disjoint graphs from the petersen family , @xmath95 and @xmath96 as subgraphs . if there are edges between the two subgraphs @xmath95 and @xmath96 such that the edges form a 6-cycle with vertices that alternate between @xmath95 and @xmath96 , then @xmath11 is minor - minimal intrinsically triple - linked in @xmath1 . _ -2.1 in connected to @xmath18 along a 6-cycle in @xmath2.,title="fig : " ] -1.7 in if @xmath95 and @xmath96 are isomorphic to @xmath18 , this result does not hold in @xmath2 , as seen in figure [ k6k6 ] . _ if @xmath95 and @xmath96 are disjoint copies of @xmath18 connected to @xmath18 along a @xmath119-cycle with vertices that alternate between the copies of @xmath18 , then @xmath120 is minor - minimal intrinsically triple - linked in @xmath121 . _ embed @xmath11 in @xmath2 . if @xmath95 or @xmath96 has all @xmath54-homologous cycles , @xmath11 will have a triple - link since @xmath18 connected to @xmath18 along a @xmath119-cycle with vertices that alternate between the copies of @xmath18 is triple - linked in @xmath1 . thus , @xmath95 and @xmath96 each have a @xmath47-homologous cycle . let @xmath95 be a graph on the vertex set @xmath122 where @xmath123 $ ] and @xmath124 $ ] are the copies of @xmath18 and the connecting edges are @xmath125 , @xmath126 , @xmath127 , @xmath128 , @xmath129 , and @xmath130 . up to isomorphsim , there are five @xmath37-cycle equivalence classes in @xmath95 . consider @xmath131 , which contains one representative from each @xmath37-cycle class . we assume , without loss of generality , one cycle in @xmath132 is @xmath47-homologous . consider @xmath124 $ ] . if there is a one homologous cycle in @xmath133 $ ] then this cycle will link with the cycle in @xmath132 that is @xmath47-homologous . since the cycle from @xmath132 links with the @xmath47-homologous cycle in @xmath96 , we have a triple - link in @xmath11 . thus , we assume every cycle in @xmath133 $ ] is @xmath54-homologous and so @xmath124 $ ] has a pair of linked cycles . by the pigeon - hole principle , at least two edges connecting vertices from the set @xmath134 are in a linked cycle in @xmath124 $ ] , so , without loss of generality , we may assume @xmath77 and @xmath78 are in one cycle . if the @xmath47-homologous cycle is in the subset @xmath135 , then there are disjoint edges from the @xmath119-cycle that connect the cycle from @xmath136 to the cycle containing @xmath77 and @xmath78 . so , by lemma [ ll ] , @xmath11 has a triple - link . if @xmath137 is the @xmath47-homologous cycle , consider @xmath123 $ ] . if there is a one homologous cycle in @xmath138 $ ] then this cycle will link with @xmath137 and the @xmath47-homologous cycle in @xmath96 , so @xmath11 will have a triple - link . else , @xmath123 $ ] has a pair of linked cycles . by the pigeon - hole principle , at least two vertices in the set @xmath73 are in a linked cycle within the embedding of one copy of @xmath18 . similarly , at least two vertices of @xmath134 are in a linked cycle in the other copy of @xmath18 . as a result of the @xmath119-cycle , there are two disjoint edges between the cycles and lemma [ ll ] then applies and @xmath11 is triple - linked . to see @xmath11 is minor - minimal with respect to intrinsic triple - linking in @xmath121 , embed @xmath11 so that @xmath95 is embedded as in the drawing in figure [ k6k6 ] and @xmath96 is contained in a sphere that lies in the complement of @xmath95 . therefore , @xmath95 does not have any triple - links and no cycle in @xmath95 is linked with a cycle in @xmath96 . without loss of generality , if we delete an edge , contract an edge or delete any vertex on @xmath96 , it will have an affine linkless embedding . thus , we can re - embed @xmath96 within the sphere in each case . thus , @xmath11 is minor - minimal for intrinsic triple - linking . -1.5 in connected to @xmath24 along an edge in @xmath2.,title="fig : " ] -1 in @xcite _ let @xmath11 be a graph formed by identifying an edge of @xmath24 with an edge from another copy of @xmath24 . then @xmath11 is intrinsically triple - linked in @xmath1 . _ if @xmath11 is isomorphic to @xmath24 connected to @xmath24 along an edge , this result does not hold in @xmath2 , as seen in figure [ k7k7 ] . we will need the following lemma : [ 5v ] @xcite _ let @xmath12 be a petersen - family graph and @xmath76 be a vertex of p. if every cycle of @xmath139 is @xmath54-homologous in an embedding @xmath140 , then @xmath141 contains a non - trivial link . _ _ if @xmath95 and @xmath96 are disjoint copies of @xmath24 connected to @xmath24 along an edge , then @xmath120 is intrinsically triple - linked in @xmath121 . _ if @xmath95 or @xmath96 have all @xmath54-homologous cycles , @xmath11 will have a triple - link since @xmath24 connected to @xmath24 along an edge is triple - linked in @xmath1 . thus , @xmath95 and @xmath96 each have a @xmath47-homologous cycle . let @xmath95 be a graph on the vertex set @xmath142 where @xmath36 $ ] and @xmath143 $ ] are the copies of @xmath24 and the connecting edge is @xmath144 . up to isomorphsim , there are three @xmath37-cycle equivalence classes in @xmath95 . we consider @xmath145 , which contains one representative from each @xmath37-cycle class . we can assume , without loss of generality , at least one cycle of @xmath132 is @xmath47-homologous . * case 1 : * let @xmath55 be a @xmath47-homologous cycle in @xmath95 . then @xmath55 links with the @xmath47-homologous cycle in @xmath96 . consider @xmath146 $ ] . if @xmath147 $ ] has a @xmath47-homologous cycle , then there are three disjoint @xmath47-homologous cycles , so we assume @xmath146 $ ] must be @xmath54-homologous and so @xmath147 $ ] has a pair of linked cycles . lemma [ ll ] applies with @xmath148 connecting to the cycle that uses @xmath81 , following the proof in @xcite . * case 2 : * let @xmath149 be a @xmath47-homologous cycle in @xmath95 . @xmath149 links with the @xmath47-homologous cycle in @xmath96 . consider @xmath146 $ ] . if @xmath147 $ ] has a @xmath47-homologous cycle , then there are three disjoint @xmath47-homologous cycles , so we assume @xmath147 $ ] must be @xmath54-homologous and so @xmath146 $ ] has a pair of linked cycles . lemma [ ll ] applies with @xmath148 connecting to the cycle that uses @xmath81 , following the proof in @xcite . * case 3 : * let @xmath150 be a @xmath47-homologous cycle in @xmath95 . @xmath150 links with the @xmath47-homologous cycle in @xmath96 . consider @xmath146 $ ] . if @xmath151 $ ] has a @xmath47-homologous cycle , then there are three disjoint @xmath47-homologous cycles , so we assume @xmath151 $ ] must be @xmath54-homologous . then , by lemma [ 5v ] , @xmath146 $ ] has a pair of linked cycles . lemma [ ll ] applies with @xmath148 connecting to the cycle with @xmath81 , following the proof in @xcite . we note that if @xmath24 connected to @xmath24 along an edge is minor - minimal with respect to triple - linking in @xmath1 , then we would also have that two disjoint copies of @xmath24 connected to @xmath24 along an edge is minor - minimal intrinsically triple - linked in @xmath2 . however , the minor - minimality of this graph is still unknown in @xmath1 . we also note that @xmath152 , as defined in @xcite , is a one - component minor - minimal intrinsically @xmath153-linked graph in @xmath2 , by the same arguement given in @xcite , since @xmath21 is intrinsically linked in both @xmath1 and @xmath2 . in @xmath2 , there are intrinsically linked graphs for which there exists an embedding in which every pair of disjoint cycles has linking number less than 1 . work has been done in @xmath1 to find graphs containing disjoint cycles with large linking number in every spatial embedding . using the fact that @xmath0 is triple - linked in @xmath1 , flapan @xcite showed that every spatial embedding of @xmath0 contains a two - component link @xmath155 such that , for some orientation , @xmath156 a similar argument using theorem 8 yields the following proposition . h. sachs , _ on spatial representation of finite graphs _ , ( proceedings of a conference held in lagow , february 10 - 13 , 1981 , poland ) , lecture notes in math . , 1018 , springer - verlag , berlin , heidelberg , new york , and tokyo ( 1983 ) . h. sachs , _ on spatial representations of finite graphs , finite and infinite sets _ , ( a. hajnal , l. lovasz , and v. t. ss , eds ) , colloq . jnos bolyai , vol . 37 , north - holland , budapest , ( 1984 ) , 649 - 662 .	flapan , _ et al _ @xcite showed that every spatial embedding of @xmath0 , the complete graph on ten vertices , contains a non - split three - component link ( @xmath0 is _ intrinsically triple - linked _ ) . the papers @xcite and @xcite extended the list of known intrinsically triple - linked graphs in @xmath1 to include several other families of graphs . in this paper , we will show that while some of these graphs can be embedded 3-linklessly in @xmath2 , @xmath0 is intrinsically triple - linked in @xmath2 .	9,126	163
bose - einstein condensates ( becs ) of trapped alkali atoms have internal degrees of freedom due to the hyperfine spin of the atoms . when a bec is trapped in a magnetic potential , these degrees of freedom are frozen and the state of the bec is described at a mean - field level by a scalar order parameter . when a bec is trapped in an optical potential , however , the spin degrees of freedom are liberated , giving rise to a rich variety of phenomena such as spin domains @xcite and textures @xcite . here the order parameter has @xmath0 components that transform under spatial rotation as the spherical tensor of rank @xmath1 , where @xmath2 is the hyperfine spin of bosons . mean field theories ( mfts ) of spinor becs were put forth for both spin-1 @xcite and spin-2 @xcite cases . according to them , the @xmath3 magnetic sublevel of an antiferromagnetic bec is not populated in the presence of magnetic field for both spin-1 and spin-2 cases . however , law et al . @xcite used many - body theory to show that the @xmath3 sublevel of a spin-1 bec _ is _ macroscopically populated due to the formation of spin - singlet `` pairs `` of bosons . it was subsequently shown @xcite that the @xmath3 sublevel of a spin-2 bec is also macroscopically occupied due to the formation of spin - singlet `` trios '' of bosons . the physics common to both cases is that the spin - singlet state is isotropic and therefore each magnetic sublevel shares the equal population . a question then arises as to what extent and under what conditions mfts are applicable . it is now understood @xcite that the validity of mfts is quickly restored with increasing an applied magnetic field . thus for the many - body spin correlations to manifest themselves , the external magnetic field have to be very low . the spin - singlet pairs of bosons should be distinguished from cooper pairs of electrons or those of @xmath4he due to the statistical difference of constituent particles . the cooper pairs consist of fermions , so that the state is symmetric only under the permutations that do not break any pair . for the case of spin - singlet pairs of bosons , the state is symmetric under any permutation of the constituents . the bose - einstein statistics leads to a constructive interference among permuted terms , giving rise to a highly nonlinear magnetic response to be discussed later . in contrast with usual antiferromagnets , where antiparallel spins are alternately aligned ( neel order ) , `` antiferromagnetic " becs do not possess such a long - range spatial order because the system lacks crystal order . the antiferromagnetic phase of becs is also called polar @xcite . in refs . @xcite , only spin degrees of freedom are considered by assuming that the spatial degrees of freedom are frozen . in this paper , we relax this restriction and develop a theory of spin-2 becs that enables us to study many - body ground states and the excitation spectrum thereof on an equal footing . for spin-1 becs , this program has been carried out in ref . many - body spin correlations and magnetic response of becs , the results of which were briefly reported in ref . @xcite , are expounded . the role of symmetry of the ground state in determining the character of the excitation spectrum is also elucidated . this paper is organized as follows . section [ sec : formulation ] derives an effective hamiltonian that enables us to study many - body spin correlations and low - lying excitation spectrum of spin-2 becs on an equal footing . section [ sec : mft ] reviews mean - field properties of each phase of becs . section [ sec : mbt ] studies many - body spin correlations and magnetic response of bec . the energy eigenstate is explicitly constructed using the creation operators of boson pairs and trios . the degeneracy of the eigenstate is examined and some novel magnetic response such as a huge jump in magnetization and the robustness of the minimum - magnetization state against an applied magnetic field are discussed . section [ sec : bogoliubov ] derives excitation spectra of becs using the bogoliubov approximation . all excitation spectra are obtained analytically and the relations of their characters to the symmetry of the ground state are discussed . section [ sec : conclusions ] summarizes the main results of the present paper . appendix [ app : characterization ] recapitulates the parametrization of the order parameter of spin-2 becs , and appendix [ app : zeeman ] describes a method of calculating zeeman - level populations . consider a system of identical bosons with hyperfine spin @xmath1 and let @xmath5 ( @xmath6 ) be the field operator that annihilates at position @xmath7 a boson with magnetic quantum number @xmath8 . the field operators are assumed to obey the canonical commutation relations @xmath9 = \delta_{mn}\delta(\bbox{r}-\bbox{r } ' ) , \ \ [ \hat{\psi}_m(\bbox{r}),\hat{\psi}_n(\bbox{r}')]=0 , \ \ [ \hat{\psi}_m^\dagger(\bbox{r}),\hat{\psi}_n^\dagger(\bbox{r } ' ) ] = 0 , \label{com}\end{aligned}\ ] ] where the kronecker s delta @xmath10 takes on the value of 1 if @xmath11 and 0 otherwise . the bose - einstein statistics requires that the total spin of any two bosons whose relative orbital angular momentum is zero be restricted to @xmath12 . we may therefore use @xmath13 as an index for classifying binary interactions between identical bosons : @xmath14 where @xmath15 describes an interaction between two bosons whose total spin is @xmath13 . to construct @xmath15 , consider the operator @xmath16 that annihilates at positions @xmath7 and @xmath17 two bosons with total spin @xmath13 and total magnetic quantum number @xmath18 : @xmath19 where @xmath20 is the clebsch - gordan coefficient . we may use @xmath21 to construct @xmath15 as @xmath22 where @xmath23 describes the dependence of the interaction on the positions of the particles . because of the completeness relation @xmath24 where @xmath25 is the total density operator and : : denotes normal ordering , that is , annihilation operators are placed to the right of creation operators . integrating eq . ( [ sum1 ] ) over @xmath26 yields @xmath27 where @xmath28 is the total number of bosons . in the case of a dilute bose - einstein condensate of neutral atoms , we may to a good approximation assume that @xmath29 , where @xmath30 characterizes the strength of the interaction between two bosons whose total spin is @xmath13 , and is related to the corresponding s - wave scattering length @xmath31 as @xmath32 equation ( [ int2 ] ) then becomes @xmath33 in the following discussions we shall focus on this case and therefore denote @xmath34 simply as @xmath35 . when @xmath36 , @xmath13 can take on values 0 , 2 , and 4 . for @xmath37 , we have @xmath38 where @xmath39 . \label{spin2a00}\end{aligned}\ ] ] for @xmath40 , we have @xmath41 for @xmath42 we have @xmath43 , \label{spin2f=4}\end{aligned}\ ] ] where eq . ( [ sum1 ] ) was used in obtaining the second equality . summing eqs . ( [ spin2f=0 ] ) , ( [ spin2f=2 ] ) and ( [ spin2f=4 ] ) , we obtain the interaction hamiltonian as @xmath44 . \label{int}\end{aligned}\ ] ] to eliminate the last term in eq . ( [ int ] ) , we note the following operator identity : @xmath45 where @xmath46 represents the spin density operators defined by @xmath47 with @xmath48 @xmath49 being the @xmath50-components of spin-2 matrices @xmath51 given by @xmath52 we may use eq . ( [ spin2identity ] ) to eliminate the last term in eq . ( [ int ] ) , obtaining @xmath53 , \label{spin2int2}\end{aligned}\ ] ] where @xmath54 , @xmath55 , and @xmath56 . in the following discussions we shall assume that the external magnetic field is weak enough to ignore the quadratic zeeman effect . then the total hamiltonian @xmath57 of the system consists of the kinetic energy term @xmath58 , the trapping potential energy term @xmath59 , the linear zeeman term @xmath60 , and the interaction term @xmath61 : @xmath62 where @xmath63 @xmath64 is the product of the gyromagnetic ratio and the external magnetic field which is assumed to be applied in the z - direction . when the system is spatially uniform , i.e. @xmath65 , it is convenient to expand the field operator in terms of plane waves : @xmath66 then the single - particle part of the hamiltonian becomes @xmath67 where @xmath68 , and the interaction hamiltonian becomes @xmath69 where @xmath70 mean field theory ( mft ) of spin-2 bec was discussed in refs . we present here a brief summary of as much of this theory as is relevant to later discussions . we shall also present some new results . when the system is uniform , bec occurs in the @xmath71 state . we therefore assume the following trial wave function @xmath72 where the complex amplitudes @xmath73 are assumed to satisfy the normalization condition @xmath74 the variational parameters @xmath73 are determined so as to minimize the expectation value of @xmath57 over the state ( [ sps ] ) : @xmath75 where @xmath76 when the external magnetic field is applied in the @xmath77-direction , only the @xmath77-component of @xmath78 is nonzero . we thus obtain @xmath79 where @xmath80 with @xmath81 . the mean - field solution should be determined so as to minimize @xmath82 subject to the normalization condition ( [ norm ] ) : @xmath83 where @xmath84 is a lagrange multiplier . multiplying both sides by @xmath85 and summing over @xmath8 yield @xmath86 on the other hand , multiplying both sides of eq . ( [ cond1 ] ) by @xmath87 and summing over @xmath8 yields @xmath88 when @xmath89 , eq . ( [ cond1 ] ) gives @xmath90\zeta_m=0 . \label{fcond}\end{aligned}\ ] ] for nonzero components @xmath91 , eq . ( [ fcond ] ) gives @xmath92 the case of @xmath93 will be discussed in sec . [ sec : cbec ] . when @xmath94 , only one component can be nonzero . the mean - field solutions , magnetizations , and mean - field energies are therefore given by @xmath95 where @xmath96 is an arbitrary global phase . the ground state is degenerate with respect to the global phase @xmath96 . this represents the gauge invariance , i.e. conservation of the total number of particle and leads to a massless goldstone mode , as will be shown in sec . [ sec : fmb ] . the conservation of the spin angular momentum does not lead to a new goldstone mode because in ferromagnets all spins are aligned in the same direction and therefore the total spin angular momentum has the same piece of information as the total number of particles . the antiferromagnetic ( or polar ) phase of a bec is defined as the one having nonzero spin - singlet pair amplitude , @xmath97 . when @xmath97 , eq . ( [ cond2 ] ) gives @xmath98 . substituting this and eq . ( [ cond3 ] ) into eq . ( [ e ] ) yields @xmath99 with @xmath98 , eq . ( [ cond1 ] ) leads to @xmath100\zeta_m=0 . \label{cond4}\end{aligned}\ ] ] when @xmath94 , the solutions of eq . ( [ cond4 ] ) is that only ( @xmath101 , @xmath102 ) or ( @xmath103 , @xmath104 ) or @xmath105 is nonzero . determining these values using conditions @xmath106 and eq . ( [ norm ] ) , we obtain the mean - field solutions and the corresponding magnetizations as @xcite @xmath107 the mean - field solutions ( [ afspinor ] ) and ( [ afspin2 ] ) are degenerate with respect to two continuous phase variables , that is , the global phase @xmath96 and the relative phase @xmath108 ( @xmath109 ) between the two nonvanishing amplitudes @xmath110 . corresponding to these two continuous degeneracies , we expect to have two goldstone modes , as will be shown in sec . [ sec : afb ] . when @xmath93 , eq . ( [ cond3 ] ) with @xmath98 gives @xmath111 , which , together with ( [ cond1 ] ) , leads to @xmath112 . hence we have @xmath113 . this is possible only when the external magnetic field is zero . the corresponding order parameter is given by @xmath114 this solution is degenerate with respect to five continuous variables : one global gauge ( @xmath115 ) , two relative gauges ( @xmath116 and @xmath117 ) , and two variables @xmath118 and @xmath119 that specify the amplitudes of the order parameter . as a consequence of these degeneracies , we expect to have five ( three density - like and two spin - like ) goldstone modes , as will be shown in sec . [ sec : afb ] . the remaining possibility is the case in which @xmath89 and @xmath120 . this phase will be referred to as cyclic phase . the energy of this phase is given from eq . ( [ e ] ) by @xmath121 let us now parameterize the order parameter of the cyclic phase as it will be needed to find the bogoliubov spectrum . there are four equations ( six real equations ) that restrict the order parameter of this phase , that is , @xmath122 we use the representation ( [ order ] ) of the order parameter derived in appendix [ app : characterization ] to analyze the cyclic phase . this representation automatically satisfies the normalization condition ( [ sum ] ) . to meet the condition ( [ s- ] ) , we note that @xmath123 the condition ( [ s- ] ) therefore requires either @xmath124 on the other hand , eq . ( [ f+ ] ) becomes @xmath125=0,\end{aligned}\ ] ] whence we obtain @xmath126 from conditions ( [ cycond1 ] ) and ( [ cycond2 ] ) , we find the following three solutions and the corresponding magnetization : @xmath127 where a global phase @xmath96 , which is chosen to be a particular value in eq.([re ] ) , is recovered . while these solutions include three parameters , the condition ( [ fz ] ) leaves only two parameters free . it should be noted that these three solutions give the same ground - state energy , and hence are equally possible unless magnetization exceeds @xmath128 . when it exceeds @xmath128 , only solution ( [ mfs1 ] ) is possible . a remark is here in order . as the representation ( [ order ] ) is obtained by assuming the full isotropy of space , it does not cover the whole order parameter space in the presence of magnetic field . we should therefore keep in mind that the solutions ( [ mfs1])-([mfs3 ] ) do not necessarily exhaust the whole order parameter space of the cyclic phase . in the following discussions we shall focus on the solution ( [ mfs1 ] ) . making the absolute square of each amplitude yields @xmath129 where @xmath130 it follows from this or by direct calculation that @xmath131 because of this restriction , the ground state of the cyclic phase is degenerate with respect to at least two continuous phase variables . we therefore expect to have at least two goldstone modes , as will be shown in sec.[sec : cb ] . in the absence of external magnetic field , the ground - state energies for the three phases are given from eqs . ( [ op_f ] ) , ( [ e2 ] ) , and ( [ cyclic ] ) by @xmath132 it follows that each phase is specified by @xcite . @xmath133 in the presence of external magnetic field we define each phase as follows : @xmath134 by directly comparing the energies in eqs . ( [ e2 ] ) , ( [ op_f ] ) , and ( [ cyclic ] ) , we find that each phase is specified by the following conditions : @xmath135 in this section we study the case in which the system is so tightly confined that the coordinate part of the ground - state wave function @xmath136 is independent of the spin state and solely determined by @xmath58 , @xmath59 , and the spin - independent part of @xmath61 ; that is , @xmath136 is the solution @xmath96 to the equation @xmath137\phi=\epsilon\phi\end{aligned}\ ] ] with the lowest eigenvalue @xmath138 . this assumption is justified if the second lowest eigenvalue @xmath139 satisfies @xmath140 where @xmath141 is an effective volume which coincides with @xmath142 in eq . ( [ pwe ] ) for the spatially uniform case ( i.e. , @xmath143 ) . when the condition ( [ epsi ] ) is met , the field operator @xmath144 may be approximated as @xmath145 , where @xmath146 is the annihilation operator of the bosons that are specified by the spin component @xmath8 and by the coordinate wave function @xmath115 . the spin - dependent part of the hamiltonian can then be written as @xmath147 where @xmath148 we first make some remarks on the properties of the operators @xmath149 . the operator @xmath150 , when applied to the vacuum , creates a pair of bosons in the spin - singlet state . this pair , however , should not be regarded as a single composite boson because @xmath150 does not satisfy the commutation relations of bosons . the operator @xmath150 instead satisfies the @xmath151 commutation relations together with @xmath152 , namely , @xmath153=\pm\hat{\cal s}_\pm , \ \ [ \hat{\cal s}_+,\hat{\cal s}_-]=-2\hat{\cal s}_z,\end{aligned}\ ] ] where the minus sign in the last equation is the only distinction from the usual spin commutation relations . as a consequence , the casimir operator @xmath154 that commutes with @xmath155 and @xmath156 is given by @xmath157 consider an eigenspace @xmath158 of @xmath154 with an eigenvalue @xmath159 . the requirement that @xmath160 must be positive semidefinite means that , in @xmath158 , the eigenvalues of the operator @xmath156 has a minimum value @xmath161 , where @xmath162 is a nonnegative integer . for a state @xmath163 that belongs to the minimum eigenvalue , the norm of @xmath164 must vanish ; hence @xmath165 with @xmath166 . we thus obtain the allowed combinations of eigenvalues @xmath167 for @xmath154 and @xmath156 such that @xmath168 and @xmath169 here we have introduced quantum numbers @xmath170 and @xmath162 , where the operator @xmath150 raises @xmath170 by one and the relation @xmath171 holds . we may thus interpret @xmath170 as the number of spin - singlet ` pairs ' , and @xmath162 as that of the remaining bosons . exact energy eigenvalues of hamiltonian ( [ h2 ] ) can be obtained as follows . the operators @xmath172 are invariant under any rotation of the system , namely , they commute with the total spin operator @xmath173 . the energy eigenstates can thus be classified according to quantum numbers @xmath162 and @xmath170 , total spin @xmath13 , and magnetic quantum number @xmath174 . we thus denote the eigenstates as @xmath175 , where @xmath176 labels orthonormal degenerate states , that is , @xmath177 the number of degenerate states @xmath178 for a given set of @xmath179 will be referred to as the size of the eigenspaces for @xmath179 . it will be shown to be independent of @xmath170 and @xmath174 below . the energy eigenvalue for the state @xmath175 is given by @xmath180 + \frac{c_2}{5v^{\rm eff}}n_{\rm s}(n+n_0 + 3 ) -pf_z , \label{e2}\ ] ] where the relation @xmath181 is used . the degeneracy @xmath178 can be calculated as follows . first we show that @xmath178 is independent of @xmath170 and @xmath174 . this is seen by the following relations , @xmath182 where @xmath183 , @xmath184 and @xmath185 these relations implies that the sizes of the eigenspaces for @xmath186 are not smaller than the size of the eigenspace for @xmath179 . the degeneracy thus depends only on @xmath162 and @xmath13 . next , we introduce a generating function of @xmath178 defined by @xmath187 let @xmath188 be the total number of states with a fixed number of bosons @xmath189 and a fixed magnetic quantum number @xmath174 . this is given by the total number of combinations of nonnegative integers @xmath190 that satisfy @xmath191 and @xmath192 . it follows that @xmath193 where we assume @xmath194 and @xmath195 to ensure the convergence of the series . let @xmath196 be the total number of states for given @xmath189 , @xmath13 , and @xmath174 . because @xmath196 is independent of the value of @xmath174 , we shall denote it simply as @xmath197 . the quantity @xmath188 is written in terms of the sum of @xmath197 as @xmath198 and hence @xmath199 . let us extend the definition of @xmath197 to the negative values of @xmath13 through this relation . it follows then from eq . ( [ gf_h ] ) that @xmath200 the right hand side of this equation can be written as the sum of two fractions @xmath201 , where @xmath202 and @xmath203 making maclaurin expansions of @xmath204 and @xmath205 around @xmath206 and regrouping them in terms of the form @xmath207 , we find that @xmath204 consists only of the terms with @xmath208 , and @xmath205 of those with @xmath209 . we thus obtain @xmath210 the quantity @xmath197 is written by the sum of the degeneracy @xmath178 as @xmath211 and hence we can write @xmath212 , where we assume that @xmath213 . it follows then from eq . ( [ gf_tildeh ] ) that @xmath214 and we finally obtain an explicit form of the generating function @xmath215 defined by eq . ( [ gf_def ] ) as @xmath216 the total spin @xmath13 can , in general , take integer values in the range @xmath217 . however , from eq . ( [ def_gf ] ) we find that there are some forbidden values . that is , @xmath218 are not allowed when @xmath219 , and @xmath220 are forbidden when @xmath221 . an easier way to find the forbidden values is discussed at the end of sec . [ sec : energy eigenstates ] . the energy eigenstates @xmath175 can be constructed using one- , two- and three - boson creation operators . let us define the operator @xmath222 such that it creates @xmath223 bosons in the state with total spin @xmath224 and magnetic quantum number @xmath225 when applied to the vacuum . such states are unique when @xmath226 . among possible operators @xmath227 , we choose the following five operators for constructing the eigenstates : @xmath228 = \sqrt{\frac{2}{5}}\hat{\cal s}_+ \\ \hat{a}^{(2)\dagger}_2&= & \frac{1}{\sqrt{14 } } [ 2\sqrt{2}\hat{a}_2^\dagger\hat{a}_0^\dagger -\sqrt{3}(\hat{a}_1^\dagger)^2 ] \\ \hat{a}^{(3)\dagger}_0&= & \frac{1}{\sqrt{210 } } [ \sqrt{2}(\hat{a}_0^\dagger)^3 - 3\sqrt{2}\hat{a}_1\dagger \hat{a}_0^\dagger\hat{a}_{-1}^\dagger + 3\sqrt{3}(\hat{a}_1^\dagger)^2\hat{a}_{-2}^\dagger + 3\sqrt{3}\hat{a}_2^\dagger(\hat{a}_{-1}^\dagger)^2 -6\sqrt{2}\hat{a}_2^\dagger\hat{a}_0^\dagger \hat{a}_{-2}^\dagger ] \label{def_a30 } \\ \hat{a}^{(3)\dagger}_3&= & \frac{1}{\sqrt{20 } } [ ( \hat{a}_1^\dagger)^3-\sqrt{6}\hat{a}_2^\dagger\hat{a}_1^\dagger \hat{a}_0^\dagger + 2(\hat{a}_2^\dagger)^2\hat{a}_{-1}^\dagger].\end{aligned}\ ] ] note that @xmath229 and @xmath230 do not exist because of the bose symmetry . note also that the operators @xmath227 commute with @xmath231 . consider a set @xmath232 of unnormalized states , @xmath233 with @xmath234 and @xmath235 . the state @xmath236 has the total number of bosons @xmath237 and the total spin @xmath238 . when @xmath189 and @xmath13 are given , @xmath239 is uniquely determined through the parity of @xmath13 , namely , @xmath240 if we introduce the following two parameters @xmath241 @xmath242 and @xmath243 are uniquely specified by them , and the remaining @xmath244 and @xmath245 are also determined as @xmath246 the parameter set @xmath247 is thus uniquely specified by the set @xmath248 . let us consider the two commuting observables @xmath249 with @xmath250 . let @xmath251 be the projection operator onto the simultaneous eigenspace of @xmath252 and @xmath253 corresponding to eigenvalues @xmath254 and @xmath255 , respectively . noting that @xmath256 includes the term @xmath257 and @xmath258 includes @xmath259 , it can be seen from eqs . ( [ def_a12])-([def_state_unit ] ) that @xmath260 and also @xmath261 if we order the pair @xmath262 in the lexicographic way , namely , @xmath263 if @xmath264 or @xmath265 and @xmath266 , the above equations imply that a state in @xmath232 specified by @xmath248 is linearly independent of the set of states specified by @xmath267 with @xmath268 . the set @xmath232 is thus a linearly independent set of states . let @xmath269 be the total number of states in @xmath232 with @xmath189 bosons , total spin @xmath13 , and magnetic quantum number @xmath270 . a generating function of @xmath269 is calculated as @xmath271 where @xmath204 is defined by eq . ( [ def_gf1 ] ) . compared to eq . ( [ gf_tildeh ] ) , we have @xmath272 . this implies that @xmath232 is complete , namely , the set @xmath232 forms a nonorthogonal basis of the subspace @xmath273 in which magnetic quantum number @xmath174 is equal to total spin @xmath13 . the energy eigenstates can be obtained by partially applying the method of schmidt s orthogonalization to the nonorthogonal basis @xmath232 . here by `` partially " we mean that eigenstates corresponding to different energies are orthogonal , but that those corresponding to the same energy are not always so . let us consider a series of subspaces @xmath274 , where @xmath275 is spanned by the states with quantum number @xmath170 [ see eq . ( [ def_n2 ] ) ] satisfying @xmath276 . let @xmath277 be the projection operator onto @xmath275 . from the relation @xmath278 , we have @xmath279 . since @xmath280 for any state @xmath281 , we have @xmath282 implying that @xmath275 is spanned by all the states @xmath283 satisfying @xmath284 . we can then construct a new basis @xmath285 made up of the states of the form @xmath286 is the projection onto the subspace with @xmath287 ( the kernel of @xmath288 ) . it is easy to see that the states belonging to @xmath285 are simultaneous eigenstates of @xmath289 , and hence energy eigenstates . the energy eigenstates with @xmath290 can be constructed by applying @xmath291 to the states of @xmath285 . to summarize , the energy eigenstates can be represented as @xmath292 with @xmath234 , @xmath235 , and @xmath293 . these parameters are related to @xmath179 as @xmath294 and the corresponding eigenenergy is given by eq . ( [ e2 ] ) . note that the states defined in ( [ eigenstates ] ) are unnormalized , and the states having the same energy ( i.e. , those belonging to the same set of parameter values @xmath179 ) are nonorthogonal . the representation ( [ eigenstates ] ) of the energy eigenstates utilizes the operator @xmath227 that creates correlated @xmath223 bosons having total spin @xmath1 . it might be tempting to envisage a physical picture that the system is , like in @xmath295he , made up of @xmath296 composite bosons whose creation operator is given by @xmath227 . however , this picture is oversimplified . first of all , the operator @xmath297 does not obey the boson commutation relation . in addition , the projection operator @xmath298 in ( [ eigenstates ] ) imposes many - body spin correlations such that the spin correlation between _ any _ two bosons must have vanishing spin - singlet component . note that two bosons with independently fluctuating spins have a nonzero overlap with the spin - singlet state in general . the many - body spin correlations of the energy eigenstates are thus far more complicated than what an intuitive picture of composite bosons suggests . on the other hand , as long as quantities such as the number of bosons , magnetization , and energy are concerned , the above simplified picture is quite helpful . by way of illustration , we provide an alternative explanation for the existence of forbidden values for the total spin @xmath13 , which were found earlier using the generating function ( [ def_gf ] ) . for example , to construct a state with @xmath37 or @xmath299 , composite particles with total spin 2 must be avoided , namely , @xmath300 . then we have @xmath301 , implying that @xmath37 or @xmath299 is only possible when @xmath219 . for a state with @xmath40 or @xmath302 , we have @xmath303 and @xmath304 implying that @xmath305 . the above simplified picture is also helpful when we consider the magnetic response as discussed below . we consider here how the ground state and the magnetization @xmath174 respond to the applied magnetic field @xmath63 . from eq . ( [ e2 ] ) , we see that the minimum energy states always satisfy @xmath270 when @xmath306 . the problem thus reduces to minimizing the function @xmath307 + \frac{c_2}{5v^{\rm eff}}n_{\rm s}(2n-2n_{\rm s}+3 ) -pf_z . \label{e3}\ ] ] for this purpose , it is convenient to consider the cases @xmath308 and @xmath309 separately . let us first consider the case @xmath308 . when @xmath310 , the energy ( [ e3 ] ) is minimized when @xmath287 , @xmath311 , and @xmath312 , and the ground state is given by @xmath313 , that is , the system is ferromagnetic . this result is consistent with that obtained from mft . when @xmath314 , let us rewrite the energy as @xmath315 the energy is thus lower when @xmath174 is closer to @xmath316 and when @xmath170 is smaller . in most of the parameter space , the ground state is @xmath317 with @xmath13 taking the allowed integer closest to @xmath316 . since these states satisfy @xmath318 , they belong to the ferromagnetic phase or the cyclic phase . the separatrix between the two phases is given by @xmath319 . we thus find that the ground state is ferromagnetic if @xmath320 and cyclic otherwise . this classification is consistent with the mean - field analysis given in sec . [ sec : boundaries ] . as seen in sec . [ sec : spectrum and degeneracy ] , the above ground state is highly degenerated in general ; this may originate from the continuous symmetry that leaves free at least two parameters characterizing the order parameter of the cyclic phase as shown in sec . [ sec : cbec ] . according to the discussions in sec . [ sec : spectrum and degeneracy ] , the degeneracy of the states @xmath317 is equal to the number of the combinations of @xmath321 satisfying @xmath322 and @xmath323 . the number of trios , @xmath243 , can take values in the range @xmath324 . when the magnetic field is nearly zero and @xmath325 , there is little degeneracy and @xmath326 , namely , almost all bosons form trios . when @xmath316 is close to the forbidden values of @xmath13 ( @xmath327 if @xmath328 ( mod 3 ) , and @xmath329 otherwise ) for the above state with @xmath287 , @xmath174 may take those values at the cost of increasing @xmath170 to 1 or 2 , since any of the three values ( 0,1,2 ) of @xmath162 mod 3 is realized by setting @xmath170 as 0 , 1 , or 2 [ recall the relation @xmath330 . whether or not the states @xmath331 and @xmath332 can be the lowest - energy state depends on the ratio @xmath333 . in fig [ fig:1 ] , we give diagrams of the ground states for small @xmath334 . in this case , it is convenient to introduce a new parameter @xmath335 and consider the energy as a function of @xmath174 and @xmath336 : @xmath337 ^ 2-\frac{c_2}{40v^{\rm eff}}l(l+2f+6)+{\rm const}. \label{e_fz_l}\ ] ] since @xmath309 , we see that @xmath338 is an increasing function of @xmath339 , namely , @xmath340 let us consider the cases @xmath341 and @xmath342 separately . _ ( a ) _ @xmath341 in this parameter region , mft predicts that the system is ferromagnetic , namely , the lowest - energy state always shows @xmath343 regardless of the magnitude of magnetic field @xmath63 . in the exact solution considered here , the magnetic response is different because of the offset term @xmath344 in eq . ( [ e_fz_l ] ) . since @xmath309 , this term counteracts the applied magnetic field . it is thus expected that magnetization is suppressed when magnetic field is weak . the exact ground state is derived as follows . when @xmath189 is even , the state @xmath345 has energy @xmath346 , and the state @xmath347 has energy @xmath348 . any other set @xmath349 gives an energy higher than one of these states . @xmath345 is thus the ground state when @xmath350 , or equivalently , @xmath351 and otherwise the ground state is @xmath347 [ see fig [ fig:2 ] ] . when @xmath189 is odd , @xmath352 is attained only when @xmath353 , @xmath354 is forbidden , and the state @xmath355 has energy @xmath356 . it is easy to confirm that @xmath357 always holds . therefore , @xmath355 is the ground state when @xmath358 , or equivalently , @xmath359 and otherwise the ground state is @xmath347 [ see fig [ fig:2 ] ] . these results indicate that in the parameter region of @xmath360 , magnetization of the ground state jumps from 0 or 2 to @xmath361 . such a huge discontinuity does not appear in mft with a linear zeeman effect . ( however , in the presence of a quadratic zeeman effect , such a jump occurs also in mft @xcite . ) @xmath342 given @xmath362 , the minimum allowed value of @xmath339 is determined as follows . note that @xmath363 is minimized when the number of singlets @xmath364 is maximized . for @xmath365 ( @xmath366 is an integer ) , the state @xmath367 gives @xmath368 . to increase @xmath174 by one ( @xmath369 ) , one singlet pair must be broken and the minimum of @xmath339 is @xmath370 given by the state @xmath371 . keeping the singlet part , @xmath174 can be further increased to @xmath372 by the state @xmath373 with @xmath374 . since @xmath375 is forbidden , @xmath376 requires one more singlet pair to break up , resulting in @xmath377 with the state @xmath378 . when @xmath379 falls between @xmath380 and @xmath381 , the lowest energy is the minimum of @xmath382 , @xmath383 , @xmath384 , @xmath385 , and @xmath386 . from eq . ( [ e_fz_l ] ) , we expect that when @xmath387 is large , nonzero @xmath339 pushes up the energy significantly and can not be the ground state , so that @xmath174 increases stepwise with @xmath388 . when @xmath387 is small , @xmath174 will increase with the step size of @xmath389 . this is confirmed by explicitly calculating @xmath338 using eq . ( [ e_fz_l ] ) , and we obtain the diagrams in fig . [ fig:3 ] . in the region @xmath390 with @xmath391 ( see broken curves in fig . [ fig:3 ] ) , @xmath174 increases by taking every integer . when @xmath392 , the values of @xmath376 are suppressed . in the region @xmath393 with @xmath394 , the values @xmath369 are suppressed , and when @xmath395 with @xmath396 , the values @xmath372 are further suppressed and the step size becomes 4 . while the averaged slope @xmath397 coincides with that in mft , the offset term @xmath344 in eq . ( [ e_fz_l ] ) ( see also the broken lines in fig . [ fig:3 ] ) makes a qualitative distinction from mft , namely , the onset of the magnetization displaces from @xmath398 to @xmath399 . note that the slope @xmath400 and the offset @xmath401 are determined by independent parameters . a typical behavior of the magnetic response when @xmath402 is shown in fig . [ fig:4 ] . we now calculate the zeeman - level populations of the ground states for @xmath309 . in mft , the lowest - energy states for @xmath309 have vanishing population in the @xmath403 levels . in contrast , the exact ground states derived in the preceding subsection , @xmath404 with @xmath405 and @xmath235 , have nonzero populations in the @xmath403 levels . the exact forms for the averaged population @xmath406 are calculated as follows . the above ground states have the form of @xmath407 with @xmath163 being a state with a fixed number ( @xmath408 ) of bosons satisfying @xmath409 . the average zeeman population for the ground states , @xmath410 is then simply related to the average zeeman populations for the state @xmath163 as @xmath411 where @xmath412 . the derivation of the formula ( [ formula_zeeman1 ] ) is given in appendix [ app : zeeman ] . the formula implies that when @xmath413 , the zeeman populations of the ground states is sensitive to the form of @xmath163 . with this formula , it is a straightforward task to calculate average zeeman - level populations for four types of ground states , @xmath404 with @xmath405 and @xmath235 . the exact result will be given in appendix [ app : zeeman ] . a striking feature appears in the leading terms under the condition @xmath414 . the results are summarized as @xmath415 and @xmath416 as seen in the preceding subsection , with the increase of magnetic field , the ground state alternates among the four types of states . while this change causes a very small difference in magnetization , it leads to large changes in the average zeeman - level populations , by a factor of 2 or 3 . the origin of this drastic change may be explained as follows . let us first consider the state @xmath417 with @xmath418 . this state has no population in zeeman levels @xmath419 . when @xmath420 is applied to this state , the operator @xmath421 that appears in @xmath420 has effectively a large amplitude of the order of @xmath422 . hence the term @xmath423 is dominant , and it approximately adds one boson to the @xmath424 level and one boson to the @xmath3 level . hence the @xmath3 population of the state @xmath425 is close to unity . this change is then amplified by a factor of @xmath426 according to the formula ( [ formula_zeeman1 ] ) , leading to eq . ( [ population_0 ] ) . similarly , applying @xmath427 effectively results in adding of two bosons to the @xmath424 level and one boson to the @xmath428 level through the dominant term @xmath429 . in this section we study the low - lying excitation spectrum of spin-2 becs in the thermodynamic limit using the bogoliubov approximation . we shall see that the symmetry of each ground state discussed in sec . [ sec : mft ] is reflected in the excitation spectrum . in the center - of - mass frame of the system bec occurs in the @xmath430 state . we therefore decompose the operators that appear in eq . ( [ spin2int2 ] ) into the @xmath71 components and the @xmath431 ones . the first term on the right - hand side ( rhs ) of eq . ( [ spin-1v1 ] ) is rewritten as @xmath432 if we ignore the terms that do not include the @xmath71 components , we may approximate @xmath433 as @xmath434 where @xmath435 we thus obtain @xmath436 similarly , we may approximate the second term on the rhs of eq . ( [ spin-1v1 ] ) as @xmath437 where @xmath438 the third term on the rhs of eq . ( [ spin-1v1 ] ) is decomposed into @xmath439 where @xmath440 and h.c . denotes hermitian conjugates of the preceding terms . substituting eqs . ( [ rhorho ] ) , ( [ ff ] ) and ( [ ss ] ) into eq . ( [ spin-1v1 ] ) , we obtain @xmath441 in the bogoliubov approximation we replace operators @xmath442 by c - numbers @xmath443 , where @xmath73 s denote the complex mean - field amplitudes introduced in sec . [ sec : mft ] and @xmath444 is the number of condensate bosons . since @xmath444 is smaller than the total number of bosons @xmath189 due to the interparticle interactions , we take into account the conservation of the total number of bosons through the relation @xmath445 then eq . ( [ h0 ] ) becomes @xmath446 \hat{n}_{\bbox{k},m } , \label{h00}\end{aligned}\ ] ] and eq . ( [ spin2v ] ) becomes @xmath447 where @xmath448 , and the definitions of @xmath449 , @xmath450 , and @xmath451 are given in eqs . ( [ f])([s- ] ) . equations ( [ h00 ] ) and ( [ spin2vv ] ) constitute our basic hamiltonian in the following discussions . we use this hamiltonian to examine low - lying excitation spectra for each phase . let us first examine the excitation spectrum of a ferromagnetic phase in which bec occurs in the @xmath424 state . then @xmath452 and eqs . ( [ h00 ] ) and ( [ spin2vv ] ) become @xmath453\hat{n}_{{\bbox{k}},m } , \label{h0f}\end{aligned}\ ] ] and @xmath454,\end{aligned}\ ] ] respectively , where @xmath455 . the total hamiltonian is therefore given by @xmath456 \nonumber \\ & & + \sum_{{\bbox{k}}\neq{\bf 0 } } \left[(\epsilon_{\bbox{k}}+p)\hat{n}_{{\bbox{k}},1 } + \left(\epsilon_{\bbox{k}}+2p-4c_1n\right)\hat{n}_{{\bbox{k}},0 } \right . \nonumber \\ & & \left . + \left(\epsilon_{\bbox{k}}+3p-6c_1n\right)\hat{n}_{{\bbox{k}},-1 } + \left(\epsilon_{\bbox{k}}+4p-8c_1n+2c_2n/5 \right)\hat{n}_{{\bbox{k}},-2 } \right ] . \label{hfm}\end{aligned}\ ] ] the second line gives the bogoliubov spectrum @xmath457 while other terms give single - particle spectra : @xmath458 for the bogoliubov excitation energy to be positive , we must have @xmath459 that is , the s - wave scattering length for the total spin-4 channel must be positive . this condition is the same as that required for the ferromagnetic mean field to be stable , that is , the first term on the rhs of eq . ( [ hfm ] ) being positive . for the single - particle excitation energies to be positive , we must have @xmath460 and @xmath461 . these conditions are the same as those in ( [ condf ] ) for which the ferromagnetic phase is the lowest - energy mean field ( note that @xmath462 ) . we note that the bogoliubov spectrum ( [ fbog ] ) is independent of applied magnetic field and remains massless in its presence . this goldstone mode is a consequence of the global u(1 ) gauge invariance due to the conservation of the total number of bosons , as discussed in sec . [ sec : mffm ] . let us next examine the excitation spectrum of an antiferromagnetic phase in which the order parameter is given by ( [ afspinor ] ) . making the replacements @xmath463 and substituting eq . ( [ fsd ] ) into eq . ( [ spin2vv ] ) , together with eq . ( [ h00 ] ) , we obtain the total hamiltonian of an antiferromagnetic bec : @xmath464\hat{n}_{{\bbox{k}},2 } \right . \nonumber \\ & & \ \ \ + \left[\epsilon_{\bbox{k}}+p(\langle\hat{f}_z\rangle+2)+c_0n\|\zeta_{-2}\|^2 + c_1n(4\|\zeta_{-2}\|^2 - 2\langle\hat{f}_z\rangle-\langle\hat{f}_z\rangle^2 ) + \frac{2c_2n}{5}(\|\zeta_{2}\|^2 - 2\|\zeta_{2 } \zeta_{-2}\|^2 ) \right]\hat{n}_{{\bbox{k}},-2 } \nonumber \\ & & \left . \ \ \ + \left [ \frac{1}{2}g_4n(\zeta_{2}^{2}\hat{a}_{{\bbox{k}},2}\hat{a}_{-{\bbox{k}},2 } + \zeta_{-2}^{2}\hat{a}_{{\bbox{k}},-2}\hat{a}_{-{\bbox{k}},-2 } ) + ( c_0 - 4c_1 + 2c_2/5)n ( \zeta_{2}^\zeta_{-2}^\hat{a}_{{\bbox{k}},2}\hat{a}_{-{\bbox{k}},-2}+ \zeta_{2 } \zeta_{-2}^\hat{a}_{{\bbox{k}},2}^\dagger\hat{a}_{{\bbox{k}},-2 } ) + { \rm h.c.}\right ] \right\ } \nonumber \\ & & + \sum_{{\bbox{k}}\neq{\bf 0}}\left\ { \left[\epsilon_{\bbox{k}}+p(\langle\hat{f}_z\rangle-1 ) + c_1n(2\|\zeta_{2}\|^2+\langle\hat{f}_z\rangle-\langle\hat{f}_z\rangle^2 ) -\frac{4c_2n}{5}\|\zeta_{2 } \zeta_{-2}\|^2 \right]\hat{n}_{{\bbox{k}},1 } \right . \nonumber \\ & & \ \ \ + \left[\epsilon_{\bbox{k}}+p(\langle\hat{f}_z\rangle+1 ) + c_1n(2\|\zeta_{-2}\|^2-\langle\hat{f}_z\rangle-\langle\hat{f}_z\rangle^2 ) -\frac{4c_2n}{5}\|\zeta_{2 } \zeta_{-2}\|^2 \right]\hat{n}_{{\bbox{k}},-1 } \nonumber \\ & & \left . \ + 2(c_1-c_2/5)n(\zeta_{2}^\zeta_{-2}^\hat{a}_{{\bbox{k}},1}\hat{a}_{-{\bbox { k}},-1 } + { \rm h.c . } ) \right\ } \nonumber \\ & & + \sum_{{\bbox{k}}\neq{\bf 0}}\left\ { \left[\epsilon_{\bf k}+p\langle\hat{f}_z\rangle - c_1n\langle\hat{f}_z\rangle^2- \frac{4c_2n}{5}\|\zeta_{2 } \zeta_{-2}\|^2\right ] \hat{n}_{{\bbox{k}},0 } + \frac{c_2n}{5}(\zeta_{2}^\zeta_{-2}^\hat{a}_{{\bbox{k}},0 } \hat{a}_{-{\bbox{k}},0}+{\rm h.c . } ) \right\}. \label{hafm1}\end{aligned}\ ] ] this hamiltonian may be simplified using the relation @xmath465 , giving @xmath466 \right\ } \nonumber \\ & & + \sum_{{\bbox{k}}\neq{\bf 0}}\left\ { \left[\epsilon_{\bbox{k } } + ( c_1-c_2/5)n+\frac{1}{2}(c_1+c_2/10)n\langle\hat{f}_z\rangle \right]\hat{n}_{{\bbox{k}},1 } + \left[\epsilon_{\bbox{k } } + ( c_1-c_2/5)n-\frac{1}{2}(c_1+c_2/10)n\langle\hat{f}_z\rangle \right]\hat{n}_{{\bbox{k}},-1 } \right . \nonumber \\ & & \left . \ + 2(c_1-c_2/5)n(\zeta_{2}^\zeta_{-2}^\hat{a}_{{\bbox{k}},1}\hat{a}_{-{\bbox { k}},-1 } + { \rm h.c . } ) \right\ } \nonumber \\ & & + \sum_{{\bbox{k}}\neq{\bf 0}}\left\ { ( \epsilon_{\bbox{k}}-c_2n/5 ) \hat{n}_{{\bbox{k}},0 } + \frac{c_2n}{5}(\zeta_{2}^\zeta_{-2}^*\hat{a}_{{\bbox{k}},0 } \hat{a}_{-{\bbox{k}},0}+{\rm h.c . } ) \right\}. \label{hafm}\end{aligned}\ ] ] this result shows that the eigenmodes are classified into three categories : the @xmath3 mode , the coupled @xmath467 modes , and the coupled @xmath468 modes . below we analyze each of them . the hamiltonian ( [ hafm ] ) shows that the @xmath3 mode is decoupled from other modes even in the presence of magnetic field . this part of the hamiltonian can readily be diagonalized to give @xmath469 we note that the spectrum ( [ af0 ] ) becomes massive in the presence of magnetic field . the @xmath470 and @xmath428 modes are coupled in the hamiltonian ( [ hafm ] ) , and the eigenenergies can be obtained by diagonalizing the following hamiltonian : @xmath471 where @xmath472 and @xmath473 the dispersion relation can be found by writing down the equations of motion for @xmath474 and @xmath475 and seeking for the solution of the form @xmath476 , with the result @xmath477 the excitation spectra ( [ spin-1 ] ) become massive in the presence of magnetic field . the positivity of this energy is guaranteed by the conditions @xmath478 and @xmath309 which are required for the antiferromagnetic phase to be the lowest - lying mean field ( see condition ( [ condaf ] ) ) . the @xmath424 and @xmath479 modes are coupled in the hamiltonian ( [ hafm ] ) , and the relevant part of the hamiltonian reads @xmath480 where @xmath481 by unitary transformations @xmath482 and @xmath483 , eq . ( [ h2af ] ) reduces to @xmath484 where @xmath485 , @xmath486 , and @xmath487 . this hamiltonian can be diagonalized by writing down the heisenberg equations of motion for @xmath488 and @xmath489 as @xmath490 by assuming that @xmath491 , we obtain the following dispersion relations : @xmath492 ^ 2\left(1-\frac{\langle\hat{f}_z \rangle^2}{4}\right ) } \right ] . \label{dis2}\end{aligned}\ ] ] the positivity of this energy is met if the conditions @xmath493 and @xmath494 are satisfied . the former condition is met whenever the antiferromagnetic phase is the lowest - energy state ( see ( [ condaf ] ) ) , while the latter condition is required for the antiferromagnetic phase to be mechanically stable , that is , the first term on the rhs of eq . ( [ hafm ] ) is positive . we note that the dispersion relations ( [ dis2 ] ) are massless even in the presence of the magnetic field . they are the goldstone modes associated with the u(1 ) gauge symmetry and the relative gauge symmetry ( the rotational symmetry about the direction of the applied magnetic field ) that are manifest in the mean - field solution discussed in sec . [ sec : mfaf ] . at zero magnetic field , ( [ af0 ] ) , ( [ spin-1 ] ) and ( [ dis2 ] ) reduces to @xmath495 , \\ ( e^{\rm af}_{{\bbox{k}},\pm 1})^2&= & \epsilon_{\bbox{k}}\left[\epsilon_{\bbox{k}}+2(c_1-c_2/5)n\right ] , \\ ( e^{\rm af}_{{\bbox{k}},\pm2})^2&= & \cases { \epsilon_{\bbox{k}}\left[\epsilon_{\bbox{k}}+2(c_0+c_2/5)n\right ] , \cr \epsilon_{\bbox{k}}\left[\epsilon_{\bbox{k}}+8(c_1-c_2/20)n\right ] , \cr}\end{aligned}\ ] ] implying that all the five excitations are goldstone modes . this reflects the fact that in the absence of the magnetic field the ground state is degenerate with respect to five continuous variables ( see eq . ( [ afspinor2 ] ) ) . we consider the case of eq . ( [ cyclic_mfs ] ) , namely , @xmath496 where @xmath497 and @xmath498 . because @xmath89 and @xmath499 , the interaction hamiltonian ( [ spin2vv ] ) reduces to @xmath500 substituting eq . ( [ cyclic_mfs2 ] ) into this , performing unitary transformations @xmath501 , where @xmath502 and @xmath503 , and combining the result with eq . ( [ h00 ] ) , we obtain @xmath504 where @xmath505 , @xmath506 , @xmath507 , and @xmath508 it can be seen from the hamiltonian ( [ hcyclic ] ) that there are two separate sets of coupled modes , that is , the @xmath467 modes and the @xmath509 modes . the equations of motion governing the @xmath467 coupled modes are given by @xmath510 where @xmath511 , @xmath512 , @xmath513 . the eigenenergies of these modes are given by @xmath514 \right\}^\frac{1}{2}. \label{cyclic1}\end{aligned}\ ] ] these excitation energies are always positive semidefinite and massive in the presence of magnetic field . we note that eq . ( [ cyclic1 ] ) has one gapless ( but not massless ) mode in the presence of external magnetic field . taking the limit @xmath515 of eq . ( [ cyclic1 ] ) , we obtain @xmath516^\frac{1}{2 } \epsilon_{\bbox{k}}. \label{cyclic2}\end{aligned}\ ] ] however , both of these modes become massless in the absence of magnetic field . in fact , eq . ( [ cyclic1 ] ) then reduces to @xmath517 the equations of motion governing the @xmath518 coupled modes are given by @xmath519 where @xmath520 , @xmath521 , and @xmath522 . substituting the expressions for @xmath523 and @xmath524 into those for @xmath525 and @xmath526 , we obtain the equations of motion for the latter set , which reduces to the cubic equation and therefore can be solved analytically . the result is given by @xmath527\right\}^\frac{1}{2 } , \label{cyclicpm2}\end{aligned}\ ] ] where we recall that @xmath505 , @xmath506 , @xmath507 . the second solutions in eq . ( [ cyclicpm2 ] ) are always positive semidefinite . the positivity of the first solution is guaranteed by the condition @xmath308 . we note that the first solution is massive and independent of the applied magnetic field and that the second solutions remain massless in the presence of external magnetic field . the latter is a consequence of the fact that the mean - field solution is degenerate with respect to at least two continuous variables , as discussed in sec . [ sec : cbec ] . in the absence of external magnetic field , the results ( [ cyclicpm2 ] ) reduce to @xmath528 it can be shown from the general analytic solutions that these results are valid up to the first order in magnetization @xmath449 . while we have been unable to complete the analysis of the cyclic phase except for the case of eq . ( [ cyclic_mfs2 ] ) , we would like to point out that the excitation spectrum of the cyclic phase always includes the first solution in eq . ( [ cyclicpm2 ] ) , even when the mean - field solution is not given by eq . ( [ cyclic_mfs2 ] ) . this can be seen directly by writing down the equation of motion for @xmath529 and @xmath530 and the corresponding eigenvalue equation . it can then be seen that @xmath531 is a solution to this equation . in this paper we have studied quantum spin correlations and magnetic response of spin-2 bose - einstein condensates ( becs ) in a mesoscopic regime , and low - lying excitation spectra of each phase of spin-2 becs in the thermodynamic regime . the ground states of spin-2 becs have three distinct phases : ferromagnetic ( fm ) , antiferromagnetic ( af ) and cyclic ( c ) phases . the former two phases appear also in spin-1 becs , while the last phase is unique to spin-2 becs . the building block of the af phase is spin - singlet pairs and that of the c phase is spin - singlet trios . these many - body features usually elude mean - field treatments that are based on the order parameter derived from the single - particle density matrix . bose symmetry restricts possible building blocks of spin-2 becs . this can be summarized in terms of the annihilation operator @xmath297 of @xmath223-bosons having total spin @xmath1 . the fundamental building block is not unique , but one minimal set is @xmath532 , @xmath256 , @xmath533 , @xmath258 , and @xmath534 . bose statistics does not allow units such as @xmath535 and @xmath536 . the unit @xmath534 is required to represent a state with odd values of the total spin . we have investigated quantum spin correlations and magnetic response in the mesoscopic regime . under the assumption that the system is so tightly confined that the spatial degrees of freedom are frozen , we derived the exact many - body ground states which are expressed in terms of the minimal set of creation operators @xmath537 , @xmath538 , @xmath420 , @xmath539 , and @xmath427 . these pairwise and trio - wise units help us understand the complicated response of the magnetization to the applied magnetic field , which stems from the fact that several values of the magnetization can not be constructed from such units and are hence forbidden . in addition to the quantization of the magnetization to discrete values , several new features which elude mean - field treatments are found , such as a sudden jump from the minimum to the maximum magnetization , and robustness of the minimum - magnetization state against a small increase in the applied magnetic field until it starts to show a linear response . the average zeeman level populations for the af - phase ground states were calculated , showing that @xmath403 populations , which stay zero in mft , vary sensitively to the applied magnetic field . we have also examined low - lying excitation spectra using the bogoliubov approximation . the excitation spectra of fm and af phases are similar to those of the spin-1 case @xcite . in the fm phase , the spectrum consists of one massless mode ( [ fbog ] ) reflecting the global gauge invariance and four single - particle modes ( [ fs1])-([fsm2 ] ) whose energy gaps are generated by the zeeman shifts as well as mean - field interactions . in the af phase , the spectrum includes two massless modes ( [ dis2 ] ) due to the global gauge invariance and the rotational symmetry about the spin quantization axis . the remaining three are also bogoliubov modes , but they all become massive in the presence of magnetic field due to the zeeman shifts . in the c phase , the spectrum have at least two massless modes ( the second term in eq . ( [ cyclicpm2 ] ) ) by the same reasons as in the af phase . the spectrum includes one peculiar single - particle mode ( the first term in eq . ( [ cyclicpm2 ] ) ) whose energy gap depends solely on the spin - dependent interactions and is insensitive to the applied magnetic field . in addition , the spectrum has one gapless mode ( the second term in eq . ( [ cyclic2 ] ) ) whose mass depends only on magnetization @xmath449 and vanishes at zero magnetic field . the remaining mode ( the first term in eq . ( [ cyclic2 ] ) ) is a bogoliubov mode which becomes massive in the presence of the magnetic field . in the present paper we have studied only static properties of spin-2 bec . with the very rich phenomena that we have found here , we may very well expect that much more remains to be revealed in their dynamics . this work was supported by a grant - in - aid for scientific research ( grant no . 11216204 ) by the ministry of education , science , sports , and culture of japan , and by the toray science foundation . m.u . acknowledges the hospitality of the aspen center for physics , where part of this work was carried out . the order parameter @xmath540 of a spin-2 bec has the same structure as that of the d - wave superconductor which was examined by mermin @xcite . we here recapitulate as much of it as is relevant to the our theory . the spin part of the order parameter , @xmath541 , of a spin-2 bec is described as a function of the azimuthal angle @xmath118 and the radial angle @xmath96 , and may be expanded in terms of the spherical harmonics of rank 2 @xmath542 as @xmath543 where @xmath73 obeys the normalization condition ( [ norm ] ) . the angle dependence of @xmath544 may be expressed in terms of components of a three - dimensional unit vector : @xmath545 as follows @xmath546 substituting these into eq . ( [ chi1 ] ) , we obtain @xmath547 where @xmath548 the order parameter is thus characterized by a @xmath549 traceless matrix tr@xmath550 with unit normalization @xmath551 we may exploit the freedom of the gauge invariance to choose the overall phase so that the real part of tr@xmath552 vanishes . @xmath553 let the real and imaginary parts of @xmath554 be @xmath555 and @xmath556 , respectively . it follows from eqs . ( [ mm ] ) and ( [ re ] ) that @xmath557 because @xmath558 is traceless , so can be @xmath555 and @xmath559 traceless . @xmath555 and @xmath559 do not commute , so they can not be diagonalized simultaneously . we follow mermin to take a representation in which @xmath555 is diagonal . then the diagonal elements of @xmath555 become @xmath560 the matrix elements of @xmath556 are given by @xmath561 substituting eqs . ( [ xx ] ) and ( [ yy ] ) into @xmath562 and comparing it with eq . ( [ m ] ) , we obtain the following characterization of the order parameter . @xmath563 let us first write @xmath406 as @xmath564 with @xmath565 . what we need is thus a general formula for calculating @xmath566 . let us first consider the decomposition of @xmath567 into a sum of eigenstates for @xmath568 , such that @xmath569 where @xmath570 is an unnormalized simultaneous eigenstate of @xmath571 with eigenvalues @xmath572 and @xmath573/4 $ ] , respectively . it follows from eqs . ( [ eigen_s ] ) and ( [ def_n2 ] ) that @xmath574 and @xmath575 . here @xmath223 is the number of bosons in @xmath567 , and @xmath576 denotes the largest integer that is not larger than @xmath577 . let us define @xmath578 and @xmath579 such that @xmath580 and @xmath581 where we have used eq . ( [ formula_s_plus ] ) and defined the coefficient @xmath582 as @xmath583 by definition , @xmath584 and @xmath585 . substituting eq . ( [ def_omega ] ) into eq . ( [ decomp_psi_s ] ) yields @xmath586 or equivalently , @xmath587 using this relation recursively , we can calculate @xmath579 as a function of @xmath588 . on the other hand , multiplying @xmath589 on both sides of eq . ( [ decomp_psi_s ] ) and taking norms , we obtain @xmath590 using this formula , we can evaluate eq . ( [ am_by_norms ] ) . if we apply @xmath591 , @xmath592 , and @xmath593 to the relations ( [ def_omega ] ) , ( [ mu_by_omega ] ) , and ( [ norm_by_mu ] ) , and noting that @xmath409 , we have @xmath594 , @xmath595 , and @xmath596 for @xmath597 , we have @xmath598\|\phi\rangle = ( -1)^m\hat{a}_{-m}\|\phi\rangle $ ] and @xmath599 . using these when we apply @xmath600 , @xmath601 , and @xmath593 to the relations ( [ def_omega ] ) , ( [ mu_by_omega ] ) , and ( [ norm_by_mu ] ) , we have @xmath602 , @xmath603 , @xmath604 , and @xmath605 combining eqs . ( [ am_by_norms ] ) , ( [ norm_phi ] ) , and ( [ norm_phi_prime ] ) , we obtain @xmath606 where @xmath607 . substituting eq.([coeff_s_plus ] ) , we obtain eq . ( [ formula_zeeman1 ] ) . \ii ) @xmath618 and @xmath609 . we take @xmath619^\dagger\|{\rm vac}\rangle$ ] and @xmath620 . then , @xmath621 @xmath622 @xmath623 and @xmath613 for @xmath624 . putting these into the formula ( [ formula_zeeman1 ] ) , we obtain @xmath625 @xmath626 @xmath627 @xmath628 @xmath629 \iii ) @xmath608 and @xmath630 . we take @xmath631^\dagger\|{\rm vac}\rangle$ ] and @xmath632 . then , @xmath633 @xmath634 @xmath635 @xmath636 and @xmath637 . putting these into the formula ( [ formula_zeeman1 ] ) , we obtain @xmath638 @xmath639 @xmath640 @xmath641 @xmath642 \iv ) @xmath618 and @xmath630 . we take @xmath643^\dagger\|{\rm vac}\rangle$ ] and @xmath644 . then , @xmath645 @xmath646 @xmath647 @xmath648 and @xmath637 . putting these into the formula ( [ formula_zeeman1 ] ) , we obtain @xmath649 @xmath650 @xmath651 @xmath652 @xmath653	the ground states of bose - einstein condensates of spin-2 bosons are classified into three distinct ( ferromagnetic , `` antiferromagnetic `` , and cyclic ) phases depending on the s - wave scattering lengths of binary collisions for total - spin 0 , 2 , and 4 channels . many - body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime , while low - lying excitation spectra are investigated in the thermodynamic regime . in the mesoscopic regime , where the system is so tightly confined that the spatial degrees of freedom are frozen , the exact , many - body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations . these pairwise and trio - wise units are shown to bring about some unique features of spin-2 becs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum - magnetization state against an applied magnetic field . in the thermodynamic regime , where the system is spatially uniform , low - lying excitation spectra in the presence of magnetic field are obtained analytically using the bogoliubov approximation . in the ferromagnetic phase , the excitation spectrum consists of one goldstone mode and four single - particle modes . in the antiferromagnetic phase , where spin - singlet `` pairs '' undergo bose - einstein condensation , the spectrum consists of two goldstone modes and three massive ones , all of which become massless when magnetic field vanishes . in the cyclic phase , where boson `` trios " condense into a spin - singlet state , the spectrum is characterized by two goldstone modes , one single - particle mode having a magnetic - field - independent energy gap , and a gapless single - particle mode that becomes massless in the absence of magnetic field . = -0.8truecm	21,017	490
hydromagnetic dynamos can be understood as magnetic instabilities driven by a special flow pattern in fluid conductors . there are , however , strong restrictions on the characteristics of such flows ( see dudley & james 1989 ) as well as on the geometry of the resulting magnetic fields @xcite . the restrictions even exclude any dynamo activity for a number of flows . we mention as an example that differential rotation alone can never maintain a dynamo ( elsasser 1946 ) . an open question is whether magnetic instabilities are able to excite a sufficiently complicated motion that together with a ( given ) background flow can generate magnetic fields . @xcite suggested that nonuniformly rotating disks can produce a dynamo when magnetorotational ( mri ) and magnetic buoyancy instabilities are active . later on , numerical simulations of @xcite and @xcite have shown that mri alone may be sufficient for the accretion disk dynamo . it remains , however , to check ( at least for the case of low magnetic prandtl number ) whether the mri - dynamo has physical or numerical origin @xcite . another possibility was discussed by @xcite who suggested that differential rotation and magnetic kink - type instability @xcite can jointly drive a dynamo in stellar radiation zones . the dynamo if real would be very important for the angular momentum transport in stars and their secular evolution . it taps energy from differential rotation thus reducing the rotational shear . radial displacements converting toroidal magnetic field into poloidal field are necessary for the dynamo . the dynamo , therefore , unavoidably mixes chemical species in stellar interiors that may have observable consequences for stellar evolution . such a dynamo , however , has not yet been demonstrated to exist . the doubts especially concern the kink - type instability that in contrast to mri exists also without differential rotation . the tayler instability develops in expense of magnetic energy . estimations of dynamo parameters are thus necessary to assess the dynamo - effectiveness of this magnetic instability . the basic role in turbulent dynamos plays the ability of correlated magnetic ( @xmath3 ) fluctuations and velocity ( @xmath4 ) fluctuations to produce a mean electromotive force along the background magnetic field @xmath5 and also along the electric current @xmath6 , i.e. @xmath7 we estimate the @xmath0 effect by tayler instability in the present paper . we do also find indications for the appearance of the turbulent diffusivity @xmath8 in the calculations but we do not follow them here in detail . for purely toroidal fields we did _ not _ find indication for the existence of the term @xmath9 which can appear in the expression ( [ 1 ] ) in form of a rotationally induced anisotropy of the diffusivity tensor . the fluctuating fields for the most rapidly growing eigenmodes and the azimuthal averaging are applied in the lhs of eq.([1 ] ) to estimate the @xmath0 effect and its relation to the kinetic and magnetic helicity @xmath10 and @xmath11 . our linear stability computations do not allow the evaluation of the @xmath0 effect amplitude but its latitudinal profile and its ratio to the product of rms values of @xmath12 and @xmath13 ( i.e. the correlation coefficient ) can be found . as the differential rotation is necessary for dynamo , we estimate also the influence of differential rotation on tayler instability . next , a dynamo model with the parameters estimated for the magnetic instability is designed to find the global modes of the instability - driven dynamo . the model and the stability analysis of this paper are very close to that of @xcite and will be discussed here only briefly . the basic component of the magnetic field inside a star is normally assumed to be the toroidal one . this toroidal field can be produced by differential rotation from even a small poloidal field . the background toroidal field of our model consists of two latitudinal belts of opposite polarities , i.e. @xmath14 ( see spruit 1999 ) with @xmath15 as the alfvn frequency of the toroidal field . spherical coordinates are used with the axis of rotation as the polar axis and @xmath16 as the azimuthal unit vector . the latitudinal profile of ( [ 2 ] ) peaks in mid - latitudes at @xmath17 and @xmath18 . the background flow is simply @xmath19 with @xmath20 as the equatorial rotation rate . the @xmath15 and @xmath20 are radius - dependent but this dependence is not of basic importance for the stability analysis . the reason is that the stratification of the radiative core is stable with positive @xmath21 where @xmath22 is the entropy and @xmath23 is the specific heat at constant pressure . the buoyancy frequency @xmath24 is large compared to @xmath20 ( @xmath25 in the upper radiative core of the sun ) . then the radial scale of unstable disturbances is short and the dependence of the disturbances on radius can be treated in a local approximation , i.e. in the form of @xmath26 . the parameter controlling the stratification influence on the instability is @xmath27 and the most unstable disturbances have @xmath28 @xcite . this means that the radial scale of the disturbances , @xmath29 , is short compared to the radial scale @xmath30 of toroidal field or angular velocity variations . in the solar tachocline where @xmath20 strongly varies in radius , the scale ratio is smaller than unity , @xmath31 @xcite . for such small scale ratio the radial derivatives in the linear stability equations are absorbed by the disturbances so that the local approximation in the radial coordinate can be applied . note that the unstable modes remain global in horizontal dimensions . in the radiation zones of the sun ( _ left _ ) and of a 100 myr old @xmath32 star rotating with 10 days period ( _ right _ ) . the model of the massive star is computed with the ez code of @xcite for @xmath33 . , title="fig:",width=154,height=170 ] in the radiation zones of the sun ( _ left _ ) and of a 100 myr old @xmath32 star rotating with 10 days period ( _ right _ ) . the model of the massive star is computed with the ez code of @xcite for @xmath33 . , title="fig:",width=154,height=170 ] the equation system of the linear stability analysis include @xmath34 as the momentum equation for the velocity fluctuations and the equations @xmath35 for the induction of the magnetic fluctuations and for the disturbances ( @xmath36 ) of the entropy @xmath37 the equations ( [ 6])([8 ] ) were reformulated in terms of scalar potentials for toroidal and poloidal parts of the magnetic and velocity fields to reduce the number of equations , i.e. @xmath38 ( chandrasekhar 1961 ) . the resulting system of five eigenvalue equations can be found elsewhere @xcite and it will here only be extended to the application of helical background fields . again the eigenvalue problem is solved numerically . the equations include finite diffusion in opposition to the otherwise similar equations of cally ( 2003 ) . the thermal diffusion is especially important because of its destabilizing effect . the tayler instability requires radial displacements . it does not exist in 2d case of strictly horizontal motion . the radial displacements in radiation zones are opposed by buoyancy . the thermal diffusion smooths out the entropy disturbances to reduce the effect of stable stratification . this largely increases the growth rates for the instability of not too strong , @xmath39 , fields @xcite . the diffusivities enter the normalized equation via parameters @xmath40 in our computations we used the values @xmath41 , @xmath42 , and @xmath43 characteristic for the upper part of the solar radiation zone . as it must be the magnetic prandtl number ( @xmath44 ) exceeds the ordinary prandtl number ( @xmath45 ) . the stability problem allows two types of equatorial symmetry of unstable eigenmodes . we use the notations sm ( symmetric mode with azimuthal wave number @xmath46 ) and a@xmath46 ( antisymmetric mode ) for these types of symmetry . s@xmath46 modes have vector field @xmath13 which is mirror - symmetric about equatorial plane ( symmetric @xmath47 and antisymmetric @xmath48 ) and mirror - antisymmetric flow @xmath12 ( symmetric @xmath49 and antisymmetric @xmath50 ) . am modes have antisymmetric @xmath13 and symmetric @xmath12 . instability can only be found for nonaxisymmetric disturbances with @xmath51 in agreement with the stability criteria of @xcite when applied to the toroidal field model of eq . ( [ 2 ] ) . of s modes as function of the amplitude of the toroidal background field . rigid rotation , @xmath51 . note the slowness of the modes for weak fields with @xmath52 and the jump at @xmath53 . the wave numbers are always optimized with respect to the maximum growth rates . @xmath54 , @xmath55.,width=264,height=264 ] the perturbations are considered as fourier modes in time , in azimuth and radius in the form @xmath56 . only the highest - order terms in @xmath57 are used so that in radial direction the theory is a local one ( short - wave approximation ) . the wave number @xmath58 enters the equations in the normalized form @xmath59 as a ratio of two large numbers . if only toroidal fields are considered with @xmath60 kitchatinov & rdiger ( 2008 ) found maximal growth rates of order 10@xmath61 ( normalized with the rotation rate ) at radial scales of @xmath62 for s1 modes . as described in kr08 , the eigenvalue @xmath63 possesses a positive imaginary part for an instability ( the growth rate is @xmath64 ) . the equations have only be solved for the nonaxisymmetric ( ` kink ' ) modes with @xmath51 . one can easily show that the equation system possesses a symmetry with respect to the change of sign of the azimuthal wave number @xmath46 . the equations are invariant under the transformation @xmath65 where the asterisks mean the complex conjugate . if an eigenmode exists for @xmath66 with a certain growth rate @xmath67 then the same @xmath67 holds for the eigenmode with @xmath68 . we shall demonstrate the importance of this finding for the generation of helicity and @xmath0 effect . figure [ f1 ] shows the growth rates for s@xmath69 modes in dependence of field strength . the results for a@xmath69 modes are very similar . for weak fields , @xmath70 the growth rates are closely reproduced by the parabolic law @xmath71 . for strong fields , @xmath72 , they are proportional to the field strength , @xmath73 and do not depend on the rotation rate . all averaging procedures in the present paper are realized by integration over the azimuth coordinate . consider the formation of the kinetic helicity ^kin = u , [ hkin ] ( only the real parts of both factors ) whose definition depends on the handedness of the used coordinate system . we shall prefer the righthand system . let the expressions w= ( w_r + i w_i)e^i , v= ( v_r + i v_i)e^i , [ wv ] represent the potential functions @xmath74 and @xmath75 . then the real part of ( [ hkin ] ) after the integration over the azimuth results to @xmath76 . \label{hkin1}\end{aligned}\ ] ] it is easy to show with the transformation rules ( [ trans ] ) that this expression for the modes with negative @xmath46 has the opposite sign as for the modes with positive @xmath46 . hence , ^kin(m=-1 ) = - ^kin(m=1 ) [ hkin2 ] ( see fig . [ f10 ] , bottom ) . it means that for every unstable mode with finite helicity there is another unstable mode with the same growth and drift rates but with opposite helicity so that the resulting net helicity should vanish . obviously , if all modes are excited the instability of a purely toroidal axisymmetric field can not produce finite values of the kinetic helicity . the same argument leads to the same conclusion for the current helicity ^curr = b. [ hcurr ] there is also a more straightforward argument with the same result . the mode @xmath68 is identical to the mode @xmath66 but considered in a lefthand coordinate system . the sign of the helicity is equal in both lefthand systems and righthand systems . hence , the mode @xmath68 which gives the same helicity in the lefthand system as the mode @xmath66 in the righthand system yields a negative helicity in the righthand system if the mode @xmath66 yields a positive helicity in the righthand system . the net helicity in both the righthand system and the lefthand system , therefore , vanishes . again the same is true for the current helicity ( [ hcurr ] ) . we find ( see , however , cally 2003 ) that from symmetry reasons unstable purely toroidal fields do not produce a net helicity . an additional poloidal component of the background field , however , breaks the symmetry as then , for example , the two modes possess two different growth rates so that a certain sign of helicity will be preferred ( gellert , rdiger & hollerbach 2011 ) . in the short - wave approximation only the radial component of the poloidal field is important . we write @xmath77 with @xmath78 . using the notation by kr08 with the operator @xmath79 the equation for the azimuthal flow reads @xmath80 the equation includes the poloidal background field via its last line . the parameter measuring the effect of the poloidal field is @xmath81 equation ( [ om_p ] ) shows that the characteristic strength of the field that can influence the tayler instability ( @xmath82 ) is @xmath83 times the toroidal field amplitude . this factor is of the order @xmath84 . for the latitudinal profile of the radial field component the simplest choice , i.e. @xmath85 , is used . hence , both the background field components @xmath86 and @xmath87 are antisymmetric with respect to the equator . the large - scale current helicity @xmath88 , therefore , is also antisymmetric with respect to the equator . for positive amplitudes @xmath89 and @xmath90 it is _ positive _ at the northern hemisphere and negative at the southern hemisphere ( it runs with @xmath91 ) . we shall see that the pseudoscalar @xmath88 alone determines the behavior of the pseudoscalars @xmath92 and also of @xmath93 . the basic rotation which via @xmath94 can also form a pseudoscalar does not play here an important role . the ratio @xmath95 of the toroidal field amplitude and the radial field amplitude is @xmath96 the equation for the meridional flow is @xmath97 the equations for the magnetic fields components are @xmath98 and @xmath99 the entropy equation is not influenced by the poloidal field . we computed the helicity @xmath92 of the critical modes always for the amplitude @xmath100 and @xmath101 with and without basic rotation . the results are given in fig . [ fff ] . the solid ( dashed ) lines give the helicity profiles for @xmath66 ( @xmath68 ) . indeed , the helicity of the single modes is antisymmetric with respect to the equator ( as also the current helicity of the background field ) . the plot at the bottom of fig . [ fff ] for the two modes s1 is for @xmath102 . note the extremely small differences to the top plot for the same modes under the influence of rotation . it is obviously the pseudoscalar @xmath88 directing the formation of the helicities rather than the pseudoscalar @xmath94 formed by the global rotation . figure [ gr1 ] shows the normalized growth rates @xmath103 vs the normalized wave length @xmath104 for three different poloidal field amplitudes . from top to bottom : @xmath105 , @xmath106 , @xmath107 . the peaks for the modes with @xmath68 drift from @xmath108 to @xmath109 . the corresponding peak values of the growth rates dramatically grow from @xmath110 to @xmath111 demonstrating the strong stabilization of the tayler instability for helical fields . an increase of the poloidal field amplitude by a factor of 6 ( from bottom to top ) leads to a reduction of the growth rate by three orders of magnitude . the main information of fig . [ gr1 ] is that the growth rates of the modes with @xmath66 and @xmath68 strongly differ . obviously , they always produce helicity of opposite signs and with different growth rates . the growth rate for the s1 mode with @xmath68 exceeds the growth rate of @xmath66 by a factor of four . it is , however , much smaller than the rotation rate . the unstable tayler modes are thus very slow , their corresponding growth times are much longer than the rotation periods . the helicity by the @xmath68 mode is _ negative _ at the north pole opposite to the current helicity of the background field . the small - scale kinetic helicity and the large - scale current helicity are anticorrelated ( cf . gellert , rdiger & hollerbach 2011 , for a similar result in cylinder symmetry ) . linear stability computations do not provide the absolute value of @xmath0 . only its latitudinal profile and its relative magnitude can be evaluated . we computed the normalized electromotive force @xmath112 which in opposition to the helicity ( fig . [ fff ] ) is symmetric with respect to the equator . @xmath113 is the rms velocity fluctuation after horizontal averaging ( longitude and latitude ) , @xmath114 is the rms magnetic fluctuation . the factor @xmath115 is introduced because the horizontal ( velocity and magnetic ) fluctuations are larger than the radial fluctuations by just this factor . with these normalizations the expression ( [ 10a ] ) is of order unity independent of the actual value of @xmath116 . ( bottom ) but for the normalized electromotive force after eq . ( [ 10a ] ) for the modes with @xmath66 ( solid ) and @xmath68 ( dashed ) . , width=302,height=264 ] figure [ f9 ] gives the main results for purely toroidal fields with ( [ 2 ] ) . the azimuthal component of the electromotive force i ) vanishes at the equator , ii ) the profile is symmetric with respect to the equator and iii ) it is highly concentrated to the poles . the first finding is naturally for the @xmath0 effect but it is _ not _ for the term @xmath9 which could appear in the expression of the turbulence - induced electromotive force as a consequence of a rotationally induced anisotropy of the diffusivity tensor . we find , therefore , this effect not existing due to the tayler instability of toroidal fields . for more complex field pattern its existence of can not be excluded but it remained small in any case . for purely toroidal fields even the @xmath0 effect does not exist as the modes with opposite sign of @xmath46 ( which have the same growth rates ) do cancel each other not only with respect to their helicities but also with respect to the resulting emf . in the nonlinear regime a spontaneous parity breaking may happen as it has been described by chatterjee et al . ( 2010 ) and by gellert et al . ( 2011 ) . in this case , however , it might be impossible to predict the sign and the amplitude of the @xmath0 effect . the concentration of helicity and emf towards the poles reflects a basic property of the tayler instability , i.e. that the instability pattern is more present in polar regions rather than in equatorial regions ( spruit 1999 ; cally 2003 ) . figure [ gr1 ] shows the normalized growth rates @xmath118 as a function of the radial scale @xmath119 under the influence of poloidal field components . the plots shows drastic differences of the growth rates between the modes of @xmath66 ( dotted lines ) and @xmath68 ( solid lines ) . e.g. , for @xmath106 both modes with @xmath68 possess the maximum growth rates @xmath118 with @xmath120 at a wavelength of @xmath121 . the figure shows a very strong influence of the poloidal field amplitude on the growth rates . the growth rates are small for strong poloidal field but they are large for weak poloidal field . we are thus confronted with the dilemma that only background fields with finite current helicity originate fluctuations with @xmath0 effect but the corresponding poloidal field components suppress the tayler instability ( see rdiger , schultz & elstner 2011 ) . for the further discussion the quantity @xmath122 is introduced which is antisymmetric with respect to the equator as expected for the ( normalized ) @xmath0 effect . the @xmath123 in the denominator eliminates the latitudinal profile of the toroidal field and @xmath124 is its amplitude ( see eq . ( [ 2 ] ) ) . also the resulting @xmath125-profiles are highly concentrated to the poles ( fig . the plots show the profiles for the weak - field cases @xmath126 with @xmath105 , @xmath127 and @xmath128 . the modes with the fastest growth produce an @xmath0 effect which is always positive ( negative ) for positive ( negative ) current helicity at the northern ( southern ) hemisphere of the background field . the result is an @xmath0 effect anticorrelated with the small - scale helicity and positively correlated with the large - scale pseudoscalar @xmath88 . exactly the same relations have been derived by nonlinear simulations of the kink - type instability for an incompressible fluid in a cylindric setup by gellert et al . we are thus encouraged to favor the results for the modes with the highest growth rates . for very weak poloidal field ( @xmath129 , not shown ) the @xmath0 effect is already so small ( and its sign fluctuates ) that the @xmath130 seems to form the lower limit of the helicity production by poloidal fields . in the weak - field regime ( @xmath131 ) the instability excites stronger magnetic fluctuations rather than flow fluctuations . the ratio of fluctuating alfvn velocity @xmath132 to @xmath113 results as @xmath133 for @xmath134 and @xmath135 for @xmath136 . hence , we find @xmath137 for weak fields . for strong fields ( @xmath138 ) the fluctuations become close to equipartition , i.e. @xmath139 . on the basis of the obtained informations about the @xmath0 effect ( amplitude and latitudinal profile ) we have to probe the possible existence of a dynamo mechanism . we start to estimate the typical velocity perturbation and the eddy magnetic - diffusivity . we write @xmath140 and use the growth time @xmath141 as the timescale @xmath142 . hence , @xmath143 the first step holds for purely toroidal fields ( @xmath144 ) . from fig . [ f10 ] ( top ) one finds @xmath145 so that with solar values @xmath146 mm / s results . the estimate @xmath147 leads to @xmath148 @xmath149/s as the order of magnitude of the eddy diffusivity . the value increases for fast rotating giants by one or two orders of magnitudes . it perfectly fits the diffusion coefficients for chemicals which are needed to explain the weak lithium depletion of solar type stars ( barnes , charbonneau & macgregor 1999 ) . the same estimate @xmath150 yields @xmath151 the three examples for helical background fields given in fig . [ gr1 ] lead to a common value of @xmath152 . hence , @xmath153 @xmath149/s results for the upper radiation zone of the sun . this value is very close to the above estimation for purely toroidal fields . it is increased by more than three orders of magnitudes if more massive stars are considered . figure [ f5 ] shows the details of a model with a mass of @xmath32 and 10 days rotation period . the @xmath0 effect enters the dynamo theory via the dimensionless dynamo number @xmath154 equation ( [ 10 ] ) together with the heuristic relation @xmath155 directly provides @xmath156 hence , the product @xmath157 determines the effectivity of the @xmath0 effect . with @xmath158 a value of @xmath159 results for the solar model . note that the buoyancy frequency linearly enters the equation . also for the @xmath32 star of fig . [ f5 ] the @xmath160 remains small . the operation of a ( stationary ) @xmath1 dynamo is thus excluded for radiation stellar zones . the existence of an ( oscillating ) @xmath2 dynamo , however , remains possible . all @xmath2 dynamos can work with very small @xmath0 effect if only @xmath161 @xmath162 is high enough and this is always possible for sufficiently small eddy diffusivity @xmath8 . the only consequences of very small @xmath0 effect are i ) that the ratio of toroidal and poloidal magnetic field components becomes very large and ii ) also the growth time of the dynamo instability becomes large . the growth time of the weakly supercritical @xmath2 dynamo for @xmath148@xmath149/s is extremely long ( order of gyrs for the sun ) . with ( say ) 3% differential rotation this value of the eddy diffusivity leads for the sun to @xmath161 of order 10@xmath163 . there is another possibility to proceed . for all @xmath2 dynamos the ratio @xmath95 of the toroidal and the radial field amplitudes is @xmath164 the scaling parameter @xmath165 is about 0.05 ( see table [ tab ] ) . hence , @xmath166 . the excitation condition for such dynamos can be written as @xmath167 where @xmath168 runs inversely with the shear . with ( [ beta ] ) @xmath169 results as excitation condition . one finds @xmath170 for fixed shear . note that the condition ( [ aa ] ) does no longer contain the stellar parameters like @xmath171 and/or the radius . the following findings are thus valid for all stellar radiation zones . our first example is formed by @xmath172 and @xmath173 . the condition ( [ aa ] ) then reads @xmath174 . as we shall see below , a typical value for the dynamo number is @xmath175 so that for dynamo excitation @xmath176 this relation must be read as an equation for the supercritical value of @xmath177 . the excitation is thus formally more easy for smaller poloidal field . after the numerical results in the figs . [ gr1 ] ( top ) and [ f2 ] ( top ) one finds @xmath178 for @xmath173 which is by far too small . for @xmath179 it is @xmath180 which is also not supercritical . the comparison of the figs . [ gr1 ] and [ f2 ] demonstrates that for large poloidal fields the @xmath125 grows but in the same time the growth rate drastically sinks which indicates the stabilizing action of the poloidal fields . an @xmath2 dynamo in radiation zones can not work , therefore , with too strong poloidal fields . note that with @xmath181 for @xmath182 the condition ( [ cond ] ) can be fulfilled due to the increase of the growth rate . for smaller @xmath177 the ability of the poloidal field to create a coherent @xmath0 effect sinks . for @xmath183 the function @xmath125 has already two signs at either hemisphere . however , as the condition ( [ cond ] ) can be fulfilled the existence of an @xmath2 dynamo can not be excluded by the numerical results . we must thus solve the dynamo equations in order to probe the existence of such dynamos which work with very small values of the @xmath0 effect . the @xmath0 effect plotted in fig . [ f2 ] shows another complicating characteristics , i.e. its concentration at the poles . in the next section nonlinear @xmath2 dynamo models are thus constructed with a weak solar - type latitudinal differential rotation and an @xmath0 effect which is concentrated at the poles . it is known from the theory of @xmath2 dynamos operating with @xmath184 profiles of the @xmath0 effect that they produce too polar butterfly diagrams . we must thus expect that the existence of @xmath2 dynamos to produce mid - latitude belts like in eq . ( [ 2 ] ) with pole - concentrated @xmath0 effect is unlikely . the influence of the polar concentration of magnetic - induced @xmath0 effect on a possible @xmath2 dynamo can easily be probed . the stability of an axisymmetric toroidal background field has been considered so that only the numerical values of an axisymmetric @xmath0 are known . this scenario is only consistent if the possible @xmath2 dynamo produces axisymmetric toroidal field belts at the latitude of the original background field . we know that this is true for an @xmath0 effect ( plus solar - type differential rotation ) with the standard @xmath185-profile . we have to check , therefore , whether this result remains true for the @xmath0 effect concentrated close to the poles . to this end a simple model is constructed . in a spherical shell between the normalized radii 0.6 and 1 the rotation frequency is @xmath186 . the small value of the relative shear allows to consider the fluid as hydrodynamically stable ( see watson 1981 ) . possible production of hydrodynamic - induced @xmath0 effect in radiative zones such as that by dikpati & gilman ( 2001 ) is therefore excluded . the shear peaks at 45@xmath187 which is the same latitude as that of the peak of the toroidal field . to model the polar concentration of the magnetic - induced @xmath0 effect the expression @xmath188 is used with the free parameter @xmath189 fixing the latitudinal profile of the @xmath0 effect . the polar concentration presented in fig . [ f2 ] leads to rather high values of @xmath189 . a perfect - conductor boundary condition is used at the inner radius . the outer computing domain is extended to the radius 1.2 . outside the stellar surface the diffusivity is increased by a factor of 10 . for the outermost boundary a pseudovacuum condition is used . the method of a global quenching of the @xmath0 effect is used in order to find the characteristic eigenvalue of marginal dynamo instability ( elstner , meinel & rdiger 1989 ) . table [ tab ] presents the numerical results for a reference dynamo model with 3% differential rotation and for @xmath190 . only the numbers are given for marginal dynamo instability . if the real @xmath191 differs from the reference @xmath191 by a factor @xmath192 then the resulting @xmath95 also differs by the factor @xmath67 , so that @xmath193 this condition is automatically fulfilled for all dynamo models which fulfill the excitation condition ( [ aa ] ) independent on the particular stellar model . all the dynamos listed in table [ tab ] provide positive ( negative ) values of the large - scale current helicity @xmath88 at the northern ( southern ) hemisphere . as we have assumed the same constellation for the above background field producing the @xmath0 effect the theory is thus consistent under this aspect . we have also to probe whether the star rotates fast enough to produce sufficiently large @xmath161 for the required dynamo numbers of the reference values of table [ tab ] . it is @xmath194 note at first that the excitation is much more easy for massive stars as their @xmath116 is smaller ( fig . [ f5 ] ) . inserting the characteristic numbers of our models the result is @xmath195 which is fulfilled by both the considered stellar models by their estimated values @xmath196 . either the sun as the considered hot stars with rotation periods of a couple of days rotate fast enough the excite an @xmath2 dynamo with a very weak shear ( 3% ) . to discuss the influence of the quantity @xmath189 we present models with @xmath197 and @xmath198 . the latter value describes the strongest concentration of the @xmath0 effect to the poles . for @xmath199 ( i.e. @xmath200 ) the dynamo - generated toroidal field peaks exactly at @xmath17 so that the presented stability analysis which bases on a toroidal field which also peaks at @xmath17 ( see eq . ( [ 2 ] ) ) would be consistent . this case together with the excitation condition ( [ cond ] ) which is fulfilled by the magnetic amplitudes @xmath201 and @xmath128 would strongly suggest the existence of an @xmath2 dynamo in magnetized radiation zones of hot stars . however , the models listed in table [ tab ] provide the result that the induced toroidal field belts become more and more concentrated to the poles for increasing @xmath189 . this behavior is insofar not trivial as the generation of the toroidal field by the differential rotation does _ not _ depend on @xmath189 . such polar belts resulting for @xmath202 can never reproduce the mid - latitudinal belts of eq . ( [ 2 ] ) on which the @xmath0 effect bases . the first example of fig . [ f6 ] ( top ) is the standard @xmath2 dynamo with the latitudinal profile @xmath200 . it produces axisymmetric toroidal magnetic belts of dipolar symmetry at the same latitude where the shear @xmath203 has a maximum . for growing @xmath189 the belt position drifts more and more polewards ( the field position becomes @xmath204 ) . it is thus _ not possible _ to maintain toroidal field belts in mid - latitudes if the @xmath0 effect mainly exists in the polar region ( @xmath205 ) . ccccccc @xmath189 & @xmath168 & @xmath191 & @xmath95 & @xmath165 & @xmath206 & @xmath207 + + 1 & 118,000 & 0.46 & 56 & 0.07 & 26 & 45@xmath208 + 5 & 347,000 & 1.35&23 & 0.05 & 31 & 38@xmath187 + 15 & 976,000 & 3.8 & 6.3 & 0.02 & 24&23@xmath187 + it is demonstrated with a linear theory that the tayler instability of a toroidal magnetic field in a density - stratified radiation zone of hot stars does not produce helicity and/or @xmath0 effect . if , however , a weak poloidal field component is added forming a large - scale current helicity @xmath209 then the instability leads to small - scale helicity , current helicity and also to @xmath0 effect . if @xmath209 is _ positive _ in the northern hemisphere then the helicity @xmath210 is _ negative _ at the northern hemisphere if only the modes are considered which grow fastest . hence , the small - scale kinetic helicity and the large - scale current helicity of the background field are anticorrelated . as it is often the case , the corresponding @xmath0 effect is anticorrelated with the kinetic helicity and , therefore , positively correlated with the large - scale current helicity @xmath209 . the calculations lead to two basic properties of this @xmath0 effect . it is i ) small , i.e. the normalized value @xmath191 is only of order @xmath84 and ii ) concentrated at the poles . the first property excludes the existence of @xmath1 dynamos in radiative zones and the second property makes the existence of @xmath2 dynamos ( with weak differential rotation in mid - latitudes ) unlikely . as we have shown only a dynamo with @xmath200 produces the toroidal fields in mid - latitudes . for @xmath202 , however , the belts are more and more shifted into the polar region . such fields can not close the loop of field amplification by reproducing the original axisymmetric toroidal field . or , with other words , the rotation law with the considered profile ( @xmath211 ) mainly induces toroidal fields at mid - latitudes where for high values of @xmath189 almost no @xmath0 effect exists . hence , an @xmath2 dynamo could only work for very fast rotation . by very fast rotation the instability is suppressed ( see fig . this topological problem seems to be the key problem with the magnetic - driven dynamo rather than the excitation conditions for @xmath2 dynamos . it is , however , not yet clear whether this argumentation also holds with the same power with other than the used rotation laws and/or in fully nonlinear simulations . all the presented @xmath2 dynamo models working with a weak solar - type differential rotation are reproducing the assumed sign of the large - scale current helicity @xmath88 . if thus the dynamo produces its own @xmath0 effect by the magnetic instability then the signs will be consistent . a basic deficit of the presented theory is the fact that both helicity and @xmath0 effect have been computed in rigidly rotating stars while for an @xmath2 dynamo the rotation must be nonrigid . it might be the case that growth rates and magnetic patterns of the tayler instability are strongly modified by even weak differential rotation . the discussion of this new subject , however , is not the scope of the present paper . we have thus considered only cases with a rather weak differential rotation ( 3 % ) . another deficit is formed by the order - of - magnitude estimation of the eddy diffusivity . using the characteristic scales of the modes with the highest growth rates the typical velocity is of order 1 mm / s and the typical diffusivity value is about @xmath212 @xmath149/s ( both for solar values ) . estimations of the radial mixing produced by the same instability provide an important test of the theory . the observed content of light elements at the sun impose restrictions on radial mixing in the upper radiative core ( barnes et al . 1999 , and references therein ) . the chemical mixing in the deep stellar interior can also be compatible with observations only if its characteristic time is longer than the evolutionary time scale @xcite . only very slightly supercritical kink - type instability can satisfy this restriction @xcite . all the material values in the paper are solar values . the formulation of the dynamo theory is such that the material parameters of the stellar interior do not appear ( see eq . ( [ aa ] ) ) . the conclusions about the dynamo activity for hot stars do thus not depend on the mass of the star . this is not true , however , for the above mentioned order - of - magnitude estimations of the characteristic values of velocity and diffusivity which are running with @xmath171 and @xmath213 , resp . from fig . [ f5 ] one finds that the given solar values increase by factors of 30 or 900 , resp . , for stars with three solar masses . only nonlinear simulations can demonstrate whether the system indeed prefers the modes with the highest growth rates . only if not then the radiative - zone dynamo has a chance to work in the solar interior but if it works then it should be only marginally supercritical . llk is grateful to the deutsche forschungsgemeinschaft for the support of the project ( 436 rus 113/839 ) . 99 barnes g. , charbonneau p. , macgregor k.b . , 1999 , apj , 511 , 466 brandenburg a. , nordlund . , stein r.f . , torkelsson u. , 1995 , apj , 446 , 746 cally p.s . , 2003 , mnras , 339 , 957 chandrasekhar s. , 1961 , hydrodynamic and hydromagnetic stability . clarendon press , oxford chatterjee , p. , mitra , d. , brandenburg a. , rheinhardt , m. , 2010 , pre , ( in press ) cowling t.g . , 1933 , mnras , 94 , 39 dikpati m. , gilman , p.a . , 2001 , apj , 559 , 428 dudley m.l . , james , r.w . , 1989 , royal soc . london a , 425 , 407 elsasser w.m . , 1946 , phys . , 69 , 106 elstner d. , meinel r. , rdiger g. , 1989 , geophys . fluid dyn . , 50 , 85 fromang s. , papaloizou j. , 2007 , a&a , 476 , 1113 fromang s. , papaloizou j. , lesur g. , heinemann t. , 2007 , a&a , 476 , 1123 gellert m. , rdiger g. , hollerbach r. , 2011 , mnras , 414 , 2696 goossens m. , biront d. , tayler r.j . , 1981 , ap&ss , 75 , 521 hawley j.f . , gammie c.f . , balbus s.a . , 1996 , apj , 464 , 690 . kippenhahn r. , weigert a. , 1994 , stellar structure and evolution . springer , berlin kitchatinov l.l . , rdiger g. , 2008 , a&a , 478 , 1 ( kr08 ) kitchatinov l.l . , rdiger g. , 2009 , a&a , 504 , 303 paxton b. , 2004 , pasp , 116 , 699 pitts e. , tayler r.j . , 1985 , mnras , 216 , 139 rdiger g. , kitchatinov l.l . , 2010 , geophys . fluid dyn . , 104 , 273 rdiger g. , gellert m. , schultz m. , hollerbach r. , 2010 , phys . e , 82 , 016319 rdiger g. , schultz m. , elstner d. , 2011 , a&a , 530 , 55 spruit h.c . , 1999 , a&a , 349 , 189 spruit h.c . , 2002 , a&a , 381 , 923 tayler r.j . , 1973 , mnras , 161 , 365 tout c.a . , pringle j.e . , 1992 , mnras , 259 , 604 watson m. , 1981 , geophys . fluid dyn . , 16 , 285	helicity and @xmath0 effect driven by the nonaxisymmetric tayler instability of toroidal magnetic fields in stellar radiation zones are computed . in the linear approximation a purely toroidal field always excites pairs of modes with identical growth rates but with opposite helicity so that the net helicity vanishes . if the magnetic background field has a helical structure by an extra ( weak ) poloidal component then one of the modes dominates producing a net kinetic helicity anticorrelated to the current helicity of the background field . + the mean electromotive force is computed with the result that the @xmath0 effect by the most rapidly growing mode has the same sign as the current helicity of the background field . the @xmath0 effect is found as too small to drive an @xmath1 dynamo but the excitation conditions for an @xmath2 dynamo can be fulfilled for weak poloidal fields . moreover , if the dynamo produces its own @xmath0 effect by the magnetic instability then problems with its sign do not arise . for all cases , however , the @xmath0 effect shows an extremely strong concentration to the poles so that a possible @xmath2 dynamo might only work at the polar regions . hence , the results of our linear theory lead to a new topological problem for the existence of large - scale dynamos in stellar radiation zones on the basis of the current - driven instability of toroidal fields . [ firstpage ] magnetohydrodynamics ( mhd ) instabilities stars : magnetic fields stars : interiors .	11,767	394
like the fundamental plane for early type galaxies @xcite the tully - fisher relation for disc galaxies embodies fundamental implications for the relationship between the mass of the galaxy , its star - formation history , specific angular momentum and dark matter content and distribution . broadly speaking , there are two competing models to explain the tully - fisher relation . the first of these is that it is a consequence of self - regulated star formation in discs with different masses ( e.g. , * ? ? ? * ) , i.e. , the competition of disc instability ( which promotes star formation ) with supernovae induced porosity ( which inhibits star - formation ) . the model is not the complete answer , however , since it does not explain the mass - to - light ratios or the scale - lengths of the discs . in the second model the tully - fisher relation is a direct consequence of the cosmological equivalence between mass and circular velocity ( e.g. , * ? ? ? * ; * ? ? ? this formalism is part of what has become the standard model for the growth of structure - the hierarchical merging model in which the gravitational effects of dark matter drive the evolution of galaxies and large - scale structure ( e.g. , * ? ? ? models of this type have the advantage of providing testable predictions about the sizes , surface densities , and rotation curves of galaxies as a function of redshift . however , as emphasized by @xcite , although the tully - fisher relation can naturally be explained by hierarchical merging models , the normalization and evolution of the tully - fisher relation depend strongly on the prescription used for the star formation and on the cosmological parameters . it is now well established that massive disc galaxies exist out to redshifts @xmath4 @xcite . for a given size scale , the number density of these disc galaxies is approximately the same at @xmath5 as is observed locally . overall , the results at moderate redshift ( @xmath6 ) are rather mixed . depending on the sample selection , the technique used to estimate the rotation speed , the median redshift of the sample , and the wavelength at which comparisons are made , there are claims in the literature that the tully - fisher relation either brightens or dims with redshift ( see e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . to help resolve this controversy and to push to higher redshift where evidence for evolution of disc galaxies will likely be stronger and more obvious , we set out to obtain spatially resolved rotation curves at the highest redshifts where large samples are available . obtaining spatially resolved rotation curves becomes quite difficult at such high redshifts since [ oii]@xmath73727 is redshifted into a region with many strong night sky lines and the sensitivity of ccds declines rapidly . instead we have chosen to observe the h@xmath1 line , reshifted to the infrared j - band . h@xmath1 is typically 2.5 times stronger than [ oii ] @xcite and being at longer wavelength , is less affected by dust extinction . to gauge the evolution of the tully - fisher relation we compare a local with a high redshift sample of highly inclined , `` normal '' spiral galaxies . the high redshift sample consists of objects with measured spectroscopic redshifts . to be able to measure h@xmath1 in the near - infrared we required @xmath8 . targets were selected from the cfrs / ldss redshift surveys , the clusters ms1054 , ac103 and the hawaii deep field ssa22 ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * respectively , and references therein ) . furthermore , we included two targets from the vlt science archive . for the majority of these sources we have hst i - band or at least vlt j - band images . for all of them we have obtained h@xmath1 isaac long - slit spectra to determine spatially resolved rotation curves . for this purpose we have selected these sources to be spatially extended on the images ( diameters mainly @xmath92 ) and to have high inclination angles ( @xmath10 , @xmath11 ) . furthermore , we chose only objects with exponential light profiles to ensure that we were observing disc galaxies . the mean redshift of our high - z sample is @xmath12 ( see tab . [ tbl-1 ] ) . to construct a complementary sample of local galaxies we searched the literature for objects with accessible ccd b - band images ( obtained from the nasa extragalactic database , ned , or eso science archive ) , published distances based on primary distance indicators or mean group velocities and an estimate for the rotation speed based on either an hi or a resolved h@xmath1 rotation curve . total magnitudes for all local sample galaxies where obtained from rc3 @xcite or leda . as a consistency check , the object magnitudes , sizes , rotation speeds and the resulting tully - fisher relation were compared to the rc3 catalogue and the data from @xcite . from this comparison , we conclude that our sample of low redshift galaxies is in good agreement with a random subsample of a magnitude limited complete sample from the rc3 . we obtained j- and h - band spectroscopy during four observing runs with the facility near - infrared spectrometer isaac at the vlt @xcite . we used the medium resolution spectroscopic mode with a slit - width resolution product of @xmath13 and a spatial pixel scale of 0.1484 . in eso - period 65 and 66 ( april - september 2000 and october 2000 - march 2001 respectively ) we obtained the first set of data in service mode . with total integration times of 3 hours and a 0.6-slit ( @xmath14 ) , which we rotated according to the position angle of the objects , we could successfully observe four targets . the target selection and successive estimation of position angles for this set we performed on 1 resolution images acquired in the j - band with isaac ( 20min integration time ) before the spectroscopic observations . in addition to these j - band images , we obtained i - band images taken with the canada france hawaii telescope ( cfht ) from that telescope s archive . furthermore , we conducted three other runs in visitor mode in february and september 2001 and september 2002 . this time the set - up included a 1-slit ( @xmath15 ) and integration times varied , depending on the h@xmath1 flux of the targets , between 1 and 3 hours . moreover , we selected targets on the basis of hst / wfpc2 f814w images , which we also used to orient the slit along the galaxy major axis . since most of the targets were too faint to be acquired directly we used a blind off - set from a``bright '' star , calculated from the hst images . using this prescription we observed another 17 objects successfully . the seeing during these four runs ranged from 0.4 to 0.8 . to reduce the isaac spectra we used standard data reduction packages ( eso eclipse & iraf ) . the data were flat fielded and dark - subtracted . after dark - subtraction and cosmic ray rejection we checked all frames individually for shifts in the wavelength direction due to flexure and applied a reverse shift if necessary . before combining the frames we calculated a wavelength solution and distortion correction . finally , we subtracted a background and smoothed the spectra with a gaussian of fwhm approximately equal to the number of pixels per resolution element . firstly , we extracted one dimensional h@xmath1-profiles across the spectral and spatial regions of the array that contained h@xmath1-flux . typically , even with the increased signal - to - noise gained in the continuum by summing up the flux in this way , we could not detect the continuum significantly . we then fitted the one dimensional h@xmath1-profiles ( typically assuming no galaxy continuum ) by one or two ( in the case of a double horned profile ) gaussians . at the level where @xmath16 of the total integrated flux was reached we measured the width of the profile . applying the corrections given by @xcite we obtained a first estimate for the terminal rotation velocity of our sample galaxies . in all cases but one , however , we could derive a terminal rotation velocity from a resolved rotation curve model . to extract the data points for the model rotation curve we developed a special adaptive curve tracing algorithm . we then fitted the resulting position - velocity diagram with a simple model of a flat rotation curve ( which is appropriate for not too - late disc galaxies ) . this step - function we convolved with a gaussian with a full - width at half maximum of the seeing . moreover , the convolution included weighting with an exponential function with a scale - length as measured from our disc surface brightness fits on the hst or vlt images assuming that line and continuum have the same light distribution . tests show that our modeling delivers robust rotation velocities , which are not very sensitive to the exact weighting scheme , the impact of nuclear bulges , the signal - to - noise ratio and in particular , how well the observationally `` flat part '' of the rotation curve is sampled by the data . we have measured the seeing from the j - band acquisition images taken before the observations , the acquisition images of a subsequent target , and the values from the visible seeing monitor . given our long integration times and the limited sampling of the seeing at the observing wavelength , the seeing is somewhat uncertain . we have therefore let it be a free parameter in the fitting process . the values obtained from an unconstrained fit correlate well with the seeing values estimated from the acquisition images and seeing monitor , though . some of the final fits to the data are shown in fig . [ rotation - profile ] . although only @xmath17 of the data show a clear turn - over or extend out to 2.2 disc scale lengths the remaining data do not exclusively consist of slow rotators only . in fact , the distributions of rotation speeds of high and low quality data can not be distinguished from each other ( see barden et al . 2003 ) . furthermore , the rotation speeds derived from the two dimensional data are not significantly smaller than the ones from the one dimensional spectra , which have much higher signal - to - noise ( see barden et al . 2003 ) . moreover , according to the parametric fits to over 2000 local disc galaxies of @xcite at radii varying from @xmath18 one should already measure at least @xmath19 of the asymptotic rotation velocity . since the majority of our high galaxies extend further out we estimate rotation velocities within @xmath20 . therefore , we are confident that we do not underestimate the rotation velocities significantly . finally , the rotation velocities have to be corrected for the inclination of the galaxies . we have derived inclinations from surface brightness profile fits to our hst or isaac images . to estimate rest - frame b - band magnitudes we compiled all available photometric information for our sources ( two to five independent measurements ) . for each magnitude we calculated the corresponding rest - frame wavelength and corrected the magnitudes for galactic foreground extinction @xcite . then we fitted the extinction corrected magnitudes and corresponding rest - wavelengths with spectra from the atlas of @xcite . we note that our final rest - frame b - band magnitudes are relatively insensitive to the models used since all of the galaxies had a photometric band available that was very near to the b - band rest wavelength . finally , we corrected the resulting rest - frame b - band magnitudes for inclination dependent internal extinction according to the method of @xcite and determined a face - on b - magnitude ( table [ tbl-1 ] ) . the parameters for the local comparison sample were determined and corrected in exactly the same manner as for the high redshift galaxies . rotation speeds were extracted from hi or optical rotation curves . only one object was included where a spatially - resolved rotation curve was not available . comparison of the resolved data with the hi line width from rc3 usually agree within @xmath21 km s@xmath22 ( see barden et al . 2003 ) . we calculated inclination corrections based on inclinations from our own profile fits in agreement with the values from rc3 or leda . total magnitudes were obtained directly from rc3 or leda and were corrected for galactic foreground extinction and internal extinction . since differences in the isophotal extent between the images of the local and high redshift galaxies can be quite dramatic ( up to two magnitudes ) , the light profiles of the local galaxy sample were fit only out to a characteristic limiting radius , which was approximately the average isophotal depth of the images of the high redshift galaxies ( taking into account cosmic surface brightness dimming ; see barden et al . 2003 ) . comparing the isophotal - size - culled local ( our comparison sample ) and high redshift galaxy samples ( see barden et al . 2003 ) we find an offset for the high - z data from the tully - fisher relation of @xmath24 magnitudes in a @xmath25-dominated universe ( @xmath26 , @xmath27 , @xmath28 ; fig . [ tully - fisher ] ) . this `` brightening '' of the distant galaxies is significant and holds for any reasonable cosmology ( @xmath29 for @xmath30 and @xmath31 ; @xmath32 for @xmath33 and @xmath31 ) . the limited range in isophotal size makes measuring a slope of the tully - fisher relation impossible . the observed offset in the tully - fisher relation at @xmath34 for large galaxies is partially a consequence of the fact that we observe an increase of central surface brightness of 1.7 mag arcsec@xmath35 and a slight decrease of @xmath36 in the disc scale lengths of the high redshift sample compared to the local control sample ( see barden et al . 2003 ) . using the relationship between disc magnitude , central surface brightness , and disc scale length , namely , @xmath37 , we find that the differences in average central surface brightness and average disc scale lengths imply an offset in the zero - point of the t - f relation between the two samples of about @xmath38 magnitudes . in addition , by comparing the rotation speeds of the high redshift galaxies and the local comparison sample , we find an average offset of @xmath39 km s@xmath22 of the high redshift sample ( see barden et al . thus , between @xmath40 and 0 the mean mass of galaxies has increased by a factor of @xmath41 ( assuming mass @xmath42 , and estimating the total mass out to 2.2 disc scale lengths measured from fitting the light profile ) . therefore , if we use a parameterization of the tully - fisher relation as given by @xcite , adjusting the zero - points by -0.3 mag to account for the average differences in surface brightness and disc scale length and decreasing the average rotation speed by @xmath43 yields an offset from the local tully - fisher relation consistent with that observed ( @xmath44 mag ) . having the highest redshift sample puts us in a unique position to discuss the evolution of the tully - fisher relation from a new perspective . all of the studies to date have found a significantly smaller amount of evolution compared to the results presented here . the most obvious way to reconcile this difference is that the offset in the tully - fisher relation grows with increasing redshift . the studies with the highest median redshift currently are @xcite and @xcite with a mean redshift of @xmath45 , compared with our mean redshift of @xmath46 . the time difference is therefore about 2 - 3 gyrs for any reasonable cosmology , clearly sufficient for galaxies to undergo significant evolution . unfortunately , comparing these results with model predictions is not very straight - forward since the exact offset predicted depends on the assumed physics and phenomenology , such as the details of star - formation ( e.g. , * ? ? ? * ; * ? ? ? recently , within the frame work of galaxy evolution ( e.g. , * ? ? ? * ) , a number of authors have made predictions about the general evolution of the galaxy population . for example , @xcite predict that from @xmath47 to 1 , using the cosmological parameters we have assumed here , that the overall luminosity function of galaxies in the b - band will brighten by about 1.15 mag and that the characteristic mass of galaxies will decline by about a factor of 2.4 . assuming that the galaxies selected for study here are a subset of this population , our numbers are in reasonable agreement with the predicted values . unfortunately , the resolution of the simulations by @xcite are too low to check whether the sizes of discs evolve self - consistently in this context . we conclude that large disc galaxies have undergone substantial evolution from @xmath40 to the present . the offset in the tully - fisher relation is about @xmath48 mag in the rest - frame b - band , and is related to an overall increase in the central surface brightnesses of disc galaxies and a decrease in rotation speed . we also find that the slope of the tully - fisher relation , due to size constraints that inevitably result from needing to obtain spatially - resolved rotation curves , can not meaningfully be estimated . in a subsequent paper ( barden et al . 2003 ) , we will discuss in detail the change in rotation speeds , disc masses , disc scale lengths , and disc angular momenta of this sample of high redshift galaxies relative to the local comparison sample , and make quantitative comparisons with theoretical models of galaxy evolution . , l. , ford , h. c. , huchra , j. , kennicutt , r. c. , mould , j. r. , sakai , s. , freedman , w. l. , stetson , p. b. , madore , b. f. , gibson , b. k. , graham , j. a. , hughes , s. m. , illingworth , g. d. , kelson , d. d. , macri , l. , sebo , k. , and silbermann , n. a. : 2000 , , 431 , w. l. , madore , b. f. , gibson , b. k. , ferrarese , l. , kelson , d. d. , sakai , s. , mould , j. r. , kennicutt , r. c. , ford , h. c. , graham , j. a. , huchra , j. p. , hughes , s. m. g. , illingworth , g. d. , macri , l. m. , and stetson , p. b. : 2001 , , 47 , a. , cuby , j .- , biereichel , p. , brynnel , j. , delabre , b. , devillard , n. , van dijsseldonk , a. , finger , g. , gemperlein , h. , gilmozzi , r. , herlin , t. , huster , g. , knudstrup , j. , lidman , c. , lizon , j .- l . , mehrgan , h. , meyer , m. , nicolini , g. , petr , m. , spyromilio , j. , and stegmeier , j. : 1998 , , 7 , b. l. , b " ohm , a. , fricke , k. j. , j " ager , k. , nicklas , h. , bender , r. , drory , n. , gabasch , a. , saglia , r. p. , seitz , s. , heidt , j. , mehlert , d. , m " ollenhoff , c. , noll , s. , and sutorius , e. : 2002 , , l69 lcccccccrr cfrs-00.0174&0.7838&12.9&3.58&79&0.11&0.14&23.22&-20.22@xmath490.15&116@xmath5040 + cfrs-00.0308&0.9704&17.8&1.33&43&0.06&0.13&23.72&-20.30@xmath500.20&86@xmath4920 + ms1054 - 1403&0.8133&40.7&3.21&76&0.54&0.15&20.74&-22.81@xmath500.10&232@xmath4950 + ms1054 - 1733&0.8347&14.9&1.88&60&0.22&0.15&22.97&-20.65@xmath500.10&136@xmath4920 + ldss2 - 03.219&0.6024&13.9&1.73&57&0.18&0.34&23.25&-19.49@xmath490.20&101@xmath5050 + cfrs-03.0999&0.7049&30.3&3.53&78&0.62&0.42&21.45&-21.71@xmath490.30&223@xmath5010 + cfrs-03.1393&0.8554&22.6&4.23&83&0.69&0.42&22.26&-21.43@xmath490.10&187@xmath5010 + cfrs-03.1650&0.6341&19.6&2.78&72&0.43&0.42&22.33&-20.56@xmath490.20&165@xmath5040 + cfrs-22.0953&0.9787&23.9&2.41&68&0.34&0.28&22.87&-21.17@xmath490.10&144@xmath5030 + cfrs-22.1313&0.8173&23.9&2.92&74&0.47&0.28&22.29&-21.27@xmath490.10&120@xmath5010 + cn84 - 023&0.6389&25.4&1.47&49&0.12&0.17&20.78&-22.13@xmath500.35&172@xmath5050 + cn84 - 123&0.6776&20.3&1.98&62&0.24&0.17&21.48&-21.58@xmath500.10&212@xmath5040 + csh96 - 68&1.5625&17.5&1.87&60&0.22&0.29&23.56&-21.73@xmath500.35 & 96@xmath5050 + hdfs-0620&1.2850 & 3.6&1.49&49&0.12&0.12&28.00&-16.77@xmath500.90&123@xmath5020 + sa68 - 5155&1.0521&25.1&2.35&68&0.33&0.22&22.98&-21.26@xmath490.10&160@xmath5050 + cfrs-00.0137&0.9512&18.1&1.39&45&0.06&0.14&22.37&-21.60@xmath490.10&237@xmath5040 + csh96 - 32&1.0215&13.1&1.49&49&0.12&0.30&22.85&-21.31@xmath500.10&209@xmath5020 + csh96 - 74&1.3633&25.8&3.78&80&0.68&0.29&23.30&-21.63@xmath500.10&112@xmath5030 + cfrs-03.0776&0.8835 & 7.6&1.22&36&0.06&0.42&23.69&-20.08@xmath490.10&144@xmath5010 + cfrs-03.1056&0.9392&13.8&1.50&50&0.13&0.42&22.16&-21.77@xmath490.10&140@xmath5010 + cfrs-03.1284&0.9393 & 8.1&1.18&33&0.04&0.42&23.58&-20.36@xmath500.10&167@xmath4920 + cfrs-22.0599&0.8856&13.9&1.76&57&0.19&0.29&22.29&-21.48@xmath490.10&159@xmath5030 +	we describe the first results of a programme to obtain rotation curves of @xmath0 disc galaxies in the near - infrared using the h@xmath1 emission line in order to study the tully - fisher relationship . to put any observed evolution into perspective and to investigate any possible selection biases , we constructed a control sample of low redshift galaxies that had rotation velocities and images available for measuring their dynamical , photometric , and morphological properties . compared to local objects with isophotal sizes similar to the high redshift targets , we find that our sample of galaxies with spatially resolved rotation curves , the most distant sample so far ( @xmath2 ) , clearly reveals a brightening of @xmath3 mag in the rest - frame b - band . the observed offset can be explained by a combination of increasing surface brightness , decreasing rotation speeds , and slightly smaller disc scale lengths of the high redshift galaxies .	6,695	243
the nonlinear differential equation @xmath1 appears in the modeling of certain phenomena in plasma physics @xcite . in @xcite , mickens calculates the period of its periodic orbits and also uses the @xmath2-th order harmonic balance method ( hbm ) , for @xmath3 , to obtain approximations of these periodic solutions and of their corresponding periods . strictly speaking , it can be easily seen that neither equation , nor its associated system @xmath4 which is singular at @xmath5 , have periodic solutions . our first result gives two different interpretations of mickens computation of the period . the first one in terms of weak ( or generalized ) solutions . in this work a weak solution will be a function satisfying the differential equation on an open and dense set , but being of class @xmath6 at some isolated points . the second one , as the limit , when @xmath7 tends to zero , of the period of actual periodic solutions of the extended planar differential system @xmath8 which , for @xmath9 has a global center at the origin . [ main1 ] a. for the initial conditions @xmath10 the differential equation has a weak @xmath11-periodic solution with period @xmath12 b. let @xmath13 be the period of the periodic orbit of system with initial conditions @xmath14 then @xmath15 and @xmath16 recall that the @xmath2-th order hbm consists in approximating the solutions of differential equations by truncated fourier series with @xmath2 harmonics and an unknown frequency ; see for instance @xcite or section [ hbm ] for a short overview of the method . in @xcite the author asks for techniques for dealing analytically with the @xmath2-th order hbm , for @xmath17 . in @xcite it is shown how resultants can be used when @xmath18 . here we utilize a more powerful tool , the computation of grbner basis ( ( * ? ? ? 5 ) ) , for going further in the obtention of approximations of the function @xmath19 introduced in theorem [ main1 ] . notice that equation is equivalent to the family of differential equations @xmath20 for any @xmath21 . hence it is natural to approach the period function , @xmath22 by the periods of the trigonometric polynomials obtained applying the @xmath2-th order hbm to . next theorem gives our results for @xmath23 here @xmath24 $ ] denotes the integer part of @xmath25 [ main2 ] let @xmath26 be the period of the truncated fourier series obtained applying the @xmath2-th order hbm to equation . it holds : a. for all @xmath21 , @xmath27 + 1}{2[\frac{m+1}2]+2}}\,a.\ ] ] b. for @xmath28 @xmath29 c. for @xmath30 @xmath31 d. for @xmath32 @xmath33 moreover , the approximate values appearing above are roots of given polynomials with integer coefficients . whereby the sturm sequences approach can be used to get them with any desirable precision . notice that the values @xmath34 for @xmath35 given in items ( ii ) , ( iii ) and ( iv ) , respectively , are already computed in item ( i ) . we only explicite them to clarify the reading . observe that the comparison of with the value @xmath19 given in theorem [ main1 ] shows that when @xmath36 the best approximations of @xmath19 happen when @xmath37 . for this reason we have applied the hbm for @xmath38 and @xmath39 to elucidate which of the approaches is better . in the table [ tperror ] we will compare the percentage of the relative errors @xmath40 the best approximation that we have found corresponds to @xmath41 our computers have had problems to get the grbner basis needed to fill the gaps of the table . [ ta1 ] @xmath42 the paper is organized as follows . theorem [ main1 ] is proved in section [ solus ] . in section [ hbm ] we describe the @xmath2-th order hbm adapted to our purposes . finally , in section [ sec sys ] we use this method to demonstrate theorem [ main2 ] . @xmath43 we start proving that the solution of with initial conditions @xmath44 , @xmath45 and for @xmath46 is @xmath47 where @xmath48 is the inverse of the error function @xmath49 notice that @xmath50 and @xmath51 . to obtain , observe that from system we arrive at the simple differential equation @xmath52 which has separable variables and can be solved by integration . the particular solution that passes by the point @xmath53 is @xmath54 combining and we obtain @xmath55 again a separable equation . it has the solution @xmath56 which is well defined for @xmath46 since @xmath57 is defined in @xmath58 . finally , by replacing @xmath59 in we obtain , as we wanted to prove . by using @xmath60 and @xmath59 given by and , respectively , or using , we can draw the phase portrait of which , as we can see in figure [ figura1].(b ) , is symmetric with respect to both axes . notice that its orbits do not cross the @xmath61-axis , which is a singular locus for the associated vector field . moreover , the solutions of are not periodic ( see figure [ figura1].(a ) ) , and the transit time of @xmath60 from @xmath62 to @xmath5 is @xmath63 . [ cols="^,^ " , ] its period function is @xmath64 where @xmath65 is the energy level of the orbit passing through the point @xmath66 . therefore , @xmath67 where we have used the change of variable @xmath68 and the symmetry with respect to @xmath69 then , @xmath70 if we prove that @xmath71 then @xmath72 and the theorem will follow . therefore , for completing the proof , it only remains to show that holds . for proving that , take any sequence @xmath73 with @xmath74 tending monotonically to infinity , and consider the functions @xmath75 we have that the sequence @xmath76 is formed by measurable and positive functions defined on the interval @xmath77 . it is not difficult to prove that it is a decreasing sequence . in particular , @xmath78 for all @xmath79 . therefore , if we show that @xmath80 is integrable , then we can apply the lebesgue s dominated convergence theorem ( @xcite ) and will follow . to prove that @xmath81 note that for @xmath82 close to 1 , @xmath83 since this last expression is integrable the result follows by the comparison test for improper integrals . this section gives a brief description of the hbm applied the second order differential equations @xmath84 with @xmath85 , and adapted to our interests . notice that if @xmath60 is a solution of then @xmath86 also is a solution . suppose that equation has a @xmath87-periodic solution @xmath60 with initial conditions @xmath44 , @xmath88 and period @xmath89 if @xmath60 satisfies @xmath90 it is clear that its fourier series has no sinus terms and writes as @xmath91 as we have seen in previous section , the weak periodic solutions of equation @xmath92 that we want to approach satisfy the above property . moreover @xmath93 and @xmath94 does not exist . in any case , if we are searching smooth approximations to this @xmath60 , they should also satisfy @xmath95 and hence @xmath96 for this reason , in this work we will search fourier series in cosinus , not having the even terms @xmath97 , @xmath98 which do not satisfy this property . this type of a priori simplifications are similar to the ones introduced in @xcite for other problems . hence , in our setting , the hbm of order @xmath2 follows the next five steps : \1 . consider a trigonometric polynomial @xmath99 \2 . compute the @xmath100-periodic function @xmath101 , which has also an associated fourier series , @xmath102 where @xmath103 @xmath104 with @xmath105 . find all values @xmath106 and @xmath107 such that @xmath108 where @xmath109 is the value such that consists exactly of @xmath2 non trivial equations . notice also that each equation @xmath110 is equivalent to @xmath111 \4 . then the expression , with the values of @xmath112 and @xmath113 obtained in point 3 , provide candidates to be approximations of the actual periodic solutions of the initial differential equation . in particular , the functions @xmath114 give approximations of the periods of the corresponding periodic orbits . choose , as final approximation , the one associated to the solution that minimizes the norm @xmath115 \(i ) notice that going from order @xmath2 to order @xmath116 in the method , implies to compute again all the coefficients of the fourier polynomial , because in general the common fourier coefficients of @xmath117 and @xmath118 do not coincide . \(ii ) the above set of equations is a system of polynomial equations which usually is not easy to solve . for this reason in many works , see for instance @xcite and the references therein , only the values of @xmath3 are considered . for solving system for @xmath119 we use the grbner basis approach @xmath120@xcite@xmath121 . in general this method is faster that using successive resultants and moreover it does not give spurious solutions . \(iii ) as far as we know , the test proposed in point 5 to select the best approach is not commonly used . we propose it following the definition of accuracy of an approximated solution used in @xcite and inspired in the classical works @xcite . we start proving a lemma that will allow to reduce our computations to the case @xmath122 consider @xmath125 , with @xmath126 given in . we have to solve the set of @xmath116 non - trivial equations @xmath127 with @xmath116 unknowns @xmath128 and @xmath107 and @xmath129 the lemma clearly follows if we prove next assertion : @xmath130 and @xmath131 is a solution of with @xmath132 if and only if @xmath133 and @xmath134 is a solution of . this equivalence is a consequence of the fact that the change of variables @xmath135 writes the integral equation in as @xmath136 and from the structure of the right hand side equation of . hence , @xmath137 as we wanted to prove . @xmath43 following section [ hbm ] , we consider @xmath138 as the first approximation to the actual solution of the functional equation @xmath139 . then @xmath140 when @xmath141 the above expression writes as @xmath142 using for @xmath143 we get @xmath144 where @xmath145 . by using integration by parts we prove that @xmath146 combining this equality and we obtain that @xmath147 or equivalently , @xmath148 that in terms of @xmath149 coincides with . the case @xmath149 odd follows similarly . the only difference is that instead of condition to find @xmath150 we have to impose that @xmath151 because @xmath152 when @xmath156 , we take an approximation @xmath157 the vanishing of the coefficients of 1 and @xmath158 in the fourier series of @xmath159 provides the nonlinear system @xmath160 by solving it and applying point 5 in the hbm we get that @xmath161 therefore , @xmath162 as we wanted to prove . for the third - order hbm we use as approximate solution @xmath163 imposing that the coefficients of 1 , @xmath164 , and @xmath165 in @xmath166 vanish we arrive at the system @xmath167 since all the equations are polynomial , the searching of its solutions can be done by using the grbner basis approach , see @xcite . recall that the idea of this method consists in finding a new systems of generators , say @xmath168 of the ideal of @xmath169 $ ] generated by @xmath170 and @xmath171 . hence , solving @xmath172 is equivalent to solve @xmath173 @xmath174 . in general , choosing the lexicographic ordering in the grbner basis approach , we get that the polynomials of the equivalent system have triangular structure with respect to the variables and it can be easily solved . when @xmath182 we consider @xmath183 and we arrive at the system @xmath184 the grbner basis of @xmath185 with respect to the lexicographic ordering @xmath186 $ ] is a new basis with five polynomials , being one of them an even polynomial in @xmath187 of degree 16 with integers coefficients . solving it we obtain that the best approximation is @xmath188 which gives @xmath189 for @xmath190 and @xmath191 we have done similar computations . in the case @xmath190 one of the generators of the grbner basis is an even polynomial in @xmath192 with integers coefficients and degree 32 . when @xmath191 the same happens but with a polynomial of degree 64 in @xmath193 . solving the corresponding polynomials we get that @xmath194 and @xmath195 , and consequently , @xmath196 and @xmath197 when @xmath201 doing similar computations that in item @xmath153 , we arrive at @xmath202 again , by searching the grbner basis of @xmath203 with respect to the lexicographic ordering @xmath204 $ ] we obtain a new basis with three polynomials , being one of them @xmath205 notice that the equation @xmath206 can be algebraically solved . nevertheless , for the sake of shortness , we do not give the exact roots . following again step 5 of our approach we get that the best solution is @xmath207 , or equivalently that @xmath208 computing the grbner basis of @xmath175 with respect to the lexicographic ordering @xmath176 $ ] we get that one of the polynomials of the new basis is an even polynomial in @xmath210 of degree 26 with integer coefficients . by solving it we obtain that the best approximation is @xmath211 , which produces the value @xmath212 of the statement . when @xmath182 we arrive at five polynomial equations , that we omit . once more , using the grbner basis approach we obtain a polynomial condition in @xmath187 of degree 80 . finally , @xmath213 and @xmath214 the two authors are supported by the miciin / feder grant number mtm2008 - 03437 , feder - unab10 - 4e-378 and the generalitat de catalunya grant number 2009-sgr 410 . the first author is also supported by the grant fpu ap2009 - 1189 d. cox , j. little , d. oshea,ideals , varieties , and algorithms : an introduction to computational algebraic geometry and commutative algebra " , third edition . undergraduate texts in mathematics . springer , new york , 2007 .	we prove that the differential equation @xmath0 has continuous weak periodic solutions and compute their periods . then , we use the harmonic balance method until order six to approach these periods and to illustrate how the sharpness of the method increases with the order . our computations rely on the grbner basis method .	4,121	69
disk galaxy mergers are believed to be responsible for triggering a variety of global _ and _ nuclear responses in galaxies . the global effects have been well documented , both with observations , starting with zwicky s extensive work ( 1950 , 1956 , 1964 ) , and with numerical simulations , first convincingly demonstrated by @xcite . work during the last decades strongly suggests that a fraction of elliptical galaxies has formed as a result of disk galaxy merging ( see * ? ? ? * and references therein ) . the morphology of tidal tails has been used to trace the mass distribution in galactic halos @xcite , and the tails themselves are possible birthplaces of some dwarf galaxies @xcite . mergers have also been closely connected to luminous and ultraluminous infrared galaxies @xcite . perhaps the most dramatic physical process associated with disk galaxy merging is the inflow of gas into the nuclear region and the consequent excitation of nuclear starburst and agn activity . this process has been reported from simulations @xcite and from observations @xcite . whether the gas `` hangs up '' and forms stars in the inner kiloparsec , or continues to flow inward towards a putative agn , has a strong impact on the luminosity and evolution of the merger . the details of the nuclear gasdynamics will depend on the structure of the host galaxies and the dynamical stage of the interaction @xcite . several scenarios have been suggested in which interactions evolve from starburst - dominated to agn - dominated regimes as the galaxies merge @xcite . unfortunately , until now , both observations and numerical simulations have lacked the spatial resolution needed to study the evolution of the merging nuclei on scales smaller than a few hundred parsecs . for this reason , our understanding of how mergers fuel nuclear starburst and agn activity , and drive galaxy evolution from the nucleus out , has remained woefully incomplete . theoretical arguments suggest that the evolution of the merging nuclei is where the merger hypothesis for the formation of elliptical galaxies through disk galaxy mergers faces its most stringent test . a wide variety of physical processes may shape the nuclei . purely stellar dynamical merging alone would tend to produce diffuse nuclei with large cores and a very shallow nuclear surface brightness gradient @xcite unless the progenitor nuclei were dense @xcite . in the case of gas - rich disk galaxies , the dissipative flow of gas into the nuclei and accompanying star formation would tend to result in a steep luminosity profile @xcite and a large central density , as seen in several young merger remnants ( r. p. van der marel et al . , in preparation ; see also figure 2 in @xcite @xcite ) . the presence of a central supermassive black hole would also induce a strong nuclear power - law cusp @xcite . yet if both galaxies contain supermassive central black holes which merge , the resulting black hole binary would act as a dynamical slingshot and eject stars from the center , thus lowering the stellar density there @xcite . for these reasons we are undertaking a high resolution _ hubble space telescope ( hst ) _ survey of the nuclear regions in a sequence of merging galaxies . the questions we want to answer include the following : 1 . what is the morphology of the ionized gas distribution around the nucleus ? is it clumpy , diffuse and extended , ring - like , or compact and disk - like ? does this morphology depend on the interaction stage ? how is the current star formation , as revealed by the h@xmath0 line emission , distributed with respect to the young stellar populations as revealed by their blue color ( and , eventually , spectra ) ? 3 . how do the nuclear starbursts affect the radial color gradients of the merger remnants ? the spatial resolution of the _ hst _ ( 1050 pc in nearby systems ) is required to investigate these questions , which are the focus of the current paper . in future papers , we will study the stellar populations , kinematics , and evolution of the merger - induced starbursts of the toomre sequence nuclei , using _ hst _ stis spectra and nicmos imaging . the toomre sequence @xcite is a sample of 11 relatively nearby ( within @xmath1120mpc ) interacting and merging disk galaxies , which have been arranged into a sequence according to the _ putative _ time before or since merging . these systems were chosen because they exhibit conspicuous tidal tails and `` main bodies that are nearly in contact or perhaps not even separable '' @xcite . they span a range of dynamical phases , from galaxies early in the merging process ( e.g. , ngc4038/39 and ngc4676 ) to late - stage merger remnants ( e.g. , ngc3921 and ngc7252 ; see figure [ f : mosaic ] and table [ t : sample ] ) . we want to emphasize the word `` putative '' , since the placing of a system in this sequence was based on the apparent degree of coalescence of the progenitor main bodies in low - resolution optical photographical plates , assuming that the progenitors resembled normal disk galaxies in the pre - encounter stage . because of the requirement of well - developed tidal tails , these systems have suffered , or are suffering , encounters that are most likely of prograde sense ( with the disks rotating in the same sense as they orbit each other ) . in such a scenario , the orbits of interacting galaxies decay as the galaxies lose orbital energy and angular momentum via dynamical friction with their dark halos . ultimately , the galaxies coalesce and form a single merger remnant . on these grounds we can call the toomre sequence an evolutionary sequence . because it is an _ optically _ selected sample of merging galaxies , it suffers less from dust obscuration than infrared - luminous samples , allowing the nuclei to be studied at optical wavelengths . the toomre sequence has been widely investigated by ground - based observations @xcite , space - based observations @xcite , and numerical studies @xcite . ccccccccc sequence number & galaxy & r.a . ( j2000.0 ) & dec . ( j2000.0 ) & @xmath2 & dist . & @xmath3 & 01 & dynamical + & & ( hh mm ss.ss ) & ( ddmmsss ) & ( km s@xmath4 ) & ( mpc ) & ( ) & ( pc ) & model + 1 . & ngc4038 & 12 01 53.06 & @xmath518 52 01.3 & 1616 & 21.6 & 61.4 & 10 & y + 1 . & ngc4039 & 12 01 53.54 & @xmath518 53 09.3 & 1624 & 21.6 & 61.4 & 10 & y + 2 . & ngc4676 nuc1 & 12 46 10.06 & + 30 43 55.5 & 6613 & 88.2 & 37.1 & 43 & y + 2 . & ngc4676 nuc2 & 12 46 11.17 & + 30 43 21.2 & 6613 & 88.2 & 37.1 & 43 & y + 3 . & ngc7592 nuc1 & 23 18 21.73 & @xmath504 24 56.7 & 7280 & 97.1 & 13.0 & 47 & n + 3 . & ngc7592 nuc2 & 23 18 22.60 & @xmath504 24 57.3 & 7280 & 97.1 & 13.0 & 47 & n + 4 . & ngc7764a & 23 53 23.74 & @xmath540 48 26.3 & 9162 & 122.2 & 0 & 59 & n + 5 . & ngc6621 & 18 12 55.25 & + 68 21 48.5 & 6191 & 84.4 & 41.7 & 41 & n + 5 . & ngc6622 & 18 12 59.74 & + 68 21 15.1 & 6466 & 84.4 & 41.7 & 41 & n + 6 . & ngc3509 & 11 04 23.59 & + 04 49 42.4 & 7704 & 102.7 & 0 & 50 & n + 7 . & ngc520 nuc1 & 01 24 34.89 & + 03 47 29.9 & 2281 & 30.4 & 40.3 & 15 & y + 7 . & ngc520 nuc2 & 01 24 33.30 & + 03 48 02.4 & 2281 & 30.4 & 40.3 & 15 & y + 8 . & ngc2623 & 08 38 24.11 & + 25 45 16.6 & 5535 & 73.8 & 0 & 36 & n + 9 . & ngc3256 & 10 27 51.17 & @xmath543 54 16.1 & 2738 & 36.5 & 0 & 18 & n + 10 . & ngc3921 & 11 51 06.96 & + 55 04 43.1 & 5838 & 77.8 & 0 & 38 & n + 11 . & ngc7252 & 22 20 44.78 & @xmath524 40 41.8 & 4688 & 62.5 & 0 & 30 & y + all images were taken with the wfpc2 camera onboard the _ hst_. we used the f555w and f814w filters , which mimic the better - known @xmath6- and @xmath7-bands , with integration times of 320 seconds in each band , split into two exposures of 160 seconds to allow for cosmic ray rejection ( see table [ t : expo ] for more information on the exposure times ) . images in these bands for ngc4038/39 @xcite , ngc3921 @xcite , and ngc7252 @xcite already existed in the _ hst _ archive . we used those images in our current study . narrow - band images covering the h@xmath0 + [ ] lines were taken with the f673n narrow - band filter in cases where the line emission from the target fell within the wavelength range covered by this filter ( ngc3509 ; ngc 7592 ) , or otherwise with the linear ramp filter ( lrf ) . the lrfs have a bandwidth of about 1.3% of the central wavelength . the position of the galaxy on the ccd chip depends on the central wavelength and limits the field of view to about @xmath8 @xmath9 @xmath8 for the lrf . the list of the adopted filters , central wavelengths , and fwhm values of the filters together with integration times are given in table [ t : expo ] . the antennae , ngc4038/39 , was imaged earlier in the _ hst _ f658n h@xmath0 filter by @xcite , and we used these data in our work . we employed the stsdas task wfixup to interpolate ( in the x - direction ) over bad pixels as identified in the data quality files . we also used the stsdas task warmpix to correct consistently warm pixels in the data , using the most recent warm pixel tables . the stsdas task crrej was used to combine the two 160 second exposures . this step corrects most of the pixels affected by cosmic rays in the combined image . in general , a few cosmic rays remain uncorrected , mostly when the same pixel was hit in both exposures . also , a small number of hot pixels remain uncorrected because they are not listed even in the most recent warm pixel tables . we corrected these with the iraf task cosmicrays , setting the `` threshold '' and `` fluxratio '' parameters to values selected by a careful comparison of the images before and after correction , to ensure that only questionable pixels were replaced . the rest of the reduction was done with the standard wfpc2 pipeline tasks using the best reference files available . the lrf exposures were flat - fielded using the narrow - band f656n and f673n ( whichever was closer in wavelength ) flat fields . we also created mosaiced f555w and f814w images with the iraf / stsdas task wmosaic . color composite mosaics of these images , which show the environment around the nuclei , and tidal tails in some systems , are displayed in figure [ f : mosaic ] . the photometric calibration , and conversion to johnson @xmath6- and @xmath7-bands was performed according to guidelines in @xcite . a @xmath10-correction has not been applied to any of our measurements . since the throughput of the narrow - band filters does not vary much at the wavelengths of the lines , it was possible to calibrate the line fluxes by assuming zero width for the lines ( monochromatic ) . we used the iraf / stsdas synphot task bandpar to compute the conversion from counts sec@xmath4 to ergs sec@xmath4 @xmath11 . the results were the same within errors to those produced by the synphot task calcpar . using the known relative orientation of the various chips on wfpc2 , we rotated the chips which contained the line emission and the pc - chip @xmath6 and @xmath7 images to the same orientation . we used stars in the images to register the frames . the final emission - line images were constructed by accounting for the different scales of images on various chips ( usually the pc chip and wf2 chip ) , and scaling down the @xmath6- and @xmath7-band images to the expected count levels corresponding to the wavelength and band of the line image , with the help of the wfpc2 exposure time calculator . we then combined the broad - band images by taking their geometric mean and performed a final adjustment to this combined continuum image by selecting areas well outside emission - line regions ( with pure stellar emission ) and comparing fluxes in corresponding areas in the continuum and line images . finally , we subtracted the scaled continuum image from the line image . using the combination of @xmath6 and @xmath7 images eliminates the extinction terms in the case of foreground extinction , as explained in detail by @xcite . in some cases , the residual image shows negative pixels , which are most likely due to strong color gradients near the nuclei . we estimate that the final uncertainty in the line fluxes is no better than 50% , based on varying the scaling of the continuum image within acceptable limits and comparing the derived total line fluxes . color index images were created by dividing the f555w image by the f814w image , taking the logarithm of the result , and correcting for differences in the color terms , using the synthetic calibration from @xcite . for our new imagery ( i.e. , all systems except ngc 4038/39 , ngc 3921 , and ngc 7252 ) , many regions in the raw color index images are of low signal - to - noise . we therefore constructed smoothed color index maps using an adaptive filtering procedure , as described by @xcite , where areas with lower signal - to - noise ratios were smoothed by a larger boxcar . these images were then masked so as to only show areas with adequate signal - to - noise ( signal - to - noise greater than two in both @xmath6- and @xmath7-images after being smoothed with an 11@xmath911 pixel boxcar median filter ) . the resulting smoothed color index images , together with the unsmoothed broad - band images and h@xmath0 + [ ] images , are shown in figures [ f : n4038][f : n7252 ] . in the following we give a brief description of each system , based on figures [ f : n4038][f : n7252 ] , together with a few references to earlier work . we use a hubble constant of @xmath12 throughout this paper . perhaps the best - known system in the sequence is the antennae . we use the spectacular _ hst _ @xmath13 , and h@xmath0 images of this system from @xcite ( @xcite ; see also @xcite and figure [ f : mosaic ] ) . these images show the chaotic dust lanes and prevalent star formation in the interface between the two galaxies . they also reveal the redder color of the underlying population in the bulge component of each galaxy . the focus of the @xcite papers was on the abundant young star clusters which have apparently formed as the result of the interaction of the two galaxies . the properties and evolution of such clusters in a merger event have been discussed extensively in a series of papers by whitmore and his collaborators ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the nuclear regions of merging galaxies are potential formation sites of new super star clusters @xcite . we instead investigate the morphology and star formation activity in the central few hundred parsecs . the nuclear region of the northern component of the antennae , ngc4038 , is bracketed by star clusters in the @xmath6 and @xmath7 images . the position of the nucleus is most likely near the knot `` j '' in the classification scheme of @xcite , and close to the nuclear position identified in the radio continuum by @xcite ( @xcite ; using the radio continuum data of @xcite @xcite ) and in the near - infrared by @xcite . in our i - band image , this is located near the center of the frame displayed in fig . [ f : n4038 ] . there we see an elongated ring - like structure , with an axis ratio of about 0.3 and a diameter of about 25 ( 260 pc ) , pointing north - northeast . the coordinates of the optical knot that we identified as the nucleus of ngc4038 in the wfpc2 images are given in table [ t : sample ] . they agree with the radio continuum position to within one arcsecond and they are within about 3 arcseconds of the estimated near - infrared position of the nucleus . the absolute accuracy of the coordinates reported in this paper is @xmath11 . this is the intrinsic accuracy of the _ hst _ guide star coordinate system . the h@xmath0 image ( see fig . [ f : n4038 ] ) also shows the elongated ring structure with a peak in the north - northeastern corner at the position of the star clusters seen in the @xmath6- and @xmath7-band images . the @xmath14 colors of the clusters in the ring - like structure vary from 0.4 to 1.0 , while a typical color outside and inside the ring is 1.5 , reaching up to 1.9 in the dust lanes . most of the dust is concentrated on the southeastern side of the elongated ring - like structure . near - infrared high resolution _ nicmos observations of the nucleus will be reported in a future paper . they promise to shed more light on the issue of the true location of the nucleus in ngc4038 . previous ground - based near - infrared imaging @xcite does not have high enough spatial resolution to indicate the exact location of the nucleus . the nucleus of ngc4039 , the southern component of the antennae , is much easier to recognize in the @xmath6 and @xmath7 images ( see fig . [ f : n4039 ] ) . our identified nuclear position ( the position of the optical peak ) in table [ t : sample ] agrees to within about two arcseconds of the estimated nuclear positions from the radio continuum data @xcite and near - infrared data @xcite . the nucleus is surrounded by dust patches and dust lanes . there is also a minor peak of h@xmath0 emission at the location of the nucleus , which could be due to continuum subtraction uncertainties . much more intense h@xmath0 emission is seen in the arm connecting to the nucleus . the @xmath14 color of the nucleus of ngc4039 is about 1.0 , and the surrounding area has a color around 1.3 , with values of 1.7 in a dusty region close to the nucleus . neither ngc4038 nor ngc4039 is classified as a seyfert or liner nucleus . these nuclei are instead hosts of mild starbursts ( e.g. , * ? ? ? * ; * ? ? ? further evidence for the starburst nature of these nuclei comes from x - ray observations with the rosat high - resolution imager @xcite and with chandra @xcite . the northern x - ray nucleus ( ngc4038 ) has a soft spectrum , which hints at thermal emission and is probably related to a hot wind , whereas the thermal + power - law spectrum of the southern x - ray nucleus ( ngc4039 ) indicates a hot ism and a contribution from x - ray binaries . the mice is a pair of spiral galaxies with only moderately active nuclei ( classified as liner - type by @xcite ) . the nucleus of ngc4676a or nuc 1 ( the northern component of the pair ) is covered by dust in our optical @xmath6 and @xmath7 images ( fig . [ f : n4676n1 ] ) . this dust is likely associated with the dense edge - on molecular disk imaged in co(1 - 0 ) by @xcite . adopting standard conversion factors , the observed peak co flux density suggests that the nucleus of ngc 4676a is hidden beneath @xmath15 60 magnitudes of extinction . just south of the area of heavy dust obscuration near the base of the northern tail is a v - shaped structure ( with the v being sideways and opening to the east , and the tip of the v at r.a . 12@xmath1646@xmath17101 and dec . @xmath1843550 ) of brighter emission from young stellar clusters near the center of the main body . about 7 ( 3 kpc ) to the south one can see a triangle - shaped structure of young clusters . the tip of the central v - shaped cluster lies near the peak emission in a ground - based k image ( j. hibbard , unpublished ; see also @xcite @xcite ) , so we assume that the nucleus lies near this position . the h@xmath0 emission from ngc4676a is weak . in general , the emission is elongated in the north south direction of the main body . the v - shaped structure has very red @xmath14 colors above 2.2 . perhaps even more surprisingly , the color of the heavily extinguished dust patch near the base of the tail is actually _ bluer _ than its surroundings , having an average @xmath14 value less than 1 . we interpret this region as scattered light from young stars which are mostly hidden by the dust . a hint of this young star population is seen in the @xmath6 image ( not shown here ) . such a young star forming region can be seen even more clearly in the new , deeper acs images of the mice @xcite . in contrast to the v - shaped structure , the triangle - shaped structure of clusters near the southern end of the main body has bluer colors than its surroundings , with @xmath14 values around 1 or slightly below it . the nucleus is easily identified in ngc4676b or nuc 2 . it is surrounded by a rather amorphous disk , and a dust lane that seems to wrap around the whole disk , ending near the location of the nucleus ( fig . [ f : n4676n2 ] ) . there is a peak of h@xmath0 emission offset by about 02 from the nucleus of ngc4676b , but in general the h@xmath0 emission is weak and patchy . the nucleus has @xmath14 colors close to 1.3 . a dust patch near the nucleus has a @xmath14 color as red as 1.7 . the majority of the disk outside the dust lanes and dust patches has a @xmath14 color of 1.21.3 . the overall appearance of ngc4676b , particularly the well - defined bulge / nucleus , is consistent with it being of an earlier hubble - type than its interacting partner , ngc4676a @xcite . this is the third most distant system in the toomre sequence , and the main bodies of both galaxies fit within the pc chip of the wfpc2 . ngc7592a or nuc 1 , further to the west , has an active seyfert 2 nucleus , whereas ngc7592b or nuc 2 is classified as a starbursting system @xcite . ngc7592a has a much better defined nucleus or bulge than ngc7592b , consistent with the suggestion of @xcite , who argue that ngc7592a is of an earlier hubble - type than ngc7592b . @xcite claim that ngc7592a is seen almost face - on , because of its roundish appearance in ground - based optical and @xmath10-band images . if so , this galaxy appears to possess a bar - like structure in the bulge , although the bar classification is uncertain due to patchy emission and one - sided dust . similarly , @xcite argued that ngc7592b is more highly inclined to the line - of - sight . this , together with its later hubble - type , explains why no clear nucleus is seen in this system . a comparison to the ground - based @xmath10-band image of @xcite suggests that the nucleus of ngc7592b lies near the center of the ring - like structure of bright clumps seen in figure [ f : n7592 ] . we are not able to identify any specific clump with the nucleus . the h@xmath0 image shows abundant emission around the nucleus and the `` bar '' of ngc7592a , and emission from the clumps in the ring of ngc7592b , with a strong central depression in emission . the nucleus of ngc7592a has blue colors with @xmath14 around 0.8 , contrasting to a dust lane lying on one side of the bar with a @xmath14 color close to 2.0 . the southern side of the ring structure in ngc7592b has very blue colors down to @xmath14 of 0.3 , but there is a region of red colors of @xmath14 up to 1.9 in the northern part of the ring . although there is dust visible at that location as well , the red color may suggest that the underlying red nucleus lies at this location . this is the most distant system in our sample at @xmath19 = 122 mpc . the location of ngc7764a in the toomre sequence , among galaxies with clearly separate nuclei ( fig . [ f : mosaic ] ) suggests that toomre thought the nuclei of the two systems were still distinct . however , our _ hst _ images ( figure [ f : n7764a ] ) do not obviously support the presence of two distinct nuclei . future nicmos near - infrared data will be critical in identifying the likely remains of the interacting galaxies . for now we only identify one main component from which both the tails are emanating . there is also a barred spiral galaxy , presumably at a similar redshift , displaced about 40 ( 23.7 kpc in projected distance ) to the southeast of the main system ( imaged on one of our wf chips , see fig . [ f : mosaic ] ) , and a third system with apparent tidal streamers displaced about 40 to the northwest , just outside our pc image . our pc image reveals a shred or linear feature about 7 ( 4.1 kpc ) to the northwest of the main body of ngc7764a . about a dozen bright , blue ( @xmath20 0.60.8 ) star clusters lie within this filament . between this filament and the center of fig . [ f : n7764a ] is a very bright , very blue ( @xmath20 0.22 ) cluster . this cluster has associated h@xmath0 emission with a``head - tail '' morphology , with the cluster at the `` head '' , and the `` tail '' pointing to the north - northeast . following the `` head '' towards the center of the image in fig . [ f : n7764a ] , there is a string of regions , crossed by dust lanes . the centralmost region is weaker than the regions to its northwest and southeast , and is associated with two adjacent bright optical clusters . these two clusters lie near the region of very red colors in the @xmath14 map , with the reddest region ( @xmath202.4 ) associated with the eastern cluster . this strong dust concentration may indicate that this is the location of one of the nuclei . the western clump of the double cluster is bluer ( @xmath20 1.5 ) and brighter . there are more bright hii regions about 3 ( 1.8 kpc ) to the northeast of the double cluster , associated with another bright star cluster embedded in dust . further to the east of this is an amorphous luminous region with rather blue colors ( @xmath20 0.60.8 ) . the northern component of this interacting galaxy pair , ngc6621 , has a dusty central region punctuated by patchy star clusters and ionized gas emission ( fig . [ f : n6621 ] ) . two major dust lanes lead into the central region . one of them comes in from the northwest , following the long tidal tail at larger radii . the other dust lane intersects the nuclear region from the north , cutting a bright ridge of emission regions into two parts . emission is associated with the bright ridge of emission regions seen in the @xmath6 and @xmath7 images , and also with another emission patch across the dust lane coming in from the northwest . the emission ridge has @xmath14 colors around 1.0 , but in the surrounding dusty area the @xmath14 colors are as red as 2 . the southwestern emission region across the dust lane from the emission ridge has a blue @xmath14 color of 0.5 . the nuclear spectrum of ngc6621 has been classified to be of starburst type @xcite . the ground - based near - infrared data of @xcite show that the nucleus likely lies near the ridge of bright emission seen in our optical images , but it is not possible to constrain the true nuclear location from these images . ngc6622 has a well - defined nucleus and either a strong stellar bar , or a close to edge - on orientation to the line - of - sight ( fig . [ f : n6622 ] ) . the @xmath21-band infrared image of @xcite shows that the outer isophotes are not too far from being circular , making it unlikely that this is an edge - on galaxy . there is an obvious dust lane on the northern and northeastern side of the galaxy , most likely signifying perturbations caused by ngc6621 . there is practically no h@xmath0 emission associated with ngc6622 , suggesting that it is an early hubble - type galaxy . the nucleus and the bar have red @xmath14 colors up to 2.3 . the typical color in the remainder of the disk is 1.3 . there is a star forming region in the area between ngc6621 and ngc6622 which has a blue @xmath14 color with typical values around 0.8 ( near the upper right edge of figure [ f : n6622 ] ) . the second - most distant galaxy of this sample , ngc3509 , falls in the middle of the toomre sequence . the sketch of this system by @xcite suggests that he envisioned a large tail curving to the northwest and a shorter tail extending to the southwest . deep ground - based ccd imagery obtained by one of us ( jeh ) suggests that the southeastern feature is not a tail , but rather the bright ridge of an inclined disk . there is a very bright rectangular - shaped feature at the head of the southeastern feature , directly south of a central bulge ( see also * ? ? ? * ) , which may have been interpreted as the bulge of the second system . our hst / wfpc2 broad - band imagery shows that this region resolves into a number of bright , blue stellar associations ( figure [ f : n3509 ] ) , and we believe it is much more likely to be a collection of bright star forming regions within the perturbed disk than the nucleus of a second system . the extremely blue colors of this region ( @xmath14 colors around 0.5 ; the tip of this region is seen at the bottom center of figure [ f : n3509 ] ) are in spectacular contrast to the rest of the galaxy ( typical @xmath14 values from 1.1 to 1.4 ) , and indicate active star formation . in our images we can clearly identify a single relatively undisturbed nucleus , surrounded by a swirl of dust lanes ( center of figure [ f : n3509 ] ) . the h@xmath0 image reveals a peculiar double - peaked h@xmath0 structure straddling the nucleus in an orientation which is perpendicular to the main body of the nuclear region . h@xmath0 emission , indicating ongoing star formation , is also seen in the blue region south of the nucleus . in the absence of an obvious second tidal tail in the ground - based imagery , we find little evidence that ngc 3509 is the obvious result of a major disk - disk merger . it may instead be the result of a minor merger , or an ongoing interaction with a smaller companion . in support of the minor merger hypothesis , we note the appearance of a roughly circular diffuse red concentration just to the north of the nucleus ( 12 @xmath1 6 kpc ; see fig . [ f : mosaic ] , just off of the pc chip ) which could be the remains of a smaller galaxy . in support of the interaction hypothesis , on a deep r - band ccd image taken by one of us ( jeh ) we note the presence of a compact object lying @xmath1 2 towards southeast , along the minor axis of this system . this possible companion also has faint low surface brightness features pointing both toward and away from ngc 3509 . the southeastern component of ngc520 is hidden behind a prominent and intricate dust lane ( figure [ f : n520n1 ] ) . this dust is likely associated with the dense edge - on molecular disk imaged in co ( 10 ) by @xcite and @xcite . adopting standard conversion factors , the observed peak co flux density suggests that the nucleus of ngc 520 is hidden beneath @xmath22300 magnitudes of extinction . it is thus not surprising that the position of the primary nucleus is impossible to determine from our data . high - resolution future nicmos observations may point out where this nucleus lies , but based on earlier ground - based near - infrared images ( e.g. , * ? ? ? * ; * ? ? ? * ) the nucleus likely lies in the center of the heaviest dust absorption . radio continuum images ( e.g. , * ? ? ? * ) show a disk - like morphology coincident with the co disk , suggesting on - going star formation within the central molecular disk . practically no h@xmath0 emission is seen in our h@xmath0 image at the position of the primary nucleus . red @xmath14 colors are seen along most of the primary nucleus . however , at the location of the most dust - affected regions , as seen in the @xmath7 image , we see relatively blue colors of 0.8 in the color index image . again , as in the case of ngc4676a , we interpret this as scattering of emission from young stars , hinted at in the @xmath6-band image ( not shown here ) . the secondary , northwestern nucleus ( fig . [ f : n520n2 ] ) comes presumably from an earlier hubble - type disk galaxy that is now embedded within an extended atomic gas disk associated with the primary galaxy @xcite . the secondary nucleus is very well defined , although its surrounding disk seems to have been mostly disrupted in the merger , and probably contributes now to the extended optical tails . the only significant h@xmath0 emission near the secondary nucleus is in clumps within 1 ( 150 pc ) , and displaced to the northwest . the secondary nucleus appears to have a bluish color , with @xmath14 of 0.6 . this nucleus is observed to be in a post - starburst phase @xcite . the @xmath14 color towards the northwest of the nucleus has bluish values around 0.8 , but dust reddens the color to 1.2 at about 3 ( 450 pc ) southeast of the nucleus . consistent with its classification as a fairly advanced merger system , only one galaxy body is evident in ngc2623 ( fig . [ f : n2623 ] ) . again , based on the high spatial resolution optical _ hst _ images it is impossible to tell the exact location of the nucleus or nuclei . a comparison to the @xmath23 nicmos images published by @xcite suggests that the position of the brightest optical peak in the center of the main body of ngc2623 is the true position of the nucleus , near j2000.0 coordinates of r.a.=8@xmath1638@xmath17241 and dec.=25@xmath2445@xmath25166 . this nucleus is classified as a liner by @xcite and @xcite . dust surrounds the nucleus mostly on the southeastern , northern , and northwestern sides , although numerous dust lanes criss - cross the whole nuclear region ( see fig . [ f : n2623 ] ) . a study by @xcite confirms the coalescence of two galaxies . based on their near - infrared observations , they conclude that the merger is complete , because only one nucleus is visible ( which is surrounded by a single , symmetric nuclear region ) . this impression is confirmed by co observations , which reveal a single centrally concentrated compact molecular disk with simple rotational kinematics @xcite . there are also numerous young star clusters scattered around the main body and the surrounding shreds . most of the h@xmath0 emission comes from a disk around the nuclear position and from a location north of the nucleus , but also from a position in the dust patch southeast of the nucleus . the nucleus has a very red @xmath14 color at 2.5 , consistent with the reddish near - infrared colors found for the nucleus by @xcite . the red near - infrared @xmath26 color , together with a relatively large co absorption index value @xcite suggest that there has been a recent starburst in the nuclear region . the red colors continue for about 2 ( 720 pc ) to the north and northwest . the main body of ngc2623 has @xmath14 colors close to 1.0 or just below it . our optical _ hst _ images show one obvious nucleus located in the center of figure [ f : n3256 ] ( see also @xcite ) . this nucleus is surrounded by a relatively symmetric spiral morphology . near - infrared , radio continuum , and x - ray observations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) have discovered a second nucleus about 5 ( 885 pc ) south of the primary nucleus ( marked with an `` x '' in fig . [ f : n3256 ] ) , presumably hidden beneath the dark ( red ) dust feature directly south of the primary nucleus in the @xmath14 image of fig . [ f : n3256 ] ( see also @xcite ) . we do not see an optical source at the position of the second near - infrared , radio , and x - ray source . our data therefore do not shed any new light on the nature of this source . neither nucleus shows any evidence of being active . studying the question of how many nuclei exist in this system ( even three nuclei have been suggested by @xcite ) , would benefit from a detailed dynamical simulation of this merger system . the measurements for ngc3256 in this paper refer only to the one nucleus we see in our wfpc2 images . archival nicmos near - infrared images suggest that the position of the primary nucleus is close to the location of the brightest emission in our f814w image @xcite . however , the optical _ hst _ images alone only show that this position is part of a ring - like structure , and there is nothing noticeably different about this location other than that it is the brightest clump in the image . other bright regions are likely to be young star clusters or globular clusters , as reported by @xcite . the h@xmath0 emission is extended and follows the spiral structure , but the brightest emission occurs at the position of the primary nucleus . most previous studies regard this nucleus to be of starburst type , although @xcite were unable to resolve its true nature . while the primary nucleus has a blue @xmath14 color near 0.7 , the dusty region covering the secondary nucleus to the south of the primary nucleus has a @xmath14 color of 2.5 or redder . in contrast , the spiral arms visible in the @xmath6 and @xmath7 images have a blue @xmath14 color between 0.5 and 1.0 . only one nucleus is visible in ngc3921 ( figure [ f : n3921 ] ) . this galaxy is a post - merger system that has developed an elliptical - like central region ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , although numerous dust lanes still criss - cross this area ( see fig . [ f : n3921 ] ) . the evidence for a recent merger can be seen in the outer region where tails and plumes prevail @xcite . several compact open star clusters or globular clusters can also be seen in the _ hst _ optical images kindly provided by b. whitmore ( see also * ? ? ? the h@xmath0 emission is centered in a disk around the nucleus , but extends to larger radii . the nucleus is classified as a liner by @xcite and @xcite . it lies in a location where the @xmath14 color gradient is very steep as seen in the @xmath14 image , but the nucleus itself has normal values of @xmath14 around 1.1 . to the southwest of the nucleus the @xmath14 color increases to @xmath1 1.8 at about 05 ( 190 pc ) distance from the nucleus , and then decreases and stays around 1.11.2 on that side of the galaxy . to the northeast , the @xmath14 color decreases to values as low as 0.7 at about 05 from the nucleus , then stays around 0.9 for another 15 ( 570 pc ) . dust extinction is at least partly responsible for this effect . @xcite noticed that the nucleus lies substantially off - centered from the outer stellar isophotes of the main body . we use the _ hst _ @xmath6 and @xmath7 images from @xcite to study the most advanced merger in the toomre sequence . the remnant from the merger is probably between 0.5 and 1 gyrs old @xcite . it has a single nucleus and is characterized by a @xmath27 profile typical of elliptical galaxies @xcite . it was quite a surprise , therefore , when _ _ revealed a central spiral disk ( @xcite and fig . [ f : n7252 ] ) . the disk extends to a radius of about 6 ( 1.8kpc ) , and is therefore coincident with the molecular disk imaged by @xcite . our new _ h@xmath0 image of this galaxy shows the disk to be actively forming stars , while the @xmath14 color map shows dust to be intertwined between the arms . in light of the results from recent hydrodynamical simulations @xcite , and the observation of tidal gas streaming back into the central regions of this remnant from the tidal tails @xcite , it seems likely that this central disk structure results from the re - accretion of tidally raised gas into the now - relaxed central potential . the nucleus appears resolved , and is very bright , as we show in section [ s : mergerstage ] . h@xmath0 emission largely follows the spiral arms , although there appears to be a central h@xmath0 source as well . however , our continuum subtraction , unfortunately , is uncertain in this area due to an optically very bright nucleus . similarly , the central @xmath14 color is difficult to determine reliably due to the same problem , but the color there appears to be close to 1.3 , whereas the surrounding spirals have @xmath14 colors as blue as 0.5 . recent x - ray observations with asca revealed x - ray emission that can be described by a two - component model @xcite , with soft emission ( kt = 0.72 kev ) , and a hard component ( kt @xmath28 5.1 kev ) . the hard component might indicate the existence of nuclear activity or even an intermediate mass black hole . we performed aperture photometry of the toomre sequence galaxies to better understand the evolution of the nuclear region of merging galaxies . because the isophotes in the nuclear region of merging galaxies are highly twisted , we did not attempt to fit ellipses and obtain radial surface brightness profiles . instead we performed aperture photometry within circular apertures , with radii of 100 pc and 1 kpc . the lower value was dictated by the smallest resolvable physical scale in the most distant of our sample galaxies , whereas the upper value is somewhat arbitrary , but set because of our emphasis on the properties of the nuclear region . we have already characterized the morphology of the nuclear region at _ hst _ resolution in the individual galaxies above . here we only report on the bulk photometric nuclear properties of each identified merger component galaxy . visual inspection of the images quickly reveals that none of the galaxies in the sample possess an extremely bright , point - like active nucleus . based on spectroscopic evidence , as cited in section [ s : images ] , the only mildly active nuclei are in ngc7592a ( seyfert 2 ) , ngc4676 nuc 1 and nuc 2 ( liner ) , ngc2623 ( liner ) , and ngc3921 ( liner ) . it is difficult to assess what the exact contribution of the nucleus to the integrated luminosities is . the photometric measurements , converted to luminosity densities , are presented in figures [ f : iphot][f : haphot ] . the `` merger stage '' , plotted along the x - axis , refers to the original order of the sequence given by @xcite , where 1 corresponds to ngc4038/39 ( the earliest merger stage ) and 11 to ngc7252 ( the latest merger stage ) . the exact correspondence is given in the figure caption to figure [ f : iphot ] . our observations do not suggest an obvious reordering of this sequence , although we are hampered by our inability to unambiguously identify all of the putative nuclei . our tentative impression is that ngc7764a ( merger stage=4 ) may be interacting with the disturbed galaxy lying off of our image , in which case it belongs near the beginning of the sequence . and ngc3509 ( merger stage=6 ) is not an obvious merger at all . however , we will await nicmos observations of this sequence before reaching any firm conclusions on this matter . we note that none of our conclusions will be affected if these two systems move earlier along or off of the sequence . apart from the large scatter in figure [ f : iphot ] we see that the three latest merger stage systems ( ngc3256 , ngc3921 , and ngc7252 ) have the highest intrinsic luminosity densities . this is true for both apertures that we used ( 100 pc and 1 kpc ) . the range of luminosity density within 100 pc corresponds to a factor of more than 300 . to a large extent this reflects the varying dust extinction in the central region , but partly also the morphology of the nuclear region ( well - defined nuclei are bright , whereas components with no clearly recognizable nucleus are faint ) . with only @xmath14 colors available it is difficult to make a meaningful estimate of dust extinction . but , for example , figure [ f : viphot ] shows that in ngc 2623 the integrated @xmath14 color is very red within a 100 pc radius aperture , with a value around 2.2 . assuming that the true @xmath14 color is 1.4 ( a more typical value seen in the same figure ) , would imply almost @xmath29 = 2 mag of extinction with a typical extinction law of a@xmath30 @xcite . since the extinction in the @xmath10-band is only one tenth of that in the @xmath6-band , our future nicmos images offer some hope of revealing the true location of the nucleus . while the @xmath14 colors for the 100 pc aperture attain values anywhere between 0.8 and 2.4 , the majority of the nuclei have rather typical colors for the centers of spiral nuclei and elliptical galaxies , ranging from 1.2 to 1.5 ( cf . , e.g. , * ? ? ? the presumably latest stage merger , ngc7252 , has colors even bluer than normal ellipticals . however , the morphology of the dust lanes we see in the images suggests that dust obscuration is likely to be significant in many systems of the toomre sequence . the h@xmath0 + [ ] luminosities ( figure [ f : haphot ] ) show a marginal trend of increasing luminosities towards the late - stage mergers , most obviously in the largest 1 kpc aperture . however , the scatter is large , and based on the sample of 15 nuclei it is impossible to establish any definite trends . since mergers are expected to bring material into the center of galaxies and generate star formation there , we also study the nuclear concentration of light and line emission along the merging sequence . we measure the nuclear concentration quantitatively by taking the ratio of the luminosities within 100 pc to that within 1 kpc , for both the @xmath7-band ( figure [ f : irat ] ) and the line emission ( figure [ f : harat ] ) . we discuss the meaning of these ratios briefly in the following section . the nuclear morphologies and central concentration are expected to be strongly affected by the timing and intensity of induced inflows and star formation during the merger . numerical models of interacting galaxies have illustrated one mechanism by which interactions trigger these inflows and nuclear activity . shortly after the galaxies first collide , the self - gravity of each disk amplifies the collisional perturbation into a growing @xmath31 mode @xcite . depending on the structural properties of each galaxy , this mode can take the form of a strong bar , prominent spiral arms , or a pronounced oval distortion @xcite . these non - axisymmetric structures drive shocks in the interstellar medium of the galaxies , leading to a spatial offset between the gaseous and stellar components which robs the gas of angular momentum , and can drive it inwards well before the galaxies ultimately merge @xcite . the onset , strength , and duration of starbursts in interacting pairs can vary widely , depending on the properties of the host galaxies . ultimately , however , any dynamical stability provided by the presence of a central bulge or diminished disk surface density will be overwhelmed by the strong gravitational torques and dissipative shocks during the final merging of the pair , at which point gaseous inflow and central activity should be ubiquitous . once the coalescence of the galaxies is complete , the gravitational potential settles down and gas can resettle into nuclear and/or spatially extended disks , supporting an extended period of relatively quiescent star formation . based on these kinds of models , we might expect to see an evolutionary trend for the nuclear properties of the toomre sequence , from quiescent star formation in the early stages of the sequence to strong nuclear activity in the late stages . however , we see little evidence for such trends in the morphology , luminosity density , or colors of the stellar component , or in the morphology or intensity of the ionized gas emission , apart possibly from the fact that the latest - stage systems have some of the highest broad - band luminosity densities and hints of starbursting central gaseous disk structures . this last suggestion is further strengthened by the trend towards bluer colors among the three latest - stage merger systems in figure [ f : viphot ] . a trend towards bluer colors was also seen in a study of merger remnants by @xcite . furthermore , the latest - stage merger systems have some of the highest nuclear h@xmath0 luminosities ( figure [ f : haphot ] ) and possess some of the largest concentrations of light in the central region ( figure [ f : irat ] ) . such a concentration is also seen to a certain extent in the h@xmath0 + [ ] emission ( figure [ f : harat ] ) . are our observations of the toomre nuclei consistent with the broad picture of interaction - induced nuclear activity in galaxies evidenced in numerical simulations ? in a broad sense , yes , although this consistency is mainly due to the variety of dynamical responses available to interacting systems . the late stage mergers seem to have settled down , showing evidence for nuclear emission - line disks ( ngc2623 , ngc3256 , ngc3921 , and ngc7252 ) , more concentrated luminosity profiles and a trend towards bluer colors . toomre sequence objects at earlier stages show very diverse properties , with no clear trends along the sequence . however , selecting a merging sequence based largely on the large - scale morphology of the tidal tails biases the sample towards a specific type of interaction prograde interactions between disk galaxies but not a specific type of disk galaxy . the galaxies which make up the toomre sequence likely possess a variety of structural properties so that there is no one - to - one correspondence between observed merger stage and nuclear morphology and activity . put differently , the toomre sequence is _ not _ a true evolutionary sequence on anything but the largest scales . instead , it is composed of a variety of galaxies progressing down varied paths of interaction - driven activity . expecting to discern a clear trend in nuclear properties is likely a naive hope . aside from the variety of physical conditions sampled by the toomre nuclei , searching for trends along the toomre sequence is also hampered by a number of observational concerns . first , the orientation of the galaxies with respect to the line - of - sight and to the orbital plane of the merger complicates the interpretation of the images . second , extinction has a large effect on the observed morphologies , colors , and luminosities , as shown in section [ s : mergerstage ] ( e.g. , ngc520 nuc 1 and ngc4676 nuc 1 ) . in this context , we hope to improve upon the interpretation of the galaxy morphologies along the sequence through an analysis of nicmos images of the entire toomre sequence . these images will suffer from much less extinction , and should also reveal the isophotes of the stellar population more clearly , thereby helping us in assessing the inclination of the merger components . the optical images shown here clearly demonstrate that dust in merging galaxies , even on small scales resolved by the _ hst _ , is not patchy enough to enable the identification of the true nuclei in optical bands . in summary , these high - resolution wfpc2 images of the toomre sequence have given us a detailed view of the nuclear regions of interacting and merging galaxies . we have characterized the broad - band and emission - line morphologies of each member of the sequence , and measured the colors and luminosity densities of the nuclei . we find little evidence for clear trends in nuclear properties along the merger sequence , other than a suggestive rise in the nuclear luminosity density in the most evolved members of the sequence . the lack of clear trends in nuclear properties is likely due both to the effects of obscuration and geometry , as well as the physical variety of galaxies involved in the toomre sequence . in subsequent papers we will combine our optical imaging with nicmos imaging and stis spectroscopy of the toomre nuclei to give a detailed picture of the physical conditions in interacting and merging galaxies . we are grateful to dr . brad whitmore for providing us with fully reduced hst wfpc2 @xmath6 and @xmath7 images of ngc3921 , ngc4038/39 , and ngc7252 . we thank dr . nick scoville for providing us with his adaptive filtering code . we are grateful to zoltan levay for help with the mosaiced images , and thank dr . william keel for providing us with a copy of his hst image of ngc6621/22 for help in planning our observations . we thank the anonymous referee for constructive comments that improved the clarity of the paper . support for proposal # 8669 was provided by nasa through a grant from the space telescope science institute , which is operated by the association of universities for research in astronomy , inc . , under nasa contract nas 5 - 26555 . the research described in this paper was carried out , in part , by the jet propulsion laboratory , california institute of technology , and was sponsored by the national aeronautics and space administration.jcm acknowledges support by the nsf through grant ast-9876143 and by a research corporation cottrell scholarship . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . the leda database , operated by centre de recherche astronomique de lyon , is kindly acknowledged .	we report on the properties of nuclear regions in the toomre sequence of merging galaxies , based on imaging data gathered with the _ hubble space telescope _ wfpc2 camera . we have imaged the 11 systems in the proposed evolutionary merger sequence in the f555w and f814w broad - band filters , and in h@xmath0 + [ ] narrow - band filters . the broad - band morphology of the nuclear regions varies from non - nucleated starburst clumps through dust - covered nuclei to a nucleated morphology . there is no unambiguous trend in the morphology with merger stage . the emission - line morphology is extended beyond the nucleus in most cases , but centrally concentrated ( within 1 kpc ) emission - line gas can be seen in the four latest - stage merger systems . we have quantified the intrinsic luminosity densities and colors within the inner 100 pc and 1 kpc of each identified nucleus . we find little evidence for a clear trend in nuclear properties along the merger sequence , other than a suggestive rise in the nuclear luminosity density in the most evolved members of the sequence . the lack of clear trends in nuclear properties is likely due both to the effects of obscuration and geometry , as well as the physical variety of galaxies included in the toomre sequence .	15,486	339
in @xcite , a proof of a formula for the restriction of a discrete series representation ( see @xcite ) of a connected , linear , semisimple lie group to a maximal compact subgroup is given . this formula was first conjectured by blattner . we recall the formula and its context briefly , from the point of view of root system combinatorics . throughout the paper , @xmath3 denotes a semisimple lie algebra over @xmath4 with a fixed cartan subalgebra @xmath5 . let @xmath6 denote the corresponding root system with weyl group @xmath7 . choose a set , @xmath8 , of positive roots and let @xmath9 be the simple roots . let @xmath10 . we assume that there exists a function @xmath11 such that if @xmath12 and @xmath13 then @xmath14 . this map provides a @xmath15-gradation on @xmath16 . we set : @xmath17 given @xmath18 , set @xmath19={\alpha}(h ) x \ ; \forall h \in { \mathfrak}h \}$ ] . let @xmath20 and + @xmath21 . then , @xmath22 will be a reductive symmetric subalgebra of @xmath3 with @xmath23 the corresponding cartan decomposition of @xmath3 . as defined , @xmath5 is a cartan subalgebra for @xmath22 so rank @xmath22 = rank @xmath3 . each equal rank symmetric pair corresponds to at least one @xmath15-gradation in this manner , and conversely . we shall refer to the elements of @xmath24 ( resp . @xmath25 ) as compact ( resp . noncompact ) . the compact roots are a sub - root system of @xmath16 . let @xmath26 , @xmath27 , @xmath28 , and @xmath29 . set @xmath30 where @xmath31 and @xmath32 . if there is no subscript , we mean @xmath33 . we remark that the @xmath15-gradation @xmath34 is determined by its restriction to @xmath35 . furthermore , to any set partition @xmath36 there exists a unique @xmath15-gradation on @xmath16 such that @xmath37 and @xmath38 . we denote the killing form on @xmath3 by @xmath39 , which restricts to a nondegenerate form on @xmath5 . using this form we may define @xmath40 by @xmath41 ( @xmath42 ) , which allows us to identify @xmath43 with @xmath44 . under this identification , we have @xmath45 , where @xmath46 is the simple coroot corresponding to @xmath47 . for each @xmath18 , set @xmath48 ( for @xmath49 ) to be the reflection through the hyperplane defined by @xmath50 . for @xmath51 , let @xmath52 , be the simple reflection defined by @xmath53 . define @xmath54 to be the set of simple roots in @xmath55 and let @xmath56 denote the weyl group generated the reflections defined by @xmath54 . let @xmath57 be the parabolic subgroup of @xmath7 defined by the compact simple @xmath3-roots . note that @xmath58 , but we do not have equality in general . for @xmath59 , set @xmath60 . note that there is also a length function on @xmath7 ( denoted @xmath61 ) but @xmath62 refers to @xmath56 . a weight @xmath49 is said to be @xmath22-dominant ( resp . @xmath3-dominant ) if @xmath63 for all @xmath64 ( resp @xmath65 ) . a weight @xmath66 is @xmath3-regular ( resp . @xmath22-regular ) if @xmath67 for all @xmath18 ( resp . @xmath68 ) . the integral weight lattice for @xmath3 is denoted by the set + @xmath69 similarly we let @xmath70 denote the abelian group of integral weights for @xmath22 corresponding to @xmath54 . let the set of @xmath22- and @xmath3-dominant integral weights be denoted by @xmath71 and @xmath72 respectively . to each element @xmath73 ( resp . @xmath72 ) , let @xmath74 ( resp . @xmath75 ) denote the finite dimensional representation of @xmath22 ( resp . @xmath3 ) with highest weight @xmath76 . next , let @xmath77 denote the @xmath78-partition function . that is , if @xmath79 then @xmath80 is the number of ways of writing @xmath81 as a sum of noncompact positive roots . put other way : there exists an algebraic torus , @xmath82 , such that to each @xmath83 there corresponds a linear character of @xmath82 , denoted @xmath84 , with differential @xmath85 . thus , @xmath86 defines the coefficients of the product : @xmath87 finally , we define the blattner formula . for @xmath88 , @xmath89 it is convenient to introduce the notation @xmath90 for @xmath59 and @xmath49 . it is easy to see that @xmath91 . since for all @xmath92 there exists @xmath93 such that @xmath94 , we will assume that @xmath95 . historically , blattner s formula arises out of the study of the discrete series and its generalizations ( see @xcite ) . [ thm_hs ] assume @xmath96 is @xmath3-dominant and @xmath3-regular . then , @xmath97 is the multiplicity of the finite dimensional @xmath22-representation , @xmath74 , in the discrete series representation of @xmath0 with harish - chandra parameter @xmath98 . in this paper , we do not impose the @xmath3-dominant regular condition on @xmath99 . this is natural from the point of view of representation theory as it is related to the coherent continuation of the discrete series ( see @xcite , @xcite and @xcite ) . from our point of view , the goal is to understand the blattner formula in as combinatorial fashion as possible . thus it is convenient to introduce the following generating function : for @xmath95 we define the formal series : @xmath100 the main result of this paper is proposition [ prop_main ] of section [ sec_main ] , which states : for @xmath95 , @xmath101 where @xmath102 denotes the character of @xmath74 . of particular interest are the cases where @xmath103 , which we address in section [ sec_sym ] . from the point of view of representation theory these include , for example , the holomorphic and borel - de siebenthal discrete series ( see @xcite ) . more recently , the latter has been addressed in @xcite . the blattner formula for the case of @xmath104 is often particularly difficult to compute explicitly when compared to , say , the cases corresponding to holomorphic discrete series . the @xmath105 case corresponds to the _ generic _ discrete series of the corresponding real semisimple lie group . in section [ sec_g2 ] we explore this situation in some detail for the lie algebra @xmath2 . finally , in light of theorem [ thm_hs ] one may observe that if @xmath95 and @xmath99 is @xmath3-dominant regular then @xmath106 . our goal is to investigate the positivity of blattner s formula using combinatorial methods . of particular interest is the positivity when we relax the @xmath3-dominance condition on @xmath99 . some results in this direction are suggested by the recent work of penkov and the second author ( see @xcite ) . in section [ sec_sym ] we prove the existence of a skew symmetry of blattner s formula that exists whenever @xmath103 . thus , the condition that @xmath104 is necessary for @xmath106 for all @xmath95 and @xmath83 . in the situation where @xmath107 we introduce the following : we say that a semisimple lie algebra is @xmath108-positive if the blattner formula corresponding to the @xmath15-gradation with @xmath104 has the property that : @xmath109 the terminology stems from the fact that a simple lie algebra is @xmath108-positive if and only if the coefficients of @xmath110 are non - negative for all @xmath95 . since the character of @xmath74 can be written as a non - negative integer combination of characters of @xmath82 , we have that @xmath110 has non - negative integer coefficients if @xmath111 does . thus the question of @xmath108-positivity reduces to the case of @xmath112 . in section [ sec_positive ] , it is shown that the only @xmath108-positive simple lie algebras are of type @xmath113 , @xmath114 , @xmath115 , @xmath116 , @xmath117 , @xmath118 and @xmath2 . we prove this result by examining the coefficients of @xmath111 . [ prop_main ] for @xmath95 , @xmath119 where @xmath120 denotes the character of @xmath74 . from the definition of blattner s formula we have : @xmath121 first we make the substitution , @xmath122 , and reorganize the sum : @xmath123 as is well known , the character may be expressed using weyl s formula ( see @xcite ) as in the following : @xmath124 the result allows us to compute the blattner formula as follows : @xmath125 let @xmath126 and we obtain : @xmath127 thus for @xmath95 and @xmath83 we have @xmath128 note that the numbers @xmath129 are weight multiplicities for the representation @xmath130 . the main result of this section is : [ thm_sym ] for @xmath131 : @xmath132 although this is well known to experts , we include our proof as it requires very little technical machinery . for @xmath133 and @xmath49 let @xmath134 . [ lemma_q ] if @xmath135 then @xmath136 . it is enough to show @xmath137 which follows from the following calculation : @xmath138 [ lemma_roots ] for all @xmath139 , @xmath140 . note that @xmath141 is the weyl group of a reductive levi factor @xmath142 of a parabolic subalgebra @xmath143 . we have a generalized triangular decomposition @xmath144 with @xmath145 . the noncompact root spaces contained in @xmath146 are positive . furthermore , all noncompact positive roots spaces are contained in @xmath146 because @xmath147 . the lie algebra @xmath148 is an @xmath142-module and therefore the weights are preserved by @xmath141 . it is this fact that implies that @xmath141 takes positive noncompact roots to positive roots . we now need to show that @xmath141 takes noncompact roots to noncompact roots . for roots @xmath149 and @xmath68 , we have the formula @xmath150 with @xmath151 an integer . thus , the reflection of a noncompact root across a hyperplane defined by a compact root is noncompact . the reflections generate @xmath141 . we now prove the main proposition of this section . let @xmath131 : @xmath152 by definition , @xmath153 . combining this with the definition of the dot " action we obtain : @xmath154 next , we use the fact that @xmath155 using lemmas [ lemma_q ] and [ lemma_roots ] . the rest is a calculation . the following is a complete calculation of @xmath111 for the case of the lie algebra @xmath157 when @xmath104 . let @xmath158 and @xmath159 be a choice of noncompact simple roots for @xmath2 with @xmath158 long and @xmath159 short . the compact positive roots are @xmath160 , while the noncompact positive roots are @xmath161 . denote the @xmath162-partition function by @xmath163 . we have : @xmath164 let @xmath165 and @xmath166 . thus , @xmath167 let the simple reflection corresponding to @xmath158 ( resp . @xmath159 ) be @xmath168 ( resp . @xmath169 ) . we have four terms in blattner s formula for @xmath170 : @xmath171 and our goal will be to close the sum @xmath172 . we will do this by multiplying by @xmath173 and summing over @xmath85 for each of the four terms . observe that : @xmath174 thus , @xmath175 . next consider the sum : @xmath176 we make the substitution @xmath177 so that the above sum becomes : @xmath178 thus the above sum is equal to @xmath179 , which we denote by @xmath180 . similarly , we set @xmath181 and @xmath182 , and @xmath183 . thus , @xmath184 now we write the above in terms of @xmath185 and @xmath186 . note that @xmath187 , @xmath188 , @xmath189 , and from these we can easily see : @xmath190 putting everything together we see @xmath191 . or equivalently , if @xmath192 then the value of @xmath193 is the coefficient of @xmath194 in : @xmath195 note that in the latter expression , it is clear that the coefficients are the series expansion are positive . the positivity of the coefficients of @xmath196 follows from the positivity of the coefficients of @xmath111 . the question of positivity for a general semisimple lie algebra will be addressed in section [ sec_positive ] . it is important to note that as we change @xmath197 the @xmath111 changes as well . for example , when @xmath198 , @xmath199 which we can easily see does _ not _ have non - negative integer coefficients . we leave it as an exercise to the reader to compute @xmath111 when @xmath200 . the generating function for other @xmath201 involve multiplying the above product by a polynomial in @xmath202 that represents the character of the corresponding irreducible finite dimensional representation of @xmath203 . one can plot the coefficients of our formal series , as we do next . in all pictures we have labeled the scale on the axes and normalized the short root ( @xmath159 ) to have length 1 and be positioned at 3 oclock . the long root @xmath158 is at 10 oclock . the first two images ( figures 1 and 2 ) are for the case with @xmath104 ( generic ) and correspond to @xmath112 and @xmath76 the highest weight of the standard @xmath204-dimensional representation of @xmath205 . we also display the same two @xmath76 s in the case when @xmath206 ( borel - de siebenthal ) in figures 3 and 4 . these latter two figures clearly display a skew - symmetry addressed section [ sec_sym ] . we have the following : it is a consequence of the main proposition that for any rank @xmath208 simple lie algebra @xmath3 we may express @xmath111 as a rational function in @xmath209 where @xmath210 ( recall that @xmath53 is the @xmath211-simple root for @xmath3 ) . for any @xmath212 , if we set @xmath213 in @xmath111 then the resulting expression is @xmath111 for a semisimple subalgebra of @xmath3 . it is not hard to see that this subalgebra is the levi factor of the maximal parabolic of @xmath3 corresponding to @xmath214 . more generally , let @xmath215 . if we set @xmath216 in @xmath111 for all @xmath217 then the resulting expression is @xmath111 for the levi subalgebra , @xmath218 , of the corresponding parabolic . note that the terms in the series expansion for @xmath111 for @xmath218 are also terms in the series expansion of @xmath111 for @xmath3 . thus a negative coefficient in the former implies a negative coefficient in the latter . the proof of proposition [ prop_seven ] involves a cases - by - case analysis using the main proposition expressing @xmath111 as a product , lemma [ lem_levi ] and the computer algebra package , maple . recall that the @xmath108-positivity of @xmath2 was proved in section [ sec_g2 ] . for @xmath219 , we have : @xmath220 for @xmath221 : @xmath222 and for @xmath223 : @xmath224 the partial fraction on the right of each of these examples establishes that the coefficients are indeed positive . next we consider @xmath225 where we have a negative result . first we note : @xmath226 for @xmath225 . we expand in a formal power series and observe that the coefficient of @xmath227 is @xmath228 . this means that @xmath108-positivity fails for this example . we then also see failure of @xmath108-positivity for any simple lie algebra which has @xmath229 as a levi factor of a parabolic . thus , we exclude all higher rank type a examples as well as @xmath230 ( @xmath231 ) , @xmath232 ( @xmath233 ) , @xmath234 ( @xmath235 ) , @xmath236 , @xmath237 and @xmath238 . now consider @xmath240 : @xmath242 upon expansion we see that the coefficient of @xmath243 is @xmath228 . thus , we may exclude this and higher rank type @xmath244 examples as they have @xmath245 as the levi factor of a parabolic . in particular , we may exclude @xmath246 , which is the only type @xmath244 example that has not been excluded yet . we also may exclude @xmath247 for the same reason . we must examine @xmath111 for @xmath248 and @xmath249 as these are the only examples not yet addressed . for @xmath250 ( ie : @xmath251 ) , we have : @xmath252 the coefficients are positive as the above expression is equal to : @xmath253	let @xmath0 be a connected , semisimple lie group with finite center and let @xmath1 be a maximal compact subgroup . we investigate a method to compute multiplicities of @xmath1-types in the discrete series using a rational expression for a generating function obtained from blattner s formula . this expression involves a product with a character of an irreducible finite dimensional representation of @xmath1 and is valid for any discrete series system . other results include a new proof of a symmetry of blattner s formula , and a positivity result for certain low rank examples . we consider in detail the situation for @xmath0 of type split @xmath2 . the motivation for this work came from an attempt to understand pictures coming from blattner s formula , some of which we include in the paper .	5,121	209
after many years of pure academic research , the casimir effect @xcite is presently of much interest in connection with applications in nanomechanical devices @xcite , noncontact friction @xcite , carbon nanotubes @xcite , bose - einstein condensation @xcite and for constraining predictions of modern unification theories of fundamental interactions @xcite . these areas of application were made possible by extensive experimental investigation of the casimir force @xcite and the generalization to real materials of field - theoretical methods which were applicable to only idealized boundaries ( see reviews @xcite ) . the basic theory of the casimir and van der waals forces at nonzero temperature proposed by lifshitz @xcite allows one to calculate all quantities of physical interest using the dielectric permittivity of boundary materials along the imaginary frequency axis . this theory was originally developed for the configuration of two semispaces and was later extended for any layer structure @xcite . using the proximity force theorem @xcite , lifshitz - type formulas for the configuration of a sphere or a cylinder above a plate were obtained and successfully used for the interpretation of experimental data @xcite . for a long time , the lack of exact results for these configurations made it possible to question the validity of the comparison of experiment and theory based on the proximity force theorem . recently , however , both the exact analytical @xcite and numerical @xcite results for the casimir force between a sphere ( cylinder ) and a plate were obtained demonstrating that at small @xmath0 the corrections to the proximity force theorem for both configurations are in fact less than @xmath1 ( @xmath0 is the separation between a cylinder or a sphere of radius @xmath2 and a plate ) , i.e. , less than it was supposed in the comparison of experiment with theory . thus , the use of the proximity force theorem in refs . @xcite and below is substantiated on the basis of first principles of quantum field theory . the vital issue in many applications of the casimir effect is how to control the magnitude of the force by changing the parameters of the system . in this respect the possibility that the casimir force can change sign from attraction to repulsion depending on system geometry is of much importance . it may be used to prevent collapse of small mechanical elements onto nearby surfaces in nanodevices @xcite . however , the casimir repulsion has yet to be observed experimentally . an alternative method to control the magnitude of the casimir force is to change the material properties of the interacting bodies . in ref . @xcite the casimir force was measured acting between a plate and a sphere coated with a hydrogen - switchable mirror that become transparent upon hydrogenation . despite expectations , no significant decrease of the casimir force owing to the increased transparancy of the plates was observed . the negative result is explained by the lifshitz theory which requires the change of the reflectivity properties within a wide range of frequencies in order to markedly affect the magnitude of the casimir force . this requirement is not satisfied by the hydrogenation . all modern experiments on the measurement of the casimir force mentioned above @xcite used metallic test bodies . metallic surfaces are necessary to reduce and compensate the effects of residual charges and work function differences . it is , however , hard to modify their reflectivity properties over a sufficiently wide range of frequencies . the appropriate materials for the control , modification and fine tunning of the casimir force are semiconductors . the reflectivity properties of semiconductor surfaces can be changed in a wide frequency range by changing the carrier density through the variation of temperature , using different kinds of doping or , alternatively , via the illumination of the surface with laser light . at the same time , semiconductor surfaces with reasonably high conductivity avoid accumulation of excess charges and , thus , preserve the advantage of metals . in addition as semiconductors are the basic fabrication materials for nanotechnology , the use of semiconductor surfaces for the control of the casimir force will lead to many applications . the modification of the casimir force between a gold coated plate and sphere , attached to the cantilever of an atomic force microscope ( afm ) , through the variation of temperature was considered in ref . while changing the temperature to modify the carrier density in semiconductors is a good method in theory , it leads @xcite to large systematic errors in the measurement setup using the afm . in ref . @xcite the casimir force between a gold coated sphere and a single crystal b - doped si plate was measured in high vacuum . it was found that the force between a metal and a semiconductor decreases with increase of separation more quickly than between two metals . in ref . @xcite the experimental data for the casimir force between a gold coated sphere and b - doped si plate were compared with two different theoretical computations , one made for the b - doped si used and another one for high - resistivity si . it was shown that the computation using the tabulated optical data for high - resistivity si is excluded by experiment at 70% confidence while the theoretical results computed for the plate used in experiment are consistent with data . in ref . @xcite the difference in the casimir forces between a gold coated sphere and two p - doped si plates with different charge - carrier densities was directly measured at a 95% confidence level . this demonstrates that the change of carrier density due to doping leads to noticeable modification of the casimir force . the most suitable method to change the carrier density in semiconductors is through the illumination of the surfaces by laser light ( see , e.g. , @xcite ) . an early attempt to measure the van der waals and the casimir forces between semiconductors and modify them with light was reported in ref . attractive forces were measured between a glass lens and a si plate and also between the glass lens coated with amorphous si and the si plate . however , the glass lens is an insulator and therefore the electric forces such as due to work function potential differences could not be controlled . this might also explain that no force change occured on illumination at separations below 350 nm @xcite where it should have been most pronounced . the present paper contains the detailed results of our experiments on the modification of the casimir force by the irradiation of a si membrane with laser pulses . the first observation of this effect at only one absorbed power was briefly reported in ref . @xcite . here we report new measurements performed at different absorbed powers and the theoretical analysis on the accuracy of the obtained results and on the comparison of experiment with theory . in our experiments the carrier density in the si membrane is changed by the incident light , and the difference in the casimir force acting between that membrane and the gold coated sphere in the presence and in the absence of light is measured . the experimental error of difference force measurements for the different absorbed powers determined at a 95% confidence level varies between 10 to 20% at a separation of 100 nm and increases with the increase of separation . the measurement data collected at different powers of the incident light at the laboratory temperature @xmath3k were compared with the lifshitz theory at both zero and at laboratory temperatures . the data are shown to be consistent with theory at laboratory temperature if in the absence of light the static dielectric permittivity of si is assumed to be finite . the lifshitz theory at laboratory temperature taking account of the dc conductivity of high - resistivity si in the absence of light is excluded experimentally at a 95% confidence level . thus , our experiments not only demonstrate the modification of the casimir force through the irradiation of a semiconductor surface , but also lead to the important result that the inclusion of zero - frequency conductivity of high - resistivity si in the model of dielectric response results in a contradiction between the lifshitz theory at laboratory temperature and experiment . this contradiction is caused by different contributions from the reflection of the transverse magnetic mode on a si surface at zero frequency in the absence and in the presence of conductivity . the obtained conclusion supports recent theoretical results that the inclusion of dielectric dc conductivity for the dielectric - dielectric @xcite and dielectric - metal @xcite configurations at nonzero temperature leads to contradiction between the lifshitz theory and the nernst heat theorem , and thus such inclusion is impermissible . at the same time , the experimental data are shown to be consistent with the lifshitz theory at zero temperature , irrespective of whether the dc conductivity of high - resistivity si is included or not . the paper is organized as follows . in sec . ii the experimental setup and sample preparation are described . iii contains the description of the measurement procedure and obtained experimental results . this includes the calibration of the setup , the measurement of the lifetime of excited carriers , the measurement of the difference in the casimir force when the light is on and off , and the analysis of the experimental errors . in sec . iv the experimental results are compared with the theory . here the difference in the casimir force with and without incident laser light is calculated and the theoretical errors are analyzed . by combining the experimental and theoretical errors , the quantitative measure of agreement between experiment and theory at 95% confidence is presented . v analyzes the role of the dc conductivity of high - resistivity si in the casimir force . vi contains our conclusions and discussion . here we discuss the experimental setup used to demonstrate the modification of the casimir force through the radiation induced change in the carrier density . the general scheme of the setup is shown in fig . [ setup ] . a high vacuum based afm was employed to measure the change in the casimir force between a gold coated sphere of diameter @xmath4 m and a si membrane ( coloured black ) in the presence and in the absence of incident light . an oil free vacuum chamber with a pressure of around @xmath5torr was used . the polystyrene sphere coated with a gold layer of @xmath6 nm thickness was mounted at the tip of a 320@xmath7 m conductive cantilever ( see fig . [ setup ] ) . the si membrane ( see below for the process of its preparation ) was mounted on top of a piezo which is used to change the separation distance @xmath0 between the sphere and the membrane from contact to 6@xmath7 m . the excitation of the carriers in the si membrane was done with 5ms wide light pulses ( 50% duty cycle ) . these pulses were obtained from a cw ar ion laser light at 514 nm wavelength modulated at a frequency of 100hz using an acousto - optic - modulator ( aom ) . the aom is triggered with a function generator . the laser pulses were focused on the bottom surface of the si membrane . the gaussian width of the focused beam on the membrane was measured to be @xmath8 mm . the cantilever of the afm flexes when the casimir force between the sphere and the membrane changes depending on the presence or the absence of incident light on the membrane . this cantilever deflection is monitored with a 640 nm beam from an additional laser ( see fig . [ setup ] ) reflected off the top of the cantilever tip . an optical filter was used to prevent the interference of the 514 nm excitation light with the cantilever deflection signal . the transmission of this filter at 514 nm is 0.001% . including the less than 1% transmission through the si membrane and the diode solid angle of @xmath9 , the impact of the 514 nm light leakage leads to less than @xmath10pn changes in the force difference . these changes are negligibly small as compared with the measured cantilever deflection signal . the latter leads to a difference signal between the two photodiodes . the resulting modification of the casimir force in response to the carrier excitation is measured with a lock - in amplifier . the same function generator signal used to generate the ar laser pulses is also used as a reference for the lock - in amplifier . the most important part of the setup is the si membrane . it should be sufficiently thin and of high resistivity to ensure that the density of charge carriers increases by several orders of magnitude under the influence of the laser pulses . the si membrane should be thick enough to make negligible the photon pressure of the transmitted light , as the illumination is incident on the bottom surface of the membrane ( see sec . ivb for details ) . the thickness of the si membrane has to be greater than 1@xmath11 m , i.e. , greater than the optical absorption depth of si at the wavelength of the laser pulses . fabrication of the few micrometer thick si membrane with the necessary properties is described below . a commercial si grown on insulator wafer ( soi ) was used as the initial product . the insulator in this case is sio@xmath12 which is the native oxide of si and thus leads to only small reductions of the excited carrier lifetime in si . a layout of the wafer is shown in fig . 2 . the wafer consists of a si substrate of 600@xmath7 m thickness and a si top layer of 5@xmath7 m thickness ( both are single crystals and have a @xmath13 crystal orientation ) with the buried intermediate sio@xmath12 layer of thickness 400 nm ( see fig the si is p - type doped with relatively high nominal resistivity of about 10@xmath14 cm . the corresponding carrier density is equal to @xmath15 @xcite . the thickness of the si substrate is reduced to about 200@xmath7 m through mechanical polishing . then , after rca cleaning of the surface , the wafer is oxidized at @xmath16k in a dry o@xmath12 atmosphere for a duration of 72 hours . as a result , in addition to the buried sio@xmath12 layer , a thermal oxide layer with a thickness of about 1@xmath7 m is formed on both ( bottom and top ) sides of the wafer ( fig . this oxide layer serves as a mask for subsequent tetra methyl ammonium hydroxide ( tmah ) etching of the si . first , a hole with the diameter of 0.85 mm is etched with hf in the center of the bottom oxidation layer ( fig . this exposes the si substrate . next , tmah is used at 363k to etch the si substrate through the hole formed in the oxide mask ( fig . note that tmah selectively etches si as its etching rate for si is 1000 times greater than for sio@xmath12 . tmah etching leads to the formation of a hole through the si substrate . given the selectivity of the etching , the buried 400 nm oxide is the stop etch layer . finally , all the thermal oxidation layers and buried oxidation layer in the hole are etched away in hf solution to form a clean si membrane over the hole as in fig . the thickness of this membrane was measured to be @xmath17 m using an optical microscope . in order for voltages to be applied to the si membrane , an ohmic contact is formed by a thin film of au deposited on the edge of the membrane followed by annealing at 673k for 10 min . the si membrane was cleaned with nanostrip and then passivated by dipping in 49% hf for 10s . the passivated si membrane was then mounted on top of the piezo as described above . all calibrations and other measurements are done at the same period of time as the measurement of the difference of casimir forces and in the same high vacuum apparatus . the calibration of the deflection signal of the cantilever from the photodiodes , @xmath18 , and the determination of the separation on contact and residual potential difference between the gold coated sphere and si membrane is done by measuring the distance dependence of an applied electrostatic force . for this purpose the same function generator ( see fig . [ setup ] ) is used for applying voltages to the membrane . for an attractive force @xmath19 and can be measured either as a current or a voltage . in addition , a small correction has to be applied to the separation distance between the gold sphere and the si membrane due to the movement of the cantilever . the actual separation distance @xmath0 between the bottom of the sphere and the membrane is given by @xmath20 here @xmath21 is the distance moved by the piezo , @xmath22 is the deflection coefficient in units of nm per unit deflection signal , and @xmath23 is the average separation on contact of the gold surface and si membrane . @xmath23 is nonzero due to the stochastic roughness of the surfaces . the complete movement of the piezo was calibrated using a fiber optic interferometer . to extend and contract the piezo , continuous triangular voltages between 0.010.02hz are applied to it . given that the experiment is done at room temperature , applying of static voltages would lead to piezo creep and loss of position sensitivity . the deflection coefficient @xmath22 can also be measured by the application of electrostatic forces between the sphere and the membrane . in our measurements , the gold sphere was kept grounded . the electric contact to the sphere was accomplished by applying a very thin gold coating to the cantilever . the electrostatic force between the sphere and the membrane is given by @xcite @xmath24 where @xmath25 is the voltage applied to the si membrane , @xmath26 is the residual potential difference between the grounded sphere and membrane , @xmath27 , and @xmath28 is the permittivity of vacuum . the nonzero value of @xmath0 at contact , @xmath23 , is due to the surface roughness . in the complete measurement range of the electrostatic force from contact to 6@xmath11 m , eq . ( [ eq2 ] ) can be rearranged to the following more simple form within the limits of relative error less than @xmath9 @xcite : @xmath29 where @xmath30 first , 30 different dc voltages between 0.65 to 0.91v are applied to the si membrane . the cantilever deflection signal is measured as a function of the distance . the 0.02hz triangular wave was applied to the piezo to change the distance between the sphere and the membrane over a range of 6@xmath11 m . larger applied voltages lead to more cantilever deflection and , according to eq . ( [ eq1 ] ) , to a contact of the two surfaces at larger @xmath21 . the dependence of @xmath21 at contact of the sphere and the membrane on the applied voltage can then be used to measure the deflection coefficient @xmath22 . in order to determine the contact of the two surfaces precisely , 32768 data points at equal time intervals were acquired for each force measurement ( i.e. , the interval between two points was about 0.18 nm ) . in cases , where the contact point was between two neighboring data points , a linear interpolation was used to identify the exact value . the deflection coefficient was found to be @xmath31 nm per unit deflection signal . the difference in the value of @xmath22 from previous measurements @xcite is due to the use of the 514 nm filter which reduced the cantilever deflection signal . the obtained value of @xmath22 was used to correct the separation distance in all measurements in accordance with eq . ( [ eq1 ] ) . the electrostatic force resulting from the application of the dc voltages is also used in the determination of the separation on contact of the two surfaces . the fit of the experimental force - distance relation to the theoretical eq . ( [ eq3 ] ) is done as outlined in our previous work @xcite . the separation distance on contact was determined to be @xmath32 nm . the uncertainty in the quantity @xmath33 from eq . ( [ eq1 ] ) was found to be 1 nm . this leads to the same error in absolute separations @xmath34 nm because the error in piezo calibration is negligibly small . for the calibration of the deflection signal and the determination of the residual potential difference between the two surfaces , an improved method , rather than simple application of the dc voltages to the membrane was used . this was done to avoid systematic errors due to scattered laser light . in addition to the application of the dc voltage to the membrane , described above , square voltage pulse of amplitudes from 1.2 to 0.6v and time interval corresponding to a separation distance between 1 to 5@xmath11 m was also applied to the membrane . fig . [ deflsignal ] shows the deflection signal of the cantilever in response to both the applied dc voltage and the square pulse as a function of the separation distance between the gold sphere and si membrane . by measuring only the difference in signal during the pulse allows one to avoid the need for a background subtraction . the fit of the difference signal to eq . ( [ eq3 ] ) leads to the value of the signal calibration constant @xmath35nn per unit deflection signal . the same fit was used to determine the residual potential difference between the sphere and the membrane which was found to be @xmath36v . the large width of the pulse applied in addition to the dc voltage allowed confirmation of the distance independence of the obtained values of the calibration constant and the residual potential difference . an independent measurement of the lifetime of the carriers excited in the si membrane by the pulses from the ar laser was performed . for this purpose a non - invasive optical pump - probe technique was used @xcite . the same si membrane and ar laser beam modulated by the aom at 100hz to produce 5ms wide square light pulses , as used in the casimir force measurement , were employed as the sample and the pump , respectively . the diameter of the pump beam on the sample was measured to be @xmath37 mm . a cw beam with a 1mw power at a wavelength of 1300 nm was used as a probe . the probe beam photon energy is below the band gap energy of si and is thus not involved in carrier generation . this beam was focused to a gaussian width size @xmath38 mm . thus the focal spot size of the probe beam is much smaller than the focal spot size of the pump light . this allowed one to measure the lifetime in a homogeneous region of excited carriers . the change in the reflected intensity of the probe beam in the presence and in the absence of ar laser pulse was detected with a ingaas photodiode . the change in reflected power of the probe beam was monitored as a function of time in an oscilloscope and found to be consistent with the change of carrier density . near normal incidence for the pump and probe beams was used , with care taken to make sure that the ingaas photodiode was isolated from the pump beam . the time decay of the reflected probe beam in response to the square ar light pulses is shown in fig . [ lifetime ] . the change of the reflectivity of the probe is fit to an exponential of the form @xmath39 where @xmath40 is the effective carrier lifetime . by fitting the whole 5ms decay of the change in reflected power , the effective excited carrier lifetime was measured to be @xmath41ms . note that this time represents both surface and bulk recombination and is consistent with that expected for si . some dependence of the lifetime of the excited carriers on their concentration was observed . in the first 0.5ms , while the concentration is still high enough , the average value of the excited carrier lifetime was measured to be @xmath42ms . the measured values of the carrier lifetime will be used in sec . iva in the theoretical computations of the casimir force differences for the comparison with several measurements having varying power of ar laser . here we present the determination of the difference in the casimir force resulting from the irradiation of the si membrane with 514 nm laser pulses . in fact it is the difference in the total force ( casimir and electric ) which is measured . as was indicated above , even with no applied voltages there is some residual potential difference @xmath26 between the sphere and the membrane . the preliminary value of @xmath26 was determined during the calibration of the setup in the absence of laser pulses . in the presence of pulses ( even during the dark phases of a pulse train ) the values of the residual potential difference can be different . we represent these residual potential differences during the bright and dark phases of a laser pulse train ( the latter is not exactly equal to the one determined in calibration ) @xmath43 and @xmath26 , respectively . during the bright phases of the pulse train we apply to the si membrane the voltage @xmath44 and during the dark phases the voltage @xmath25 . using eq . ( [ eq3 ] ) for the electric force , we can represent the difference in the total force ( electric and casimir ) for the states with and without carrier excitation in the following form : @xmath45 + \delta f_c(z ) . \label{eq4}\ ] ] here @xmath46 is the difference in the casimir force and @xmath47 is the casimir force with ( without ) light . the difference in the total force in eq . ( [ eq4 ] ) was measured by the lock - in amplifier with an integration time constant of 100ms which corresponds to a bandwidth of 0.78hz . the measurement procedure is described below . first we kept @xmath48 and changed @xmath44 . the parabolic dependence of @xmath49 on @xmath44 in eq . ( [ eq4 ] ) was measured at different separations @xmath0 . care should be taken to apply only small voltage amplitudes ( up to a few tens of mv ) so as to keep the space charge region negligible . at every measured separation distance @xmath49 is plotted as a function of @xmath44 . as is seen from eq . ( [ eq4 ] ) , the value of @xmath44 where the parabola reaches a maximum is @xmath43 [ recall that @xmath50 . in this way the value @xmath51v was found and shown to be independent of the separation from 100 to 500 nm where the difference in the casimir force can be measured . next we kept @xmath52 , changed @xmath25 and measured the parabolic dependence of @xmath49 on @xmath25 at different separations . the value of @xmath25 where parabolas reach minima is @xmath53v . these values of the residual potential difference between the sphere and the membrane in the presence and in the absence of excitation light were substituted in eq . ( [ eq4 ] ) . the small change of around 78mv in the residual potential difference between the sphere and the membrane in the presence and in the absence of excitation light is primarily due to the screening of surface states by few of the optically excited electrons and holes . the above small value is equal to the change in band bending at the surface . it is consistent with the fact that almost flat bands are obtained at the surface with the surface passivation technique used ( see , e.g. , @xcite ) . then other voltages ( @xmath54 ) were applied to the si membrane and the difference in the total force @xmath49 was measured as a function of separation . data were collected from contact at equal time intervals corresponding to 3 points per 1 nm ( i.e. , in 1209 points within the separation interval from 100 to 500 nm ) . from these measurement results , the difference in the casimir force @xmath55 was determined from eq . ( [ eq4 ] ) . this procedure was repeated with some number of pairs ( @xmath56 ) of different applied voltages ( @xmath54 ) and at each separation the mean value @xmath57 was found . in fig . [ result1 ] the experimental data for @xmath57 as a function of separation are shown by dots for different absorbed laser powers : @xmath58mw ( @xmath59 ) , 8.5mw ( @xmath60 ) , 4.7mw ( @xmath61 ) in figures a , b , and c , respectively . the corresponding incident powers were 15.0 , 13.7 and 7.6mw , respectively . as expected , the magnitude of the casimir force difference has the largest values at the shortest separations and decreases with the increase of separation . it also decreases with the decrease of the absorbed laser powers ( the solid , short- and long - dashed lines in fig . [ result1 ] are explained in sec . iv devoted to the comparison with theory ) . now we proceed with the analysis of the experimental errors . the variance of the mean difference in the casimir force is defined as @xmath62 ^ 2\right\}^{1/2 } , \label{eq6}\ ] ] where @xmath63 is the number of point in one set of measurements changing from 1 to 1209 , @xmath64 is the number of the pair of the applied voltages . using student s @xmath65-distribution with a number of degrees of freedom @xmath66 ( or 40 and 32 for the measurements with different absorbed powers ) and choosing @xmath67 confidence , we obtain @xmath68 and @xmath69 . thus , the absolute random error in the measurement of the difference casimir force is given by @xmath70 in this experiment the random error is separation dependent . it is presented in fig . [ random ] as a function of separation for the three different measurements with different absorbed laser powers ( lines a , b , and c correspond to decreasing power indicated above ) . as is seen from fig . [ random ] , the random error is rather different for different measurements . it is the lowest for measurement ( b ) which was done with 8.5mw absorbed power . in this measurement the random error decreases from 0.32pn at @xmath71 nm to 0.23pn at @xmath72 nm and preserves the latter value at larger separations . the main systematic error is due to the instrumental noise and is equal to @xmath73pn independent of separation . the systematic error determined from the resolution error in data acquisition , @xmath74pn , also does not depend on separation . the calibration error , @xmath75 , depends on separation and is equal to 0.6% of the measured difference in the casimir force . these systematic errors are random quantities characterized by a uniform distribution . they can be combined at a given confidence probability @xmath76 with the help of statistical criterion @xcite @xmath77 ^ 2}\right\ } , \label{eq7a}\ ] ] where @xmath78 is a tabulated coefficient . in our experiment there are @xmath79 systematic errors listed above and at @xmath67 ( 95% confidence level ) @xmath80 . as a result , from eq . ( [ eq7a ] ) we arrive at the total systematic error for all three measurements varying from 0.092 to 0.095pn . the total experimental error of the force difference , @xmath81 , at 95% confidence can be found by the combination of random and systematic errors . this is done using the statistical rule described in ref . @xcite and applied to the casimir force measurements in refs . @xcite . according to this rule , the total error is equal to the random one if , as is the case in our experiments , the inequality @xmath82 is satisfied . thus , the total experimental error in the values of @xmath55 for all three measurements as a function of the separation is presented in fig . [ random ] . as a result , the relative experimental error changes from 10 to 20% at a separation @xmath71 nm and from 25 to 33% at a separation @xmath83 nm for different absorbed laser powers . this allows us to conclude that the modulation of the dispersion force with light is demonstrated at a high reliability and confidence . the observed effect can not be due to the mechanical motion of the membrane . this is because membrane movement due to heating ( in our case less than @xmath84c ) would lead to a different force - distance relationship for both electrostatic force and the casimir force in disagreement with our observation and the confirmation of the distance independence of @xmath26 and @xmath43 . the temperature rise of less than @xmath85c is estimated based on the net thermal energy increase in the si membrane . the absorption of photons during the course of the optical pulse increases the thermal energy of the membrane , while conductive and radiative heat outflow to the si around the membrane and surrounding leads to a decrease in its thermal energy . the net change results in the less than @xmath85c . the latter would lead to a negligible less than @xmath10 relative expansion in the diameter of the membrane . in order to account for roughness , the surface topography of the sphere and membrane was characterized using the afm . images resulting from the surface scan of the gold coating on the sphere demonstrate stochastically distributed roughness peaks with heights up to 32 nm . table i contains the fractions @xmath86 of the gold coating with heights @xmath87 ( @xmath88 ) . the surface scan of si surface demonstrates much smoother relief with maximum heights equal to 1.68 nm . the fractions @xmath89 of the si surface with heights @xmath90 ( @xmath91 ) are presented in table ii . the roughness data are used in sec . iv in theoretical computations . the casimir force acting between a large gold sphere of radius @xmath2 and a plane si membrane can be calculated by means of the lifshitz formula @xcite , along with the use of the proximity force theorem @xcite @xmath92\right . \nonumber \\ & & \phantom{aaaaaa}\left . + \ln\left[1-r_{\bot}^{(1)}(\xi_l , k_{\bot } ) r_{\bot}^{(2)}(\xi_l , k_{\bot})e^{-2q_lz}\right]\right\}. \label{eq8}\end{aligned}\ ] ] here @xmath93 is the boltzmann constant . the reflectivity coefficients for gold ( @xmath94 ) and si ( @xmath95 ) for the two independent polarizations of electromagnetic field ( transverse magnetic and transverse electric modes ) are defined by @xmath96 where @xmath97 are the matsubara frequencies , @xmath98 , @xmath99 are the frequency - dependent dielectric permittivities of gold and si , and the following notations are introduced @xmath100^{1/2}. \label{eq10}\ ] ] the dielectric permittivities of gold and of high - resistivity si in the absence of laser light were computed @xcite by means of the dispersion relation @xmath101 where @xmath102 are taken from the tabulated optical data for the complex index of refraction @xcite . high - precision results for @xmath103 ( gold ) are presented in ref . @xcite . for high - resistivity si the behavior of @xmath104 as a function of @xmath105 is shown by the long - dashed line in figs . [ epsilon]a and [ epsilon]b . in particular @xmath106 . on irradiation of the si membrane by light , the equilibrium value of the carrier density is rapidly established during a period of time much shorter than the duration of the laser pulse . therefore , we assume that there is an equilibrium concentration of pairs ( electrons and holes ) when the light is incident . thus , in the presence of laser radiation , the dielectric permittivity of si along the imaginary frequency axis can be represented in the commonly used form @xcite @xmath107 } + \frac{\omega_p^{(p)}{\vphantom{\omega_p^{(e)}}}^2}{\xi\left[\xi + \gamma^{(p)}\right ] } , \label{eq12}\ ] ] where @xmath108 and @xmath109 are the plasma frequencies and the relaxation parameters for electrons and holes , respectively . the values of the relaxation parameters @xmath110rad / s and @xmath111rad / s can be found in ref . the plasma frequencies can be calculated from @xmath112 where the effective masses are @xcite @xmath113 , @xmath114 , @xmath115 is the electron mass , and @xmath116 is the concentration of charge carriers . the value of @xmath116 for the different absorbed powers can be calculated in the following way . first , we note that for a membrane of @xmath117 m thickness @xmath116 does not depend on the depth . the reason is that a uniform concentration in this direction is established even more rapidly than the equilibrium discussed above @xcite . in fact the assumption on an uniform charge - carrier density in the si membrane is justified due to the long carrier diffusion lengths and the ability to obtain almost defect free surfaces in silicon through hydrogen passivation @xcite . next , we approximately model the central part of the gaussian beam of diameter @xmath118 by a uniform cylindrical beam of the same diameter . the power contained in this cylindrical beam , @xmath119 , is equal to the power in the central part of the gaussian beam with a diameter @xmath118 . elementary calculation using the gaussian distribution leads to @xmath120 . the power @xmath119 is absorbed uniformly in the central part of the si membrane of diameter @xmath118 having a volume @xmath121 . incidentally , the central region of the membrane with a diameter @xmath118 contributes almost 100% ( 99.9999% @xcite ) of the total casimir force acting between a membrane and a sphere . at equilibrium , the number of created charge carrier pairs per unit time per unit volume @xmath122 , where @xmath123rad / s is the frequency of ar laser light , is equal to the recombination rate of pairs per unit volume @xmath124 . thus , at equilibrium @xmath125 eqs . ( [ eq13 ] ) and ( [ eq14 ] ) allow us to calculate the densities of charge carriers @xmath126 , @xmath127 , @xmath128 and the respective plasma frequencies @xmath129 in all measurements a , b , and c with different powers of the absorbed laser light . in the calculations of charge carrier densities using eq . ( [ eq14 ] ) we have used @xmath130ms and @xmath131ms in accordance with the measurement results in sec . iiib , taking into account the fact that @xmath40 decreases when @xmath116 increases . recall that @xmath132 and @xmath133 were obtained from first 0.5ms of the time decay . our value for @xmath134 obtained using the whole 5ms decay may lead to a minor underestimation of the carrier density , a fact included in the resulting 21% error in the value of @xmath135 . note that the above values of the relaxation parameters @xmath136 and @xmath137 do not depend on the absorbed power @xcite and can be used in all measurements . in fig . [ epsilon]a the dielectric permittivity of si in the presence of laser radiation ( [ eq12 ] ) is shown by solid lines a , b and c as a function of imaginary frequency for the measurements with different absorbed powers a , b and c , respectively . the lines a and b in fig . [ epsilon]a almost coincide . the region around the first matsubara frequency @xmath138 at @xmath3k is shown in fig . [ epsilon]b on an enlarged scale . the obtained values of @xmath103 , @xmath104 , and @xmath139 were substituted in the lifshitz formula ( [ eq8 ] ) and the difference of the casimir forces @xmath140 from eq . ( [ eq5 ] ) in the presence and in the absence of laser light was computed at the laboratory temperature @xmath3k . note that there is discussion in the literature on the correct value of the reflection coefficient for gold @xmath141 at zero frequency ( see , e.g. , refs . our calculation , however , does not depend on chosen value of @xmath141 because in eq . ( [ eq8 ] ) it is multiplied by @xmath142 for the silicon . in the absence of light the latter equality holds for any true dielectric with finite static dielectric permittivity . in the presence of light the equality @xmath143 also holds true as is seen from the substitution of eq . ( [ eq12 ] ) in eq . ( [ eq9 ] ) . in both cases at zero frequency only the transverse magnetic mode of the electromagnetic field contributes to the result . note that for si in the absence and in the presence of light for the transverse magnetic mode @xmath144 respectively . finally the lifshitz formula ( [ eq8 ] ) was used to compute the difference in the casimir forces at all experimental separations @xmath145 ( @xmath146 ) and for the three measurements performed at different absorbed powers . the results of these calculations should be corrected for the presence of surface roughness @xcite . the stochastic roughness on our test bodies can be taken into account using the procedure presented in detail in refs . first , the zero roughness levels on both gold ( @xmath147 ) and si ( @xmath148 ) are determined from @xmath149 where the heights @xmath150 and the fractions of the surfaces covered by roughness with these heights are given in tables i and ii , respectively . from eq . ( [ eq16 ] ) it follows @xmath151 nm , @xmath152 nm . the absolute separation @xmath0 between the test bodies is in fact measured between the zero roughness levels . then the theoretical values of the difference casimir force with account of the surface roughness are calculated as the geometric averaging @xmath153 where @xmath140 was computed by the lifshitz formula for perfectly shaped bodies with and without light on a si membrane . in the present experiments the contribution from roughness correction is very small . thus , at @xmath71 nm it contributes only 1.2% of the calculated @xmath154 . at @xmath155 nm the contribution from surface roughness decreases to only 0.5% of the calculated force difference . similar to refs . @xcite it is easily seen that the contribution from the nonadditive , diffraction - type effects to roughness correction [ which is not taken into account in eq . ( [ eq17 ] ) ] is negligibly small . the results of the numerical computations of the difference casimir force between rough surfaces @xmath154 are shown as solid lines in fig . [ result1],a - c for the measurements with different powers of the absorbed laser light . they are in a very good agreement with the experimental data shown by dots in the same figures ( see the following subsections for the quantitative measure of agreement between experiment and theory ) . for completeness , we present also the results of theoretical computations using the lifshitz formula at zero temperature . they are obtained from eq . ( [ eq8 ] ) by changing the discrete matsubara frequencies @xmath156 for continuous @xmath105 and by replacement of the summation for integration @xmath157 following the same procedure as at @xmath3k , we first calculate @xmath158 using the lifshitz formula and then find @xmath159 including the effect of surface roughness with eq . ( [ eq17 ] ) . the results of these computations are shown as short - dashed lines in fig . [ result1],a - c . as is seen from the figure , in all cases the short - dashed lines describe a slightly larger magnitude of the casimir force difference than at @xmath3 in rather good agreement with the experimental data shown as dots ( see the next subsections for further discussion ) . the theoretical errors in the computation of the casimir force acting between a sphere and a membrane were discussed in detail in refs . the major source of the theoretical uncertainty in this experiment is the error in the concentration of charge carriers @xmath116 when the light is on . from sec . iva , this error is of about 20% . calculations using the lifshitz formula show that the resulting relative error in the difference casimir force , @xmath160 , i.e. , is equal to approximately 12% and does not depend on separation . the error due to uncertainty of experimental separations @xmath145 , in which the theoretical values @xmath161 should be computed , is equal to @xmath162 and takes the maximum value of 3% of the casimir force at the shortest separation of @xmath71 nm ( recall that according to sec . iiia @xmath34 nm ) . this leads to only 2% error in the difference of the casimir force at @xmath71 nm [ so that @xmath163 and to smaller errors at larger separations . the other sources of theoretical errors , discussed in refs . @xcite , like sample - to - sample variation of the tabulated optical data in au , use of the proximity force theorem , patch potentials , nonlocal effects and finite thickness of the gold coating on the sphere contribute negligible amounts to the error in @xmath161 . thus , for example , using the lifshitz formula for a polystyrene sphere covered by a gold layer of 82 nm thickness instead of eq . ( [ eq8 ] ) written for a solid gold sphere , we would get only a 0.03% decrease in the casimir force magnitude . a specific new uncertainty which is present in this experiment is connected with the pressure of light transmitted through the membrane and incident on the bottom of the sphere ( see sec . this effect is present only during the light phase of the pulse train and can be easily estimated . the maximum intensity of the laser light incident on a sphere section with radius @xmath164 parallel to the membrane is @xmath165 where @xmath166 is the fraction of the absorbed power transmitted through the membrane . the value of @xmath166 is given by @xmath167 where @xmath168 m ( see sec . ii ) and the transmission coefficient @xmath169 . the force due to light pressure acting on the sphere in spherical coordinates takes the form @xmath170 substituting eq . ( [ eq19 ] ) in eq . ( [ eq21 ] ) and integrating , one obtains @xmath171 , \label{eq22}\ ] ] where erfi@xmath172 is the imaginary error function . for the absorbed powers used in three experiments ( @xmath173 , 8.5 , and 4.7mw , respectively ) , ( [ eq22 ] ) leads to the following maximum forces which may act on the sphere due to light pressure : @xmath174 , 0.078 and 0.043pn . the force due to light pressure can be taken into account as one more error in the theoretical evaluation of the casimir force difference @xmath161 . at a separation @xmath71 nm the respective relative error , @xmath175 , is equal to 2.3 , 2.7 , and 1.5% for the three absorbed powers . at @xmath176 nm the relative theoretical error in @xmath161 due to light pressure increases up to 8.9 , 8.7 , and 5.0% , respectively . all three errors discussed above can be considered as the random quantities described by the same distribution law which is close to a uniform distribution . for this reason the statistical criterion @xcite used in sec . iiic can be applied once more , giving the total relative theoretical error in the difference casimir force @xmath177 ^ 2}\right\ } \label{eq22a}\ ] ] with @xmath79 and @xmath80 . the resulting total absolute theoretical error , @xmath178 is presented in fig . [ erth ] as a function of separation for the three experiments with decreasing power of the absorbed laser light ( lines a , b , and c , respectively ) . as is seen from this figure , the total theoretical errors for the measurements a and b are almost equal , and for the measurement c this error is slightly lower . the relative total theoretical error changes from 13.5 to 13.7% at @xmath71 nm and from 13.7 to 14.4% at @xmath179 nm for the three different absorbed powers . at @xmath176 nm the relative total theoretical error ranges from 14.9 to 17.2% for the different absorbed powers . in the foregoing we have independently found the total experimental ( sec . iiic ) and theoretical ( sec . ivb ) errors in the difference of the casimir force in the presence and in the absence of laser light excited carriers at 95% confidence . to compare experiment with theory , we consider the quantity @xmath180 and determine its absolute error @xmath181 as a function of separation at the confidence of 95% . this can be done in the same procedure as in refs . @xcite applying the statistical criterion @xcite and using the data in figs . [ random ] and [ erth ] @xmath182 ^ 2 + \left[\delta^{\!\rm tot}(\delta f_c^{\rm theor})\right]^2 } \right\}. \label{eq22b}\ ] ] here @xmath183 . the resulting confidence intervals @xmath184 $ ] are shown in fig . [ result2],a - c as the solid lines for the three measurements with the largest , intermediate , and smallest powers , respectively . the differences between the theoretical values of @xmath161 ( computed in sec . iva at @xmath3k ) and experimentally measured @xmath185 are shown in fig . [ result2 ] by dots labeled 1 ( once again dots in fig . [ result2],a - c are related to the three measurements with different power ) . as is seen from fig . [ result2 ] , practically all dots labeled 1 are well inside the confidence intervals at all separation distances . this means that the lifshitz theory at nonzero temperature , using the dielectric permittivity of high - resistivity si @xmath104 in the absence of laser light and the dielectric permittivity @xmath139 given by eq . ( [ eq12 ] ) in the presence of light , is consistent with experiment . the consistency of the experiment with the theory is preserved when the theoretical values of @xmath161 are computed at zero temperature ( see the short - dashed lines in fig . [ result1],a - c and the discussion in sec . the reason is that the thermal correction to the casimir force in the region of small separations under consideration is practically negligible and the thermal effect can not be resolved taking into consideration the experimental and theoretical errors reported above . for illustrative purposes , the agreement between experiment and theory is presented in a more standard form in fig . [ result3 ] . here a more narrow separation interval from 100 to 150 nm is considered and each third experimental point from the measurement b is plotted together with its error bars @xmath186 $ ] shown as crosses ( there are too many points to present all of them in this form ) . the theoretical force difference @xmath161 computed by the lifshitz formula at @xmath3k is shown by the solid line . it is seen that the experimental data are in a very good agreement with the theory in confirmation of the conclusion made above using fig . [ result2 ] . in sec . iva the dielectric response of high - resistivity si in the absence of excitation laser light was described by the function @xmath104 having a finite static value @xmath106 . it is common knowledge , however , that dielectrics have some nonzero dc conductivity @xmath187 at any nonzero temperature . this conductivity decreases with the decrease of temperature as @xmath188 , where @xmath189 can be expressed in terms of the band gap or dopant activation energy . to take the dc conductivity into account in the lifshitz theory , the dielectric permittivity of si along the imaginary frequency axis @xmath104 used in sec . iva should be replaced with @xmath190}. \label{eq23}\ ] ] the value of the plasma frequency in eq . ( [ eq23 ] ) is found by substituting the concentration of carrier density @xmath15 ( see sec . ii ) in eq . ( [ eq13 ] ) with the result @xmath191rad / s . note that for @xmath192 the value of the relaxation parameter has an insignificant effect on the magnitude of the casimir force @xcite . because of this in eq . ( [ eq23 ] ) the same value of @xmath137 as in eq . ( [ eq12 ] ) is used . the behavior of @xmath193 as a function of @xmath105 is plotted in fig . [ epsilon]a by the short - dashed line . the presence of some low dc conductivity in dielectric materials was used in refs . @xcite to obtain a large effect of the van der waals friction which could bring the observations of ref . @xcite in agreement with theory . in ref . @xcite for two dielectric plates and in @xcite for one metal and one dielectric plate it was proved , however , that the inclusion of the dc conductivity for dielectrics into the lifshitz theory leads to the violation of the third law of thermodynamics ( the nernst heat theorem ) . thus , it is not acceptable from a theoretical point of view . our experiments on the modification of the casimir force with laser pulses clarify the problem whether or not the dc conductivity of high - resistivity si should be taken into account in the lifshitz theory of the casimir and van der waals forces . for this purpose , we have completely repeated the theoretical computations of the difference casimir force made in sec . iva replacing the dielectric permittivity of si @xmath194 , used there , for @xmath195 given in eq . ( [ eq23 ] ) . the obtained theoretical results for @xmath196 versus separation are shown by the long - dashed lines in fig . [ result1],a - c for all the three measurements with different powers of the absorbed light . as is seen in fig . [ result1 ] , all the long - dashed lines are far outside both the experimental data shown as dots and from the solid lines calculated using the lifshitz theory disregarding dc conductivity of high - resistivity si at the laboratory temperature . notice that the computational results at @xmath197 ( shown by the short - dashed lines in fig . [ result1 ] ) do not depend on whether the dc conductivity is included in the dielectric permittivity used to describe the high - resistivity si . to make a quantitative conclusion on the measure of agreement between the data and two models with and without inclusion of dc conductivity of high - resistivity si , we have plotted in fig . [ result2],a - c the differences @xmath198 , where @xmath196 was computed including the dc conductivity according to eq . ( [ eq23 ] ) . these differences are shown as dots labeled 2 in fig . [ result2],a - c . as is seen in fig . [ result2],a , b , the model with included dc conductivity of high - resistivity si is excluded experimentally at 95% confidence within the region from 100 to 250 nm . from fig . [ result2],c it follows that this model is excluded at 95% confidence within the separations region from 100 to 200 nm . the same conclusion , that the model of high - resistivity si , which includes dc conductivity , is inconsistent with our experiments on the optically modulated casimir force , is confirmed also in fig . [ result3 ] , where the quantity @xmath196 versus separation is plotted as the dashed line . it can be clearly observed that the dashed line is not only far away from the solid line based on theory neglecting the si dc conductivity in the absence of excitation light , but is also distant from all error bars representing the experimental data . the physical explanation for the deviations of the long - dashed lines from the solid lines in figs . 5,a c and 10 is as follows . when the dc conductivity of si is taken into account , the equalities @xmath199 follow from the substitution of eqs . ( [ eq12 ] ) and ( [ eq23 ] ) in eq . ( [ eq9 ] ) . once again , at zero frequency only the transverse magnetic mode contributes to the result . here , however , for si both in the absence and in the presence of light the equations @xmath200 hold . it is exactly this change in the magnitude of the transverse magnetic reflection coefficient @xmath201 , as given in eq . ( [ eq15a ] ) , with @xmath202 in eq . ( [ eq29 ] ) leads to the deviation of the long - dashed lines from the respective solid lines in figs . 5,a c and 10 . it seems somewhat surprising that the use of the permittivity @xmath195 in eq . ( [ eq23 ] ) , which can be considered as a more exact than @xmath104 , leads to the discrepancy between experiment and theory . this is , in fact , one more observation that there are puzzles concerning the applicability of the lifshitz theory to real materials . in the case of metals , the drude description of conduction electrons in the thermal casimir force was excluded experimentally in the series of experiments @xcite . it also leads to the contradiction with the nernst heat theorem for perfect crystal lattices @xcite . for metals , the deviation of the experimental results from the drude model approach and the violation of the nernst theorem are explained by the vanishing contribution from the transverse electric mode at zero frequency . the present experiment dealing with semiconductors is not sensitive enough to detect this effect . the effect reported here is novel and arises due to the difference in the contributions of the zero - frequency transverse magnetic mode . these contributions , as was shown above , depend on whether or not the dc conductivity of si in the absence of light is taken into account . in this paper we demonstrate that it is possible to control the casimir force between the gold coated sphere and si membrane by the irradiation of si with laser pulses . on absorption of light , the carrier density increases leading to an increase in the magnitude of the casimir force . this change in the casimir forces was investigated as a function of separation between the test bodies and the power of the absorbed light . the experiments were performed with a specially prepared single crystal si membrane in an oil - free vacuum chamber using an afm . the developed calibration procedure permitted measurement of the difference casimir force of the order of 1pn with a relative experimental error at the shortest separation of 100 nm varying from 10 to 20% for the measurements performed at different absorbed powers . at a separation of 180 nm the relative experimental error in different measurements varies from 25 to 33% . all errors were determined at 95% confidence . the obtained experimental results demonstrate the ability to modulate the van der waals and casimir forces in micro- and nanoelectromechanical devices by irradiation with laser light . these are pioneering experiments where the modification of the casimir force acting between the test bodies was achieved due to the influence of some external factor other than the change of separation distance . the experimental results were compared with the results of theoretical computations using the lifshitz theory at both zero and nonzero temperature . the si membrane in the absence of laser light had a carrier density of approximately @xmath203 . in the first model , the dielectric permittivity of high - resistivity si was described with a finite static value . in the presence of laser light the si had charge carriers pair densities varying from @xmath204 to @xmath205 depending on the radiation power absorbed by the sample and was described by the permittivity in eq . ( [ eq12 ] ) . the total theoretical error varied from 13.5 to 13.7% at @xmath71 nm and from 14.9 to 17.2% at @xmath176 nm depending on the absorbed power . the main contribution to this error was given by the uncertainty in the number of charge carriers in the presence of laser light . the experimental and theoretical results were found to be consistent over the whole measurement range taking into account the experimental and theoretical errors both at laboratory temperature @xmath3k and at zero temperature . the same experimental data were compared with the lifshitz theory using a second model of high - resistivity si which includes the dc conductivity of the si membrane in the absence of laser radiation . in this case the dielectric permittivity of si in the absence of radiation is represented by eq . ( [ eq23 ] ) and goes to infinity when the frequency goes to zero . the detailed comparison leads to the conclusion that this model is excluded by the experiment at 95% confidence if computations are performed at the laboratory temperature @xmath3k . the difference in the force magnitudes when conductivity at zero frequency is absent or present arises from different contributions of the transverse magnetic modes of the electromagnetic field reflected from the si surface . the physical explanation of our results can be understood in fig . [ epsilon]a . as is seen from this figure , the short - dashed line representing the dielectric permittivity of high - resistivity si with included dc conductivity is located far to the left of the first matsubara frequency @xmath138 and does not belong to the region of frequencies contributing to the force . at the same time , the lifshitz theory at zero temperature using the model of high - resistivity si with included dc conductivity remains experimentally consistent . thus , we can infer that the lifshitz theory at nonzero temperature using the model of high - resistivity semiconductors and dielectrics with included conductivity properties at zero - frequency is inconsistent with our experiments . it is notable that just this theoretical approach was demonstrated @xcite to lead to the violation of the third law of thermodynamics ( the nernst heat theorem ) . to avoid contradictions with thermodynamics and experiment one should follow the originators of the lifshitz theory @xcite who described dielectrics by a model with a finite static dielectric permittivity in computations of the van der waals and casimir forces at nonzero temperature ( the same model was used in the recent paper @xcite on the thermal effect in the casimir - polder force ) . this suggests that the theory of van der waals and casimir forces between real materials requires further investigation . although we are still lacking a fundamental explanation of why the lifshitz theory does not admit inclusion of the conductivity properties of high - resistivity materials at zero frequency , this prescription on how to perform computations in an experimentally and thermodynamically consistent way is topical for numerous applications of the van der waals and casimir forces ranging from condensed matter physics and nanotechnology to the theory of fundamental interactions . the experimentally demonstrated phenomenon of modulation of the casimir force through optical modification of charge - carrier density will be used in the design and function of micro- and nanoelectromechanical devices such as nanoscale actuators , micromirrors and nanotweezers . g.l.k . and v.m.m . are grateful to the department of physics and astronomy of the university of california ( riverside ) for its kind hospitality . the instrumentation in this work was supported by the nsf grant phy0653657 . theoretical calculations and personnel were supported by the doe grant de - fg02 - 04er46131 . 99 h. b. g. casimir , proc . akad . wet . * 51 * , 793 ( 1948 ) . e. buks and m. l. roukes , phys . rev . b * 63 * , 033402 ( 2001 ) . h. b. chan , v. a. aksyuk , r. n. kleiman , d. j. bishop , and f. capasso , science , * 291 * , 1941 ( 2001 ) ; phys . lett . * 87 * , 211801 ( 2001 ) . a. a. chumak , p. w. milonni , and g. p. berman , phys . b * 70 * , 085407 ( 2004 ) . m. kardar and r. golestanian , rev . phys . * 71 * , 1233 ( 1999 ) . b. s. stipe , h. j. mamin , t. d. stowe , t. w. kenny , and d. rugar , phys . lett . * 87 * , 096801 ( 2001 ) . j. r. zurita - snchez , j .- j . greffet , and l. novotny , phys . a * 69 * , 022902 ( 2004 ) . b. geyer , g. l. klimchitskaya , and v. m. mostepanenko , phys . d * 72 * , 085009 ( 2005 ) . g. barton , j. phys . a : math . gen . * 37 * , 1011 ( 2004 ) ; ibid * 38 * , 2997 ( 2005 ) . e. v. blagov , g. l. klimchitskaya , and v. m. mostepanenko , phys . b * 71 * , 235401 ( 2005 ) . m. bordag , b. geyer , g. l. klimchitskaya , and v. m. mostepanenko , phys . b * 74 * , 205431 ( 2006 ) . e. v. blagov , g. l. klimchitskaya , and v. m. mostepanenko , phys . b * 75 * , 235413 ( 2007 ) . m. antezza , l. p. pitaevskii , and s. stringari , phys . a * 70 * , 053619 ( 2004 ) . j. f. babb , g. l. klimchitskaya , and v. m. mostepanenko , phys . a * 70 * , 042901 ( 2004 ) . m. bordag , b. geyer , g. l. klimchitskaya , and v. m. mostepanenko , phys . d * 58 * , 075003 ( 1998 ) ; * 60 * , 055004 ( 1999 ) ; * 62 * , 011701(r ) ( 2000 ) . j. c. long , h. w. chan , and j. c. price , nucl . b * 539 * , 23 ( 1999 ) . r. s. decca , e. fischbach , g. l. klimchitskaya , d. e. krause , d. lpez , and v. m. mostepanenko , phys . d * 68 * , 116003 ( 2003 ) . r. s. decca , d. lpez , e. fischbach , g. l. klimchitskaya , d. e. krause , and v. m. mostepanenko , ann . 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a mapping @xmath1 , where the dependence on the continuous parameter is smooth , is called a _ semidiscrete surface_. let us connect @xmath2 with @xmath3 by segments for all possible pares @xmath4 . the resulting piecewise smooth surface is a _ piecewise ruled surface_. in this paper we study infinitesimal and higher order flexibility conditions for such semidiscrete surfaces . by _ flexions _ of a semidiscrete surface @xmath5 we understand deformations that isometrically deform corresponding ruled surfaces and in addition that preserve all line segments connecting @xmath2 with @xmath3 . many questions on discrete polyhedral surfaces have their origins in classical theory of smooth surfaces . flexibility is not an exception from this rule . the general theory of flexibility of surfaces and polyhedra is discussed in the overview @xcite by i. kh . sabitov . in 1890 @xcite l. bianchi introduced a necessary and sufficient condition for the existence of isometric deformations of a surface preserving some conjugate system ( i.e. , two independent smooth fields of directions tangent to the surface ) , see also in @xcite . such surfaces can be understood as certain limits of semidiscrete surfaces . on the other hand , semidiscrete surfaces are themselves the limits of certain polygonal surfaces ( or _ meshes _ ) . for the discrete case of flexible meshes much is now known . we refer the reader to @xcite , @xcite , @xcite , and @xcite for some recent results in this area . for general relations to the classical case see a recent book @xcite by a. i. bobenko and yu . b. suris . it is interesting to notice that the flexibility conditions in the smooth case and the discrete case are of a different nature . currently there is no clear description of relations between them in terms of limits . the place of the study of semidiscrete surfaces is between the classical and the discrete cases . main concepts of semidiscrete theory are described by j. wallner in @xcite , and @xcite . some problems related to isothermic semidiscrete surfaces are studied by c. mller in @xcite . we investigate necessary conditions for existence of isometric deformations of semidiscrete surfaces . to avoid pathological behavior related to noncompactness of semidiscrete surfaces we restrict ourselves to compact subsets of the following type . an _ @xmath6-ribbon surface _ is a mapping @xmath7\times \{0,\ldots , n\ } \to\r^3 , \qquad ( i , t ) \mapsto f_i(t).\ ] ] we also use the notion @xmath8 while working with a rather abstract semidiscrete or @xmath6-ribbon surface @xmath5 we keep in mind the two - dimensional piecewise - ruled surface associated to it ( see fig . [ part ] ) . @xmath9 in present paper we prove that any 2-ribbon surface ( as a ruled surface ) is flexible and has one degree of freedom in the generic case ( theorem [ 2-ribbon flex ] ) . this is quite surprising since generic 1-ribbon surfaces have infinitely many degrees of freedom , see , for instance , in @xcite , theorem 5.3.10 . we also find a system of differential equations for the deformation of 2-ribbon surfaces ( system a and corollary [ 111c ] ) . in contrast to that , a generic @xmath6-ribbon surface is rigid for @xmath10 . for the case @xmath11 we prove the following statement ( see theorem [ inf ] and corollary [ lambdah ] ) . * infinitesimal flexibility condition . * + _ a 3-ribbon surface is infinitesimally flexible if and only if the following condition holds : @xmath12 where @xmath13 and @xmath14 _ _ remark . _ throughout this paper we denote the derivative with respect to variable @xmath15 by the dot symbol . having this condition , we also show how to construct inductively the variational isometric conditions of higher orders . finally , we show that an @xmath6-ribbon surface is infinitesimally or finitely flexible if and only if all its 3-ribbon subsurfaces are infinitesimally or finitely flexible ( see theorems [ n - ribbon - infini ] and [ n - ribbon - fini ] ) . we say a few words in the case of developable semidiscrete surfaces whose flexions have additional surprising properties . * organization of the paper . * in section 1 we discuss flexibility of 2-ribbon surfaces . we study infinitesimal flexibility questions for 2-ribbon surfaces in subsections 1.2 and 1.3 . in subsection 1.2 we give a system of differential equations for infinitesimal flexions , prove the existence of nonzero solutions , and show that all the solutions are proportional to each other . in subsection 1.3 we define the variational operator of infinitesimal flexion which is studied further in the context of finite flexibility for 2-ribbon surfaces . in subsection 1.4 we prove that a 2-ribbon surface is finitely flexible and has one degree of freedom if in general position . in section 2 we work with @xmath6-ribbon surfaces . subsection 2.2 gives infinitesimal flexibility conditions for 3-ribbon surfaces . subsection 2.3 studies higher order variational conditions for 3-ribbon surfaces . finally , subsection 2.4 shows the relations between flexibility of @xmath6-ribbon surfaces and infinitesimal and flexibility of 3-ribbon subsurfaces contained in it ( in both infinitesimal and finite cases ) . we conclude the paper with flexibility of developable semidiscrete surfaces in section 3 . in this case flexions have additional geometric properties . * necessary notions and definitions . * within this paper we traditionally consider @xmath15 as a smooth argument of a semidiscrete surface @xmath5 . the time parameter for deformations is @xmath16 . a _ perturbation _ of a semidiscrete ( @xmath6-ribbon ) surface is a smooth curve @xmath17 in the space of all sufficiently smooth semidiscrete surfaces . we assume that the curve is parameterized by @xmath18 $ ] for some positive @xmath19 such that @xmath20 . denote by @xmath21 the _ infinitesimal perturbation _ of a semidiscrete ( @xmath6-ribbon ) surface @xmath5 along the curve @xmath22 , i. e. the tangent vector @xmath23 . we say that a perturbation is a _ flexion _ if it does not change the inner geometry of the surface obtained by joining all the pairs @xmath24 and @xmath25 by straight segments . in the case of semidiscrete ( @xmath6-ribbon ) surfaces a surface is flexible if the the following quantities are preserved by the perturbation : @xmath26 ( for all possible @xmath27 and @xmath15 in the case of an @xmath6-ribbon surface ) . we say that an infinitesimal perturbation is an _ infinitesimal flexion _ if it does not change the inner geometry of the surface infinitesimally . in other words , the first derivatives of the quantities listed above are all equal to zero . in this section we describe flexions of 2-ribbon surfaces . such surfaces are defined by three curves @xmath28 , @xmath29 , and @xmath30 . our main goal here is to prove under some natural genericity assumptions that any 2-ribbon surface is flexible and has one degree of freedom . our first point is to describe the system of differential equations ( system a ) that determines infinitesimal flexions corresponding to finite flexions and find solutions to this system ( see subsections 1.1 and 1.2 ) . further via solutions of system a we define the variational operator of infinitesimal flexion @xmath31 ( in subsection 1.3 ) . finally , to show finite flexibility of 2-ribbon surfaces we study lipschitz properties for @xmath31 ( in subsection 1.4 ) . in this small subsection we collect some useful relations . for any infinitesimal flexion of a 2-ribbon surface @xmath5 the following properties hold : @xmath32 for a semidiscrete or @xmath6-ribbon surface @xmath5 and a @xmath33-curve @xmath22 the operations @xmath34 , @xmath35 , and @xmath36 commute , so we do not pay attention to the order of these operations in compositions . the first three equations follow from the fact that infinitesimal flexions preserve the norm of tangent vectors to the curves @xmath29 , @xmath37 , and @xmath30 . the invariance of the lengths of @xmath38 and @xmath39 implies the fourth and the fifth equations . equations ( [ e8 ] ) and ( [ e9 ] ) follows from invariance of angles between the vectors @xmath40 and @xmath41 and the vectors @xmath40 and @xmath41 . finally , the last two equations hold since the angles between @xmath38 and @xmath42 and @xmath39 and @xmath42 are preserved by infinitesimal flexions and therefore @xmath43 ( in addition we use equations ( [ e8 ] ) and ( [ e9 ] ) respectively ) . in this subsection we write down a system of differential equations ( system a ) which describe infinitesimal flexions of a 2-ribbon surface in general position . we show the existence of infinitesimal flexions and prove that they are proportional to each other ( theorem [ 1-d - inf ] ) . let @xmath44 denote by _ system a _ the following system of differential equations @xmath45 in proposition [ inner ] below we show an explicit formula for the function @xmath46 , it is @xmath47 in our notation of section 2 . note also that @xmath48 and @xmath49 in system a. the remaining part of this subsection is dedicated to the proof of theorem [ 1-d - inf ] on the structure of the space of infinitesimal flexions . in proposition [ condition ] we show that any infinitesimal flexion satisfies system a. then in proposition [ existence ] we prove that any solution of system a with certain initial data is an infinitesimal flexion . finally , in proposition [ 111 ] we show the uniqueness of the solution of system a for a given initial data . after that we prove theorem [ 1-d - inf ] . let us show that any infinitesimal flexion satisfies system a. [ condition ] let @xmath50 , @xmath38 , and @xmath39 be linearly independent . then for any infinitesimal flexion @xmath34 the functions @xmath51 satisfy system a. we start the proof with the following general lemma . [ additional ] for any infinitesimal flexion @xmath34 we have the equalities @xmath52 the functions @xmath53 , @xmath54 , and @xmath55 are infinitesimally preserved by infinitesimal flexions , hence @xmath56 , @xmath57 , and @xmath58 vanish . the invariance of angles between @xmath50 and @xmath38 , and @xmath50 and @xmath39 yield the equations @xmath59 and @xmath60 , respectively . _ proof of proposition [ condition ] . _ from lemma [ additional ] the functions @xmath56 , @xmath57 , and @xmath58 are equivalent to zero , thus @xmath61 , @xmath62 , and @xmath63 are equivalent to zero as well . let us prove the expression for @xmath64 and @xmath65 . note that @xmath66 thus equations ( [ e8 ] ) and ( [ e6 ] ) imply @xmath67 to obtain the expression for @xmath64 rewrite @xmath68 and @xmath69 in the basis consisting of vectors @xmath50 , @xmath38 , and @xmath39 . the same strategy works for the functions @xmath65 . now we study expressions for @xmath70 and @xmath71 . from lemma [ additional ] we know that @xmath72 and @xmath73 and hence @xmath74 and @xmath75 . therefore , the equations for @xmath70 and @xmath71 are satisfied . in order to get an expression for @xmath76 , we first note that @xmath77 , since the function @xmath78 is an invariant of an infinitesimal flexion . so we get @xmath79 rewrite @xmath80 second , we have @xmath81 third , we get @xmath82 fourth , @xmath83 after a substitution of the four above expressions and simplifications we have @xmath84 further , we get @xmath85 from the last two identities , by substituting @xmath86 and @xmath72 ( see lemma [ additional ] ) , we obtain the expression for @xmath76 . the expression for @xmath87 is calculated in a similar way . this concludes the proof . further we prove that any solution of system a with certain initial data is an infinitesimal flexion . [ existence ] let @xmath5 be a 2-ribbon surface , @xmath88\to \r^3 $ ] for @xmath89 . assume that the function @xmath90 has no zeros on @xmath91 $ ] . then any infinitesimal perturbation @xmath34 of @xmath5 satisfying system a and the boundary conditions @xmath92 is an infinitesimal flexion . by the definition of an infinitesimal flexion it is enough to check that the following 11 functions are preserved by the infinitesimal perturbation : @xmath26 ( for all possible admissible @xmath27 ) . _ invariance of @xmath53 , @xmath54 , @xmath55 , @xmath93 , and @xmath94_. from system a we have @xmath95 and hence all five functions under consideration are constants . so it is enough to show that they vanish at some point : we show this at point @xmath96 . @xmath97 _ invariance of @xmath98 and @xmath99_. note that @xmath100 hence @xmath101 . similar reasoning shows that @xmath102 . _ invariance of @xmath103 and @xmath104_. let us prove that @xmath105 . first , note that @xmath106 recall that @xmath107 . let us substitute the expression for @xmath64 from system a and rewrite @xmath69 in the basis of vectors @xmath50 , @xmath38 , and @xmath39 . one obtains @xmath108 hence @xmath109 it follows that @xmath103 is invariant under the infinitesimal perturbation . the proof of the invariance of @xmath110 is analogous . _ invariance of @xmath111 and @xmath112_. let us prove that @xmath113 . @xmath114 we have already shown that @xmath115 and @xmath116 . hence @xmath117 we rewrite the last @xmath118 in the last expression in the basis @xmath119 and get @xmath120 let us rewrite @xmath121 , @xmath122 , and @xmath123 in terms of @xmath124 . first , we have : @xmath125 the second equality holds since we have shown that @xmath126 . if we rewrite @xmath68 in the basis @xmath127 , we get the following : @xmath128 second , we have @xmath129 third , with @xmath130 and the expression for @xmath76 from system a we get : @xmath131 finally , we combine these three expressions and arrive at @xmath132 it is clear that the coefficients of @xmath133 and @xmath134 vanish identically . let us study the coefficient of @xmath135 . consider the following mixed product @xmath136 , it is identical to zero . let us rewrite @xmath68 in the second position of the mixed product in the basis @xmath137 , @xmath38 , @xmath39 . we get the relation @xmath138 we apply this identity to the first two summands of the coefficient of @xmath135 and get the following expression for the coefficient of @xmath135 : @xmath139 we rewrite this as @xmath140 let us study the expression in the brackets . @xmath141 the second equality holds by the jacobi identity . hence the coefficient of @xmath135 is zero . therefore , @xmath142 and @xmath111 is invariant under the infinitesimal perturbation . the proof of the invariance of @xmath143 repeats the proof for @xmath111 . so we have checked the invariance of all the 11 functions in the definition of an infinitesimal flexion . hence the infinitesimal perturbation @xmath34 is an infinitesimal flexion . in the following proposition we prove that system a has a unique solution for any single 2-ribbon surface @xmath5 ( not for a deformation ) and initial data for @xmath144 at one point @xmath145 . recall that @xmath15 is an argument of @xmath5 . [ 111 ] let @xmath5 be a 2-ribbon surface , @xmath88\to \r^3 $ ] for @xmath89 . for any collection of initial data @xmath146 there exists a unique solution of system a. this solution is extended for all @xmath147 , where @xmath148 the system of differential equations for @xmath149 is a system of homogeneous linear equations with variable coefficients and hence for any collection of initial data it has a unique solution . the initial conditions of the last proposition can be reformulated in terms of infinitesimal flexion @xmath150 at a single point @xmath151 itself . [ 111c ] let @xmath5 be a 2-ribbon surface , @xmath88\to \r^3 $ ] for @xmath89 . for any collection of initial data @xmath152 there exists a unique solution of system a. this solution is extended for all @xmath147 , where @xmath148 the corollary follows directly from proposition [ 111 ] after obtaining the initial values @xmath153 from the vectors @xmath154 : @xmath155 now we have all the ingredients to prove the general theorem on the structure of the space of infinitesimal flexions . [ 1-d - inf ] consider a 2-ribbon surface defined by curves @xmath88\to \r^3 $ ] for @xmath89 , where @xmath28 and @xmath30 are @xmath156-smooth and @xmath29 is @xmath33-smooth . assume that the function @xmath157 has no zeroes on @xmath91 $ ] . the space of infinitesimal flexions of such surfaces @xmath158up to isometries@xmath159 is one - dimensional . _ uniqueness . _ any infinitesimal flexion is isometrically equivalent to an infinitesimal flexion which satisfies @xmath92 consider functions @xmath144 defined by equations ( [ g_i ] ) . by proposition [ condition ] these functions satisfy system a. hence by corollary [ 111c ] , the functions @xmath144 are uniquely defined by @xmath5 and the initial conditions for infinitesimal flexions . recall that elements of an arbitrary euclidean vector @xmath160 are uniquely determined by its scalar products with an arbitrary basis : @xmath161 therefore , the infinitesimal flexion is uniquely defined by the functions @xmath144 . hence the dimension of infinitesimal flexions is at most one ( the parameter @xmath22 is the unique parameter of this flexion ) . _ existence . _ by corollary [ 111c ] there exists an infinitesimal deformation satisfying system a and the initial values @xmath162 by proposition [ existence ] this infinitesimal deformation is an infinitesimal flexion . since the function @xmath163 has no zeroes , @xmath164 is a nonzero vector and hence the infinitesimal deformation is nonvanishing . let us fix an orthonormal basis @xmath165 in @xmath166 . suppose that we know the coordinates of a 2-ribbon surface @xmath167\times \{0,1,2\ } \to\r^3 $ ] in this basis . denote the coordinate functions for @xmath50 , @xmath38 , and @xmath39 as follows @xmath168 denote by @xmath169 the banach space @xmath170)^9 $ ] with the norm latexmath:[\[\\|(h_1,\ldots , h_9)\\|=\max\limits_{1\le i \le 9}(\max(\sup @xmath5 is defined by the curves @xmath172 , @xmath38 , and @xmath39 up to a translation . so the space @xmath169 is actually the space of all 2-ribbon surfaces with one endpoint fixed , say @xmath173 . we say that a point @xmath174 is in _ general position _ if the determinant @xmath175 for any point in the segment @xmath91 $ ] . this condition obviously corresponds to @xmath176 denote by @xmath177\times \omega^1_9 \to \omega^1_9 $ ] the variational _ operator of infinitesimal flexion _ in coordinates @xmath178 : @xmath179 for ( @xmath180 ) . here @xmath181 is a solution of system a at point @xmath5 with the initial conditions corresponding to @xmath182 i. e. , @xmath183 note that the variational operator of infinitesimal flexion @xmath31 is autonomous , it does not depend on time parameter @xmath16 . let us show in brief how to find the coordinates of the perturbation @xmath21 in the basis @xmath184 satisfying @xmath185 first , one should solve system a with the above initial data , then substitute the obtained solution @xmath186 to equations ( [ e11 ] ) . now we have the coordinates of @xmath187 , @xmath188 , and @xmath189 . having the additional condition @xmath190 one can construct @xmath191 , @xmath192 , and @xmath193 : @xmath194 in previous subsection we showed that any @xmath195-ribbon surface in general position is infinitesimally flexible and the space of its infinitesimal flexions is one - dimensional . the aim of this subsection is to show that a 2-ribbon surface in general position is flexible and has one degree of freedom . we start with the discussion of the initial value problem for the following differential equation on the set of all points @xmath169 in general position ( here @xmath16 is the time parameter ) : @xmath196 to solve the initial value problem we study local lipschitz properties for @xmath31 . consider a banach space @xmath197 with a norm @xmath198 and let @xmath199 be a subset of @xmath200\times e$ ] . we say that a functional @xmath201 _ locally satisfies a lipschitz condition _ if for any point @xmath202 in @xmath199 there exist a neighborhood @xmath203 of the point and a constant @xmath204 such that for any pair of points @xmath205 and @xmath206 in @xmath203 the inequality @xmath207 holds . first we verify a lipschitz condition for the following operator . define @xmath208\times \omega^1_9 \to \omega^1_9 $ ] by @xmath209 where @xmath210 are defined by equations ( [ g_i ] ) . [ g1 ] the functional @xmath211 locally satisfies a lipschitz condition at any point in general position . consider a point @xmath212 . the element @xmath186 itself satisfies a linear system of differential equations ( system a ) . the coefficients of this system depend only on a point of @xmath213 . since the point @xmath214 is in general position , there exists an integer constant @xmath204 such that for a sufficiently small neighborhood @xmath215 of @xmath214 the dependence is @xmath204-lipschitz , i.e. , for @xmath216 and @xmath217 from @xmath215 all the coefficients satisfy the inequality @xmath218 hence the solutions for @xmath219 $ ] satisfy the lipschitz condition on @xmath215 as well : for some constants @xmath220 we have @xmath221 from system a we know that the @xmath222 linearly depend on @xmath223 , therefore , we get the lipschitz condition for the derivatives : for some constants @xmath224 we have @xmath225 thus there exists a real number @xmath226 such that for all points @xmath216 and @xmath217 in @xmath215 , @xmath227 therefore , @xmath211 satisfies a lipschitz condition on @xmath215 . lemma [ g1 ] and expression ( [ e11 ] ) directly imply the following statement . [ d1 ] the functional @xmath31 locally satisfies a lipschitz condition at points in general position . now we prove the following theorem on finite flexibility of 2-ribbon surfaces in general position . [ 2-ribbon flex ] consider a 2-ribbon surface defined by a @xmath33-curve @xmath29 and @xmath156-curves @xmath28 and @xmath30 defined on a segment @xmath91 $ ] . assume that @xmath90 does not have zeros on @xmath91 $ ] . then the set of all flexions of such surface @xmath158up to isometries@xmath159 is one - dimensional . as we show in corollary [ d1 ] , the operator @xmath31 satisfies a lipschitz condition in some neighborhood of the point @xmath216 related to @xmath50 , @xmath38 , and @xmath39 . from the general theory of differential equations on banach spaces ( see for instance the first section of the second chapter of @xcite ) it follows that this condition implies local existence and uniqueness of a solution of the initial value problem for the following differential equation @xmath228 in some neighborhood of @xmath214 . since the 2-ribbon surface @xmath229 with a fixed endpoint @xmath230 is uniquely defined by @xmath231 , we get the statement of the theorem . in this section we study necessary flexibility conditions of @xmath6-ribbon surfaces . we find these conditions for 3-ribbon surfaces , and we show how they are related to the conditions for @xmath6-ribbon surfaces . in this subsection we prove certain relations that we further use in the proof of the statement on infinitesimal flexibility conditions for 3-ribbon surfaces . as we have shown in section 1 the notions of finite flexibility and infinitesimal flexibility coincide for the 2-ribbon case . still in this subsection we say _ infinitesimal flexions _ of a 2-ribbon surface to indicate that an infinitesimal flexion of an @xmath6-ribbon surface coincides with finite flexions of all its 2-ribbon surfaces . consider the following function @xmath232 this function plays a central role in our further description of the flexibility conditions of @xmath0-ribbon and @xmath6-ribbon surfaces ( see theorem [ inf ] and theorem [ n - ribbon - infini ] ) . let @xmath233 be the infinitesimal flexion of @xmath47 . via this function we describe monodromy conditions for finite flexibility . proposition [ inner ] and corollary [ discr ] deliver necessary tools to describe continuous and discrete parts of the monodromy condition on @xmath47 . here we study the dependence of the infinitesimal flexion @xmath234 on the argument @xmath15 . [ inner ] * ( on continuous shift . ) * suppose @xmath50 , @xmath38 , and @xmath39 are linearly independent on the segment @xmath235 $ ] . then for an infinitesimal flexion @xmath236 the following condition holds : @xmath237 this is a direct consequence of the next lemma . [ rel ] let @xmath50 , @xmath38 , and @xmath39 be linearly independent , then we have @xmath238 note that @xmath239 let us prove the statement of the lemma for an arbitrary point @xmath151 . without loss of generality we fix @xmath240 and @xmath241 ( this is possible since any flexion is isometric to a flexion with such properties and isometries of flexions do not change the functions in the formula of the lemma ) . then @xmath242 is proportional to @xmath243 , and hence there exists some real number @xmath244 with @xmath245 thus we immediately get @xmath246 let us express the summands for @xmath247 . we start with @xmath248 . first we note that @xmath249 equation ( [ e8 ] ) implies @xmath250 from equation ( [ e4 ] ) we have @xmath251 the function @xmath252 is invariant of an infinitesimal flexion , therefore : @xmath253 and hence @xmath254 now we decompose @xmath255 in the last formula in the basis of vectors @xmath256 , @xmath257 , and @xmath258 : @xmath259 therefore , after substitution ( [ ei ] ) of @xmath260 we apply ( [ eii ] ) , ( [ eiii ] ) , ( [ eiv ] ) , and the last expression and get @xmath261 similar calculations for the summand @xmath262 ( applying equations ( [ e3 ] ) , ( [ e5 ] ) , and ( [ e9 ] ) and the conditions @xmath240 and @xmath241 ) show that @xmath263 further we have @xmath264 therefore , @xmath265 and consequently @xmath266 thus lemma [ rel ] holds for all possible values of @xmath151 . any 3-ribbon surface contain 2-ribbon surfaces as a subsurfaces . each of them has an infinitesimal flexion @xmath267 ( @xmath268 ) . here we show the relation between @xmath269 and @xmath270 for the same values of argument @xmath15 . first , in proposition [ curvatures ] we show a relation for @xmath271 and @xmath272 . second , in proposition [ phipsi ] we give a link between @xmath271 and @xmath267 . this will result in the formula of corollary [ discr ] on the relation between @xmath269 and @xmath270 . we start with a formula expressing @xmath272 via @xmath271 . [ curvatures ] we have the following equation : @xmath273 we do calculations at a point @xmath151 again assuming that @xmath274 and @xmath241 ( by choosing an appropriate isometric representative of the deformation ) . let us show that @xmath275 . first , note that @xmath276 secondly we show that the inner products of @xmath277 and the vectors @xmath256 , @xmath278 , and @xmath279 are all zero ( this would imply that @xmath280 ) . from equation ( [ e9 ] ) we have @xmath281 further , from equations ( [ e5 ] ) , we get @xmath282 finally , from the equation @xmath283 we obtain @xmath284 therefore , @xmath285 , and hence @xmath286 . from equation ( [ e1 ] ) and equation ( [ e7 ] ) we get @xmath287 therefore , for some real number @xmath288 we have @xmath289 by a similar reasoning ( since we have shown that @xmath290 ) we get @xmath291 since @xmath292 , at point @xmath151 we have @xmath293 hence , @xmath294 and , therefore @xmath295 . this implies @xmath296 and @xmath297 the last two formulas imply the statement of proposition [ curvatures ] . now let us relate @xmath271 and @xmath233 . [ phipsi ] suppose @xmath50 , @xmath38 , and @xmath39 are linearly independent . then the following equation holds : @xmath298 we restrict ourselves to the case of a point . without loss of generality we assume that @xmath240 and @xmath299 . so as we have seen before , there exists @xmath244 such that @xmath300 and hence @xmath301 let us calculate @xmath302 . decompose @xmath303 since @xmath304 we get @xmath305 by equation ( [ e6 ] ) we have @xmath306 hence after the substitution of @xmath242 in the first summand one gets @xmath307 therefore , we obtain @xmath308 since the statement does not depend on the choice of the basis and invariant under isometries , we get the statement for all the points . we introduce the abbreviations @xmath309 let us show a formula of a discrete shift . [ discr](on discrete shift . ) suppose @xmath50 , @xmath38 , and @xmath39 are linearly independent . then the following holds : @xmath310 the statement follows directly from propositions [ curvatures ] and [ phipsi ] . in this subsection we write down the infinitesimal flexibility monodromy conditions for 3-ribbon surfaces ( via continuous shifts of proposition [ inner ] and discrete shifts of corollary [ discr ] ) . recall that @xmath311 and @xmath312[1111 ] [ inf ] consider a 3-ribbon surface @xmath5 with linearly independent @xmath172 , @xmath38 , and @xmath39 at all admissible points . the surface @xmath5 is infinitesimally flexible if and only if for any @xmath313 and @xmath314 in the interval @xmath315 $ ] we have @xmath316 by corollary [ discr ] we get relations between @xmath317 and @xmath318 for @xmath268 . on the other hand , proposition [ inner ] relates @xmath319 and @xmath320 for @xmath268 . these four relations define the monodromy condition for @xmath321 that is the condition in the theorem and , therefore , it holds if a surface is infinitesimally flexible . suppose now the condition holds . then the flexion is uniquely defined by the value of @xmath269 at a point @xmath151 . let us simplify the expressions for @xmath322 and @xmath323 performing the following normalization for a fixed parameter @xmath16 . denote @xmath324 here the semidiscrete surface @xmath5 is flexible if and only if @xmath325 is flexible . in addition for the semidiscrete surface @xmath325 we get @xmath326 for all arguments @xmath15 . therefore we get the expressions for @xmath322 and @xmath323 as follows : @xmath327 and @xmath328 notice that this expression holds momentary , i.e. only for a fixed time parameter @xmath16 , so it can not be use for finite deformations . in this subsection we say a few words about higher order variational conditions of flexibility for 3-ribbon surfaces . we give an algorithm to rewrite these conditions in terms of the coefficients of the infinitesimal flexion defined by the system of differential equations ( system a ) . we introduce a further auxiliary function by letting @xmath329 [ lambdah ] a 3-ribbon surface is infinitesimally flexible if and only if the following condition holds : @xmath330 this condition is obtained from the condition of theorem [ inf ] by differentiating w.r.t @xmath314 at the point @xmath313 . therefore , these conditions are equivalent . from the infinitesimal flexibility condition of corollary [ lambdah ] one constructs many other conditions of flexibility . if a 3-ribbon surface has a flexion , depending on a parameter @xmath16 , then @xmath331 at all points . this implies the following statement . if a 3-ribbon surface is flexible then for any positive integer @xmath332 we have @xmath333 where @xmath334 . let us briefly describe a technique to calculate @xmath335 . + * step 1 . * to simplify the expressions we write : @xmath336 further we let @xmath337 here we are interested in derivatives at an arbitrary value of a curve argument @xmath15 but at a fixed parameter of deformation @xmath338 . note that @xmath339 . the functions @xmath340 and @xmath341 are calculated from the initial data for the @xmath0-ribbon surface @xmath5 . let us find the expressions for @xmath342 . without loss of generality we fix @xmath343 for a starting point @xmath96 . we find @xmath244 from corollary [ discr ] ( on discrete shift ) . first , we have @xmath344 therefore , from corollary [ discr ] we have @xmath345 all the functions @xmath342 are found as the corresponding solutions of two systems of differential equations ( system a for @xmath268 ) according to corollary [ 111c ] . since @xmath346 , these solutions are compatible . for the functions @xmath347 we have @xmath348 to avoid cross products in the above expression we use lagrange s formula : @xmath349 * step 2 . * define @xmath350 as before , the functions @xmath351 and @xmath352 are calculated from the initial data for the 3-ribbon surface @xmath5 . first , let us find the expressions for @xmath353 by induction on @xmath332 . _ induction base . _ for @xmath354 we get the formulae from step 1 . _ induction step . _ suppose that we know the expressions for @xmath332 let us find @xmath355 . we have @xmath356 the expression in the left hand part is a function that is known by induction . the last two summands of the hand right part are also expressed inductively after rewriting @xmath357 , and @xmath358 in the basis @xmath359 , @xmath360 , @xmath361 . secondly , decomposing @xmath362 in the basis @xmath359 , @xmath360 , @xmath361 we get @xmath363 * step 3 . * note that @xmath364 and @xmath365 [ aalg ] suppose that we get a rational polynomial expression @xmath366 in variables @xmath367 , @xmath368 , @xmath369 , and @xmath370 . then @xmath371 is also a rational polynomial expression in variables @xmath367 , @xmath368 , @xmath369 , and @xmath370 . steps 13 give all the tools to write the expression for @xmath371 explicitly . for any positive integer @xmath332 the function @xmath372 is a rational polynomial expression in variables @xmath367 , @xmath368 , @xmath369 , and @xmath370 . therefore , we can apply steps 13 and proposition [ aalg ] to calculate @xmath373 using induction on @xmath332 . we conclude this subsection with a few words on sufficient conditions for flexibility . we start with an open problem . find a sufficient condition for flexibility of semidiscrete and @xmath6-ribbon surfaces . for the case of @xmath0-ribbon surfaces we have the following conjecture . consider a @xmath0-ribbon surface @xmath5 . let @xmath374 for all non - negative integers @xmath332 ( where @xmath375 ) . then @xmath5 is locally flexible . we also conjecture that it is enough to take only a finite number of these conditions . then the following question is actual : _ what is the number of independent conditions of isometric deformation _ ? let us finally describe a relation between ( finite and infinitesimal ) flexibility of @xmath6-ribbon surfaces and flexibility of all 3-ribbon subsurfaces contained in them . we start with theorem on infinitesimal flexibility . [ n - ribbon - infini ] consider an @xmath6-ribbon surface satisfying the genericity condition : @xmath376 , @xmath377 , and @xmath378 are not coplanar at any admissible point @xmath151 and integer @xmath27 . then this surface is infinitesimally flexible if and only if any 3-ribbon surface contained in the surface is infinitesimally flexible . the proof is straightforward . all the conditions for infinitesimal flexion are exactly the conditions for 3-ribbon surfaces of theorem [ inf ] . for the finite flexibility we have the following . [ n - ribbon - fini ] consider an @xmath6-ribbon surface satisfying the genericity condition : @xmath376 , @xmath377 , and @xmath378 are not coplanar at any admissible point @xmath151 and integer @xmath27 . then this surface is flexible if and only if any 3-ribbon surface contained in the surface is flexible . we think of this theorem as of a semidiscrete analogue to the statement of the paper @xcite on conjugate nets and all @xmath379-meshes that they contain . in this paper we do not study phenomena related to non - compactness and hence we restrict ourselves to the case of compact @xmath6-ribbons surfaces . the `` only if '' part of the statement is straightforward . we prove the converse by induction on the number of ribbons in a surface . _ induction base . _ by assumption any 3-ribbon subsurface contained in the surface is flexible . it has one degree of freedom , since by theorem [ 1-d - inf ] any 2-ribbon subsurface of a 3-ribbon surface has at most one degree of freedom , while the genericity condition holds in a certain neighborhood of a starting position . _ induction step . _ suppose we know that any @xmath380-ribbon subsurface is flexible and has one degree of freedom in some neighborhood ( for @xmath381 ) . let us prove the statement for any @xmath382-ribbon subsurface . we consider a @xmath382-ribbon subsurface as the union of two @xmath380-ribbon subsurfaces that intersect in a @xmath383-ribbon subsurface . by the induction assumption this @xmath383-ribbon surface has one degree of freedom compatible with the flexions of both @xmath380-ribbon subsurfaces . therefore , the flexion of the @xmath383-ribbon subsurface is uniquely extended to the both @xmath380-ribbon subsurfaces . this implies flexibility of the @xmath382-ribbon with one degree of freedom . suppose that all ribbons of a semidiscrete surface are developable , i.e. the vectors @xmath376 , @xmath384 , and @xmath385 are linearly dependent . we call such semidiscrete surfaces _ developable_. in this section we describe an additional property for flexions of developable semidiscrete surfaces . this fact gives a surprising corollary concerning the flexion of a 2-ribbon developable surface . the degree of freedom for a flexion of a generic 2-ribbon developable surface is 1 , as can easily be seen from the genericity condition for 2-ribbon surfaces . so a flexion is unique up to the choice of a parameter . denote by @xmath392 the angle between @xmath38 and @xmath39 . consider a flexion of a 2-ribbon developable surface @xmath5 . let us choose the parameter @xmath22 of the flexion such that @xmath393 changes linearly in @xmath22 . then for any @xmath15 the value @xmath394 changes linearly in @xmath22 . for the surface of @xmath395 we get @xmath397 since @xmath398 and @xmath399 . now the statement of the corollary for @xmath395 follows from proposition [ inner ] and the inner geometry expression of proposition [ devel ] for the function under integration ( that is actually @xmath400 ) . * acknowledgement . * the author is grateful to j. wallner for constant attention to this work , a. weinmann for good remarks . this work has been performed at technische universitt graz within the framework of the project `` computational differential geometry '' ( fwf grant no . s09209 ) . a. i. bobenko , t. hoffmann , w. k. schief , _ on the integrability of infinitesimal and finite deformations of polyhedral surfaces _ , discrete differential geometry , a. i. bobenko , p. schrder , j. m. sullivan , g. m. ziegler , ( eds . ) , series : oberwolfach seminars , v. 38 ( 2008 ) , pp . 6793 . mller , j. wallner , _ semi - discrete isothermic surfaces _ , geometry preprint 2010/02 , tu graz , february 2010 . h. pottmann , j. wallner , _ infinitesimally flexible meshes and discrete minimal surfaces _ , monatshefte math . , v. 153 ( 2008 ) , pp . 347365 .	in this paper we study necessary conditions of flexibility for semidiscrete surfaces . for 2-ribbon semidiscrete surfaces we prove their one - parametric finite flexibility . in particular we write down a system of differential equations describing flexions in the case of existence . further we find infinitesimal criterions of @xmath0-ribbon flexibility . finally , we discuss the relation between general semidiscrete surface flexibility and @xmath0-ribbon flexibility .	12,267	109
the effects of absorption and scattering by interstellar dust grains on the structural parameters of galaxies has been a long - standing and controversial issue . the only way to tackle this problem is to properly solve the continuum radiative transfer equation , taking into account realistic geometries and the physical processes of absorption and multiple anisotropic scattering . over the years , many different and complementary approaches have been developed to tackle the continuum radiative transfer problem in simple geometries such as spherical or plane - parallel symmetry . while one - dimensional radiative transfer calculations have been crucial to isolate and demonstrate the often counter - intuitive aspects of important parameters such as star - dust geometry , dust scattering properties and clumping @xcite , we need more sophisticated radiative transfer models to model complicated systems such as disc galaxies in detail . thanks to new techniques and ever increasing computing power , the construction of 2d and 3d realistic radiative transfer models is now possible . a complementary and powerful way to study the content of galaxies is to use the direct emission of dust at long wavelengths . large dust grains will typically reach a state of local thermal equilibrium ( lte ) in the local interstellar radiation field ( isrf ) and re - radiate the absorbed uv / optical radiation at far - infrared ( fir ) and submm wavelengths . thanks to the spectacular advances in instrumentation in the fir / submm wavelength region , we have seen a significant improvement in the amount of fir / submm data on both nearby and distant galaxies . in particular , the launch of the herschel space observatory with the sensitive pacs and spire instruments has enabled both the detailed study of nearby , resolved galaxies and the detection of thousands of distant galaxies . whereas large grains typically emit as a modified blackbody at an equilibrium temperature of 15 - 30 k and hence dominate the fir / submm emission of galaxies , small grains and pah molecules are transiently heated by the absorption of single uv photons to much higher temperatures . the nlte emission from very small grains and pahs dominates the emission of galaxies at mid - infrared wavelengths . the iso and particularly the spitzer mission have been instrumental in uncovering the mid - infrared emission of nearby galaxies @xcite . different approaches have been developed to calculate the nlte emission spectrum due to very small grains and pahs , but the integration of nlte emission into radiative transfer codes has proven to be a challenging task . the main reason is that the computational effort necessary to calculate the temperature distribution of the different dust grains is substantial . in the general case , the calculation of the dust emissivity in a single dust cell requires the solution of a large matrix equation for each single dust population , with the size of the matrix determined by the number of temperature or enthalpy bins . in the so - called thermal continuous cooling approximation @xcite , this matrix equation can be solved recursively , but still the calculation of the emission spectrum remains a significant computational challenge . indeed , since the temperature distribution of dust grains depends strongly on both the size of the grains and the strength and hardness of the isrf , a large number of temperature or enthalpy bins is necessary to sample the temperature distribution correctly . moreover , because of this strong dependence on grain size and isrf , the choice of the temperature bins is hard to fix a priori and an iterative procedure is to be preferred . in spite of the high numerical cost , nlte dust emission has been built into several radiative transfer codes , using various approximations and/or assumptions . the most simple approach is the one followed by e.g. @xcite and @xcite , who use a set of predefined nlte dust emissivities with the simplifying assumption that the emissivity is a function only of strength and not of the spectral shape of the exciting isrf . a pioneering code in which nlte dust emission was included in a self - consistent way was the 2d ray - tracing code by . the number of temperature distribution calculations are minimized by the assumptions that grains with a size larger than about 80 are in thermal equilibrium , and by the use of a pre - fixed time dependence of the cooling of pah grains . a similar approach was adopted in the 3d monte carlo radiative transfer code dirty @xcite . the trading code by uses a different approach : this code uses a fixed ( and limited ) grid of temperature bins for all isrfs and grain sizes , which allows to precompute and tabulate a significant fraction of the quantities necessary for the calculation of the temperature distribution . yet a different approach is the work by : driven by the observation that the spectrum of the local isrf is very similar in many places in a dusty medium , they considered the idea of a dynamic library of dust emission spectra . the idea is that the intensity of the isrf at a very limited number of reference wavelengths ( they typically used only two ) suffices to make a reliable estimate of the total isrf and hence of the dust emission spectrum . in this paper we present an updated version of the skirt monte carlo radiative transfer code . this code , of which the name is an acronym to stellar kinematics including radiative transfer , was initially developed to study the effect of dust absorption and scattering on the observed kinematics of dusty galaxies @xcite . in a second stage , the skirt code was extended with a module to self - consistently calculate the dust emission spectrum under the assumption of local thermal equilibrium @xcite . this lte version of skirt has been used to model the dust extinction and emission of various types of galaxies , as well as circumstellar discs @xcite and clumpy tori around active galactic nuclei @xcite . in this present paper we present a strongly extended version of the skirt code that can perform efficient 3d radiative transfer calculations including a self - consistent calculation of the dust temperature distribution and the associated fir / submm emission with a full incorporation of the emission of transiently heated grains and pah molecules . in section [ skirt-general.sec ] we present the general characteristics of the skirt code , whereas we highlight a number of particular aspects in section [ particular.sec ] and some implementation details in section [ implementation.sec ] . in section [ applications.sec ] we describe a number of tests and applications , and section [ conclusions.sec ] we present our conclusions . skirt is a 3d continuum radiative transfer code based on the monte carlo algorithm . the key principle in monte carlo radiative transfer simulations is that the radiation field is treated as a flow of a finite number of photon packages . a simulation consists of consecutively following the individual path of each single photon package through the dusty medium . the journey or lifetime of a single photon package can be thought of as a loop : at each moment in the simulation , a photon package is characterized by a number of properties , which are generally updated when the photon package moves to a different stage on its trajectory . the trajectory of the photon package is governed by various events such as emission , absorption and scattering events . each of these events is determined statistically by random numbers , generated from the appropriate probability distribution @xmath0 . typically , a photon is emitted by a star , undergoes a number of scattering events and its journey ultimately ends when it is either absorbed by the dust or it leaves the system . a monte carlo simulation repeats this same loop for every single one of the photon packages and analyzes the results afterwards . the mathematical details and practical implementation of monte carlo radiative transfer have both been described in detail by various authors and will not be repeated here in full detail . our overall approach is comparable to the dirty @xcite and trading radiative transfer codes and we refer the interested reader to these papers for more details . we will only give a compact description of the general characteristics of the skirt monte carlo code and not describe all the details . instead , we will focus our attention to those aspects of the skirt code that are novel and/or different compared to the other codes . each skirt simulation consists of four phases : the initialization phase , the stellar emission phase , the dust emission phase , and the clean - up phase . the initialization phase consists of adopting the correct unit system , setting up the random number generator , computing the optical properties of the various dust populations , constructing the dust grid , setting up the stellar geometry and setting up the instruments of the various observers . once this initialization is finished , the actual simulation can start . in the stellar emission phase , we consider the transfer of the primary source of radiation ( usually stellar sources , but it can also include an accretion disc or nebular line emission ) through the dusty medium . the stellar emission phase consists of a set of parallel loops , each of them corresponding to a single wavelength . at every single wavelength , the total luminosity of the stellar system is divided into a very large number ( typically @xmath1 to @xmath2 ) of monochromatic photon packages , which are launched consecutively through the dusty medium in random propagation directions . once a photon package is launched into the dusty medium ( either after an emission event or following a scattering event ) , it can be absorbed by a dust grain , it can be scattered by a dust grain , or it can travel through the system without any interaction . in a naive monte carlo routine , these three possibilities are possible and it is randomly determined which of the three will happen . this is generally an inefficient procedure , though , which leads to poor signal - to - noise both in the absorption rates in the different cells and in the scattered light images . to overcome these problems , we have set up a combination of continuous absorption and eternal forced scattering ( see section [ forced.sec ] for details ) . the result is that , contrary to most monte carlo codes where the life cycle of a photon package ends when it either leaves the system or is absorbed , the photon packages in skirt can never leave the system . the life cycle of a photon package ends when the package contains virtually no more luminosity ( typically we use the criterion that it must have lost 99.99% of its original luminosity ) . whenever this happens , a different stellar photon package is launched until also this one finishes its life cycle . this loop is repeated for all stellar photon packages at a given wavelength , and subsequently for all wavelengths ( in a multi - core system , each core can handle the loop corresponding to a different wavelength at the same time ) . after the stellar emission phase , the code moves to the dust emission phase . this phase is roughly similar to the stellar phase , except that the sources that emit the radiation are now not the primary , stellar sources but the dust cells . from the stellar emission phase we know the total amount of absorbed radiation at each wavelength in each cell of the dust domain . from this absorption rate we can calculate the mean intensity of the isrf in each cell , which allows the calculation of the dust emissivity , depending on the physical processes the skirt user is interested in ( see section [ dustemissivity.sec ] ) . with the sources ( the dust cells ) and their emissivity determined , the simulation now enters a loop that is very similar to the one in the stellar emission phase . at each individual wavelength , a huge number of photon packages is generated which are launched and followed consecutively through the dusty medium . care is taken that all regions of the dusty medium , including those cells where only a small amount of luminosity has been absorbed , are well - sampled . the dust - emitted photon packages in turn increase the absorption rate in the dust cells where they pass through . this results in an increase of the mean intensity of the isrf . the result is that at the end of the dust emission phase , the absorption rates used to calculate the dust emissivity in each cell do not correspond to the mean intensity of the isrf . this naturally leads to an iterative procedure , in which the absorption rate , the mean intensity and the dust emissivity are updated until convergence is achieved . we hence repeat the dust emission phase of the code several times . we typically require a 1% level accuracy in the dust bolometric absorption rate of each cell as a stopping criterion for the iteration . it is typically reached in only a few iterations ; for all the simulations we have done so far , less then five iterations have been necessary . the last phase of the monte carlo simulation starts when the last of the photon packages emitted by the dust component has lost 99.99% of its initial luminosity . it simply consists of calibrating and reading out the instruments ( all output is written to fits files ) and other useful information , such as 3d absorption rate maps and dust temperature distributions . a critical aspect in monte carlo radiative transfer simulations is the choice of the dust grid . the dust grid consists of tiny cells , each of which have a number of characteristics that fully describe the physical properties of the dust at the location of the cell . the choice of the grid has a significant impact on both the run time and the memory requirement of the simulation . indeed , each photon package typically requires several integrations through the dust ( i.e. the determination of the optical depth along the path and the conversion of a given optical depth to a physical path length ) , and the calculation time of a single optical depth typically scales with the number of grid cells crossed . different kinds of dust grids can be applied in the skirt code . the most general grid is a 3d cartesian grid in which each dust cell is a rectangular cuboid . for simulations with a spherical or axial symmetry , we also have 1d spherical and 2d cylindrical grids ( the elementary dust cells being shells or tori respectively ) . the distribution of the grid points ( in 1d spherical , 2d cylindrical or 3d cartesian grids ) can be chosen arbitrarily ; linear , logarithmic , or power - law cell distributions have been pre - programmed , but any user - supplied grid cell distribution is possible . the main goal of the dust grid is to discretize the dust density . we assume that the density of each dust component is uniform within each individual cell . in principle , the density does not need to be constant within each dust cell . in the first versions of skirt , we have experimented with a more sophisticated kind of dust grid , where the density of the dust within each cell is not uniform but determined by trilinear interpolation of the values of the density on the eight border points of the cell ( in case of a cartesian grid with cubic cells ) . in this case , the computation of optical depths in the dusty medium take more computation time , but the accuracy is increased such that a grid with less cells and less photon packages are needed per simulation . for models in which only absorption and scattering are taken into account , we found that this kind of dust grid is computationally more efficient than a dust grid with uniform density , in particular when the system harbors a large dynamical range of dust densities @xcite . however , for radiative transfer simulations in which the thermal emission of the dust is taken into account , each dust cell needs to contain an absorption rate counter , which collects the absorbed luminosity at every wavelength . the size of the dust cells is hence the typical resolution of the simulation , and the advantage of the interpolated grid ( where the dust grid can be coarser because the density is resolved within each cell ) largely disappears . skirt therefore only uses dust grids with a uniform density in each cell . the first step in the life cycle of each stellar photon is the random generation of the location where it is emitted . this means that we have to generate random positions from the three - dimensional probability distribution @xmath3 where @xmath4 is the luminosity density at the wavelength @xmath5 of the photon package . as skirt is primarily focused towards modeling galaxies , we have done efforts to optimize the generation of random positions from arbitrary 3d probability functions . the skirt code contains a library with common geometries for which the generation of a random position vector can be performed analytically . these include spherical @xcite , @xcite or @xcite models , or axisymmetric power - law , exponential , sech or isothermal disc models . for other frequently used luminosity density profiles , e.g. flattened @xcite or @xcite models , or more general density profiles that can not be described by an analytical function , such a direct analytical inversion is not possible . two complementary approaches have been included in the skirt code to deal with generating random positions from such density profiles . the first technique is to expand the density profile into a set of subcomponents . skirt contains a routine to perform a multi - gaussian expansion ( mge ) of surface brightness distributions . an mge expansion basically expands any surface brightness distribution as a finite sum of two - dimensional gaussian components . the mge method has proven to be a very powerful tool for image analysis : even with a relatively modest set of gaussian components , @xmath6 , even complex geometries can be reproduced accurately . one of the reasons why a mge expansion is very useful for skirt is that this approach enables a straightforward determination of the 3d spatial distribution : if the euler angles of the line of sight are known , the de - projection of a 2d gaussian distribution on the plane of the sky is a 3d gaussian distribution and the conversion formulae are completely analytical . generating random positions from a sum of 3d gaussian probability distributions is straightforward . an example of this approach can be seen in figure [ sombrero.pdf ] , where we present a radiative transfer model for the sombrero galaxy based on the mge expansion of its surface brightness distribution presented by . the second approach consists of sampling random positions directly from the stellar density distribution using the so - called stellar foam . the stellar foam is a skirt structure based on the foam library developed by @xcite and @xcite . foam is a self - adapting cellular monte carlo tool aimed at monte carlo integration of multi - dimensional functions , including integrands with an arbitrary pattern of singularities . it achieves a high efficiency thanks to an intelligent division of the integration space into small simplical or hyper - rectangular form , which are created in a self - adaptive way by binary splitting . it has originally been developed for use in high - energy physics ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , but it can also be adopted as a general - purpose monte carlo event generator . we use an adapted version of the foam library for the generation of random positions from an arbitrary probability density @xmath7 . one problem is that the probability density @xmath7 from which we want to generate random positions is typically defined on the entire 3d space , whereas foam requires a probability density on the @xmath8-dimensional unit hypercube @xmath9^n$ ] . we achieve this through a coordinate transformation from the usual coordinates @xmath10 to new coordinates @xmath11 such that we map each infinite interval @xmath12 $ ] onto the unit interval @xmath9 $ ] . the probability density @xmath7 , normalized such that @xmath13 is transformed to a new probability density @xmath14 this new density will be defined and normalized to one on the unit cube , @xmath15 there are many possible transformations we can apply to achieve this . we have chosen the transformation [ barxyz ] @xmath16 with @xmath17 , @xmath18 and @xmath19 three scale parameters , which we can adapt for the specific probability density we are considering . the inverse transformation is [ xyz ] @xmath20 and the jacobian reads @xmath21 summarizing , if we want to generate random positions @xmath22 from an arbitrary probability density @xmath7 , we first determine representative scale lengths @xmath17 , @xmath18 and @xmath19 along the three dimensions , and subsequently calculate the corresponding probability density @xmath23 using equations ( [ barp ] ) , ( [ xyz ] ) and ( [ jacobian ] ) . the foam generator is applied to this new probability function to generate the random points @xmath24 , which are converted to the desired positions @xmath22 through the formulae ( [ xyz ] ) . the construction of the stellar foam takes only a few seconds in two dimensions up to about one minute in three dimensions . once a photon package has been generated at a random location ( sampled from the stellar density ) and it has been given a random propagation direction ( sampled randomly from the unit sphere ) , it is ready to start its journey through the dusty medium . it has three possible fates : it can be absorbed by a dust grain at a certain position along its path , it can be scattered by a dust grain at a certain position on its path , or it can travel along its path through the system without any interaction . the probability for each of these three options are determined by the scattering albedo @xmath25 and the optical depth @xmath26 along the path , defined as @xmath27 where @xmath28 is the extinction coefficient , @xmath29 is the dust density and the integral covers the entire path of the photon package through the dusty medium . the most straightforward way to model these different physical processes in a monte carlo radiative transfer code is to generate randomly which of these three processes will take place . in case the photon package leaves the system , its lifetime is terminated and a different package is launched . in the case of a scattering event , the position of the scattering event is determined by choosing a random path length from the appropriate probability distribution and a scattering event is simulated . finally , in case it is absorbed , the position of the absorption is determined in a similar way , the luminosity of the photon package is stored in a local absorption rate counter attached to the dust cell where the absorption event took place , and the photon package s life is terminated . these local absorption rates are used in a later stage of the simulation to estimate the local mean intensity of the isrf , necessary to calculate the thermal dust emission . this traditional method has two significant drawbacks : along paths where the optical depth is modest , many photon packages will escape from the system without interactions , which will result in bad statistics of the scattered intensity and the absorbed luminosity . even if the photon packages do interact , most interactions will take place on those sections of the path where the density is largest . many absorption events are necessary in each cell to guarantee a high - quality estimate of the local absorption rate and the corresponding mean intensity . in dust cells with a low density ( such that only few absorptions take place ) and at wavelengths where the absorption rate is low , this usually requires large numbers of photon packages . these two problems can be minimized using the efficient combination of two clever monte carlo techniques : forced scattering and continuous absorption . continuous absorption ( or lucy estimation ) is a technique to estimate the mean intensity of the isrf throughout the dusty medium . the continuous absorption technique is designed to solve the problem of poor statistics in the absorption rate in low - density regions by spreading the absorption over all cells along the photon package s path instead of concentrating it in one single cell . a different but equivalent way to see it is that the mean intensity in each cell is estimated using the sum of the path lengths covered by all photon packages that traverse that particular cell . forced scattering is an old technique that was already implemented in the first monte carlo radiative transfer codes . when applying forced scattering , photon packages are forced to interact with a dust grain before they leave the system . this incorrect behavior is corrected for by decreasing the luminosity of the photon package . in most radiative transfer codes , forced scattering is considered only after an emission event and subsequent scattering events are unforced . in the skirt code we always consider forced scattering , such that we have eternal forced scattering . the combination of eternal forced scattering and continuous absorption results in a very efficient monte carlo routine . instead of determining randomly whether a photon package with luminosity @xmath30 will escape , will be absorbed or will be scattered , we split it into @xmath31 child photon packages ( with @xmath32 the number of dust cells along the path ) : one child photon package that will leave the system without interaction , one child photon package that will be scattered by a dust grain somewhere along the path , and @xmath32 children that will be absorbed , one in each of the @xmath32 cells along the path . the luminosity of each of these children is easy to calculate : we find @xmath33 with @xmath34 the optical depth to the point on the path where the photon package would leave the @xmath35th dust cell . obviously , we have @xmath36 in our monte carlo simulation , we now consider each of these @xmath31 child photon packages . each one of the @xmath32 children with luminosity @xmath37 is absorbed in the @xmath35th cell along the path , which means that its luminosity is added to the absorption rate counter attached to this dust cell . the child photon package with luminosity @xmath38 escapes from the system , which implies that we do nt have to take this one into account anymore . finally , the child photon package with luminosity @xmath39 is scattered somewhere along the path . this is basically the only photon package that we need to follow up . we still have to randomly determine the location of the scattering event along the path . this is achieved by selecting a random optical depth @xmath40 from an exponential distribution over the finite range @xmath41 , i.e.@xmath42 and translate this randomly generated @xmath40 to a physical path length by solving the equation @xmath43 for @xmath44 . once this path length has been determined , we can determine the position of the scattering event . if we then also determine a new propagation direction , determined randomly by generating a random direction from the scattering phase function , we are back at the starting point . this child now becomes the parent , it can be split into children and we can repeat the same loop all over again . summarizing , the net result of the combination of continuous absorption and eternal forced scattering is that after each emission / scattering event , we distribute a fraction of the photon package s luminosity among all the cells along the path , and we continue the monte carlo loop with a less luminous photon package that is always scattered at some point along the path . hence , contrary to most monte carlo codes where the life cycle of a photon package ends when it either leaves the system or is absorbed , the photon packages in skirt can never leave the system . the life cycle of a photon package ends when the package contains virtually no more luminosity . typically we use the criterion that it must have lost 99.99% of its original luminosity , which immediately is the minimum level of absolute energy conservation of the simulation . the goal of a monte carlo radiative transfer simulation is to simulate observable properties of a dusty system , i.e. images and spectral energy distributions . skirt uses the technique of peel - off photon packages to create an arbitrary number of images / seds at different observing positions . peeling off is a monte carlo technique designed to create high signal - to - noise images ( in particular scattered light images ) , adopted for the first time by @xcite and included in almost all state - of - the - art monte carlo radiative transfer codes . after every emission or scattering event , the code calculates which fraction of the luminosity contained in the photon package would arrive at the observers locations and at which point on the plane of the sky , if the photon package would be emitted or scattered in the appropriate propagation direction . repeating this for every photon package at every emission or scattering event implies that the maximum available information is obtained for a fixed set of photon packages and hence strongly increases the signal - to - noise compared to the more simple monte carlo codes where only photon packages that leave the system are recorded . each skirt detector is basically an integral field detector , i.e. a data cube with two spatial and one wavelength dimension . in most monte carlo radiative transfer codes the simulated detectors are natural , idealized representations of actual ccd detectors ( or a series of them at each wavelength ) . they basically consist of a two- or three - dimensional array of pixels , which act as a reservoir for the incoming photon packages . when a photon package leaves the system and arrives at the location of the observer , the correct pixel is determined and the luminosity of the photon package is added to the luminosity at that pixel . at the end of the simulation , the detector is read out pixel by pixel and the surface brightness distribution is constructed . while this approach seems the most natural way to simulate the detection of photon packages in a monte carlo simulation , it might not be the most efficient . we must be aware that , although we are simulating a real detection as closely as possible , we have more information at our disposal than real observers . the maximum information that a real observer can obtain ( in the theoretical limit of perfect noise - free observations and instruments ) when imaging with a ccd detector is the number of photon packages that arrive in each of his pixels . as numerical simulators , we have at our disposal the full information on the precise location of the impact of each photon package on the detector . in order to use this information , we have considered the concept of smart detectors , which take full advantage of the exact location of the impact of the incoming photon packages @xcite . the principle of these smart detectors is based on the estimate of the density distribution in smoothed particle hydrodynamics simulations . while preserving the same effective resolution and having virtually no computational overhead , smart detectors realize a noise reduction of about 10 percent . a crucial aspect of the skirt code is the calculation of the dust emissivity . from the stellar emission phase we know the total amount of absorbed radiation @xmath45 at each wavelength in each cell of the dust domain . from this absorption rate we can calculate the mean intensity of the isrf @xmath46 in each cell using @xmath47 where @xmath48 is the dust absorption coefficient , @xmath49 is the dust density and @xmath50 is the volume in cell number @xmath51 respectively . knowledge of the mean intensity and the dust properties in each cell allows the dust emissivity @xmath52 to be determined . skirt contains three different modules for the calculation of the dust emissivity , depending on the physical processes that are taken into account . the simplest option is to consider the dust grains as a single species that is in local thermal equilibrium ( lte ) with the local isrf . in this case , the dust emits as a modified blackbody radiator , @xmath53 where the dust equilibrium temperature @xmath54 of the @xmath51th dust cell is determined by the condition of thermal and radiative equilibrium , @xmath55 the second , somewhat more realistic option is to still consider lte for the dust grains , but taking into account that each species and size of dust grain reaches its own equilibrium temperature . the skirt code allows to consider dust mixtures with an arbitrary number of grain species and size distributions . the size distributions of the various dust species are subdivided into different bins , resulting in a dust mixture with @xmath56 populations , each of them corresponding to a dust species and a small size bin . assuming lte for each individual population , the dust emissivity is given by a sum of modified blackbodies , where the temperature of each population is still determined by the condition of thermal and radiative equilibrium , @xmath57 with @xmath58 is the contribution of the @xmath59th dust population to the total absorption coefficient @xmath48 , and @xmath60 the equilibrium temperature of the @xmath59th population in cell number @xmath51 , determined by @xmath61 the third option , in fact the only realistic option to model the spectral energy distribution of galaxies , is to consider nlte dust emission . in theory , the transit from lte to nlte dust emission is not an enormous step . the main difference is that each dust population is not characterized by a single equilibrium temperature , but by a temperature distribution . once the temperature distribution function has been determined , the dust emissivity can easily be determined . as argued in the introduction , however , the practical inclusion of nlte dust emission in radiative transfer codes is a notoriously tough nut to crack . rather than develop our own routines to calculate the nlte emission for transiently heated grains , we have opted to couple the skirt code to the dustem code . dustem is a publicly available , state - of - the - art numerical tool designed to calculate the nlte emission and extinction of dust given its size distribution , optical and thermal properties . the code builds on the work by and uses an adaptive temperature grid on which the temperature distribution of the grains is calculated iteratively . no lte approximation is made , i.e. even for large grains the temperature distribution is calculated explicitly . one of the advantages why we have chosen to couple skirt to the dustem code is that the latter code has been designed to deal with a variety of grain types , structures and size distributions and that new dust physics ( ionization of pahs , polarized emission , spinning dust emission , temperature - dependent dust emissivity ) can easily be included . on the other hand , a consequence of choosing for a very complete and accurate nlte routine in which basically no simplifications or assumptions have been made , is that the computation load is substantial . for a typical dust model consisting of three or four dust species each with their size distribution , the calculation of the emissivity for a single isrf takes typically of the order of several seconds on a standard desktop / laptop computer . while this is compatible with 1d or 2d simulations with up to @xmath62 cells , this is excessively long for general 3d simulations for which we have designed skirt . to overcome this problem , we have adopted a strategy based on a library of dust emissivity profiles , inspired by the work of . their approach consists of three steps : they first run an exploratory radiative transfer simulation on a grid with a reduced number of grid cells , without taking dust re - emission into account . this low - resolution simulation is used to determine the range of isrfs encountered in the simulation . the second step consists of picking a small number @xmath63 of reference wavelengths @xmath64 ( typically @xmath65 ) . the different isrfs found in step one are discretized onto logarithmic intervals at each of the reference wavelengths , and at each bin in the @xmath63-dimensional parameter space , the full nlte dust emission spectrum is calculated and stored in a library . the final step in the simulation consists of running a radiative transfer simulation at the full resolution . the dust emissivity at any given cell is found by looking at the isrf at the @xmath63 reference wavelengths and interpolating the dust emissivities from the library . while valuable and inspiring , we see two drawbacks in the method as implemented by . in panchromatic radiative transfer simulations of galaxies , we typically solve the transfer equation at many uv / optical wavelengths , and hence have the isrf at all these wavelengths at our disposal at every grid cell . it would be a pity not to use this information to determine the dust emissivity and only base our estimate on the value of the isrf at a very small number of reference wavelengths . in particular , the isrf in monte carlo simulations can be noisy in certain cells ; when the dust emissivity is determined based on the value of the isrf at a small number of reference wavelengths , this noise could lead to a significant error . using an estimate that exploits the available information at all wavelengths can minimize this error . the second drawback is that the library method of requires an exploratory , low - resolution simulation in which the parameter space of isrfs is explored and the library of dust emissivities is built . this extra simulation not only requires a computational overhead , it also creates the danger that it does not cover the entire range of strengths and shapes of the isrf . for example , one can assume that the strongest isrf in a simulation is found in small dust cells very close to the heating sources . in a low - resolution simulation , with larger dust cells , this strong isrf will be smoothed over the larger grid cells . similarly , the weakest isrf ( or equivalently , the coldest dust ) in some simulations could be found in the inner regions of dense cores , and due to smoothing a low - resolution simulation might not reach these weakest isrf levels . the result is that the low - resolution grid will not cover the full range of isrfs encountered in the high - resolution grid , and hence that the library of dust emissivities must be somehow extended to incorporate this missing part in the parameter space . are aware of this inconvenience ( they discuss only the coldest spectrum as they concentrate on dark clouds illuminated by an external radiation field ) . they argue that this problem is not expected to be significant , and that it could be relieved by using a low - resolution simulation with a slightly higher density . still , it is clear that the use of a low - resolution grid leads to an additional overhead and complication and is a potential source of error , and it would be better to avoid it . to overcome both problems , we have taken a slightly different approach to implement our dust emissivity library . the first step in our library approach is to calculate a number of parameters that characterize the isrf in each dust cell after the stellar emission phase . instead of the value of @xmath66 at a number of wavelengths , we use parameters that use combined information at all available wavelengths . from the various range of possibilities , we choose the lowest - order moments of @xmath67 , the product of the isrf and the dust absorption coefficient . instead of the actual zeroth - order moment or normalization of this function , we consider the equivalent would - be equilibrium dust temperature @xmath68 of the dust mixture , found by solving the equation @xmath69 as a second parameter , we take the first - order moment @xmath67 , i.e. the mean wavelength , @xmath70 in skirt we limit ourselves to two parameters , but in principle this procedure can be extended to more parameters . with @xmath68 and @xmath71 calculated in each dust cell , we construct a 2d rectangular grid with @xmath72 pixels in the @xmath73 parameter space , based on the range of values encountered in the present simulation . in every parameter space pixel we construct a reference isrf by averaging all isfrs that correspond to those particular values of @xmath68 and @xmath71 . we experimented with different ways of averaging , including taking the straight mean , the median or the mean using sigma - clipping , but found no noticeable difference . the final step of the library construction is to feed the reference isrfs to the dustem routine , and save each of the resulting dust emissivity profiles in the library . once the library is constructed , finding the correct dust emissivity for a given dust cell is straightforward , as each dust cell is already connected to a certain pixel in the @xmath74 parameter space and hence a dust emissivity profile in the library . while the very first version of skirt were written in fortran 77 , the code is now completely written in ansi c++ and currently contains some 30000 lines of code . it uses the object - oriented nature of the c++ language extensively to support a strong modularity . the use of inheritance and abstract classes renders the inclusion of new components ( such as new density distributions for the stars or dust , or new dust mixtures ) straightforward . the dustem code is written in fortran 95 and has been slightly adapted to be coupled to skirt . the entire skirt code is driven by a graphic user interface written in pyqt . batch jobs can be run using a command line version with xml input files . an important implementation aspect of skirt is the parallelism . parallelism can typically work on two domains : data parallelism focuses on distributing the data across different parallel computing nodes , with the principle aim of enabling simulations that need more memory consumption than is available on a single node . task or control parallelism focuses on distributing execution processes ( threads ) across different parallel computing nodes with the principle aim of decreasing the run - time of a program . ideally , both approaches can be combined . monte carlo radiative transfer codes are easily parallelized in a task parallelism approach : the different levels of iterations can easily be split over different nodes . skirt uses the openmp protocol to support task parallelism on shared - memory machines . one of the advantages of openmp parallelism is the spectacularly low coding cost : less than 100 lines of code ( on a total of 30000 ) have been added to skirt to convert it from a serial into a parallel code . the main parallelism is situated in the loop over wavelength , which implies that skirt runs both the stellar and dust phases at different wavelengths simultaneously . one of the planned future developments of skirt is to look into the possibilities of using the mpi interface for data parallelism , in order to allow monte carlo simulations to be run on distributed memory systems . data parallelism is much harder to achieve for monte carlo simulations than task parallelism . the main reason is the non - local nature of the physical problem : each photon package in a simulation typically requires data from the entire physical domain ( read access to calculate the optical depth and write access to update the absorption rates ) . contrary to e.g. grid - based hydrodynamic codes , distributing the dust cells over different parts of memory would imply an enormous overhead in communication between the different nodes . in principle , data parallelism could be achieved by splitting the data in the wavelength dimension , where different nodes contain different parts of the absorption rate counters of the dust grid and different parts of the detectors . however , in this approach a large amount of data ( such as the dust density grid ) would need to be shared / copied between the different nodes and communication overheads would be significant in the dust emission phase . future work will investigate whether the benefits of distributed - memory parallelism can outweigh the communication overheads and the significant additional coding complexity . we have run extensive tests to check the accuracy of the skirt code against other radiative transfer codes . the early versions of the code were already tested against several other results , most importantly the set of spiral galaxy models by @xcite ( see e.g. * ? ? ? * ) . we have successfully tested the lte version of skirt against the 1d and 2d lte circumstellar benchmark problems of @xcite and . skirt is also one of the codes used in a new ongoing lte benchmark effort focusing on a disc galaxy environment ( baes et al . 2011 , in preparation ) . the preliminary results , based on the results from five independent radiative transfer codes indicate excellent agreement , with relative differences in the seds around the 1% level or even below . as a full nlte radiative transfer benchmark is not ( yet ) available at the moment , we have tested our nlte radiative transfer code , and particularly the library approach , using different models with gradually increasing levels of complexity . in order to run simulations in a realistic setting , we adopt the sbc galaxy ugc4754 as a template model . ugc4754 is a edge - on spiral galaxy , which has always been a favorite class of galaxies for radiative transfer modellers , as the dust is clearly visible both in absorption and emission . this galaxy was one of the first large edge - on galaxies to be observed with the herschel space observatory as part of the herschel astrophysical terahertz large area survey ( h - atlas , * ? ? ? * ) science demonstration phase observations . in , we fitted a radiative model to the observed images of ugc4754 in the sdss and ukidss bands . we subsequently used the skirt code to predict the galaxy s sed and images at fir wavelengths . while the radiative transfer model used in that paper was sufficient to serve its goal ( investigation of the dust energy balance ) , it suffered from two significant limitations : the assumptions of lte dust emission and of a smooth interstellar medium . these two assumptions prevented us from making a self - consistent model covering the entire spectral region from the uv to mm wavelengths . moreover , they also might introduce a significant source of uncertainty , as it has been demonstrated by several authors that the extinction properties of a clumpy interstellar medium can be significantly different from those of a smooth medium . in this section we will gradually refine our model for ugc4754 from a smooth , 2d , lte model to a fully 3d model that includes nlte dust emission and a clumpy structure of the dusty ism . the main objectives are to test the accuracy of our approach using a realistic setting , and to demonstrate the ability of skirt to run realistic 3d nlte radiative transfer calculations . for a full investigation of the dust energy balance in spiral galaxies , based on our refined skirt code and multi - wavelength imaging data , we refer to future work . the starting point for our models is the best fitting , smooth 2d model from . the stellar distribution consists of two components : a double - exponential stellar disc with a scale length of 4.05 kpc and a scale height of 330 pc , and a flattened srsic bulge with a major axis effective radius of 800 pc , a srsic parameter of 0.9 and an intrinsic flattening of 0.6 . both components have a similar intrinsic sed , corresponding to a population of 8 gyr old with an exponentially decaying star formation rate and an initial burst duration of 0.15 gyr . the total bolometric luminosity of the system is @xmath75 , of which the bulge contributes 8% . the dust is also distributed in a double - exponential disc with a scale length of 6.1 kpc and a scale height of 270 pc . the total amount of dust is characterized by the @xmath76-band edge - on optical depth of 0.73 , which corresponds to a total dust mass of @xmath77 . contrary to where we used the @xcite model to describe the dust optical properties , we now use the @xcite dust model , as this model is embedded in the dustem library . simulations are run on a wavelength grid with 181 grid points , with 101 grid points distributed logarithmically beween 0.05 and 5000 @xmath78 m and 80 additional grid points distributed logarithmically between 1 and 30 @xmath78 m to capture the pah peaks . for our 2d simulations , we considered an axisymmetric grid with 51 grid points in the radial direction and 51 grid points in the vertical direction , resulting in a total number of @xmath79 grid cells . the grid points are chosen to have a power - law distribution , with an extent of 30 kpc ( 2 kpc ) for the radial ( vertical ) distribution and a size ratio of 30 between the innermost and outermost bins . in all skirt runs discussed here , we used @xmath80 photon packages for each wavelength in both the stellar and the dust emission phase . figure [ totalsed.pdf ] shows the resulting seds of two different skirt 2d simulations based on this model setup , as well as the observed galex , sdss , ukidss , iras and herschel fluxes for ugc4754 . the blue curve shows the sed corresponding to a model assuming lte dust re - emission , where we took into account that different grain types and sizes reach different equilibrium temperatures . the black curve shows the sed of the model that includes nlte dust re - emission using the library approach discussed in section [ dustemissivity.sec ] . the red curve represents the flux of the model without dust extinction , the yellow line corresponds to the contribution of the dust to the sed in the nlte case . it is logical that there is no difference between the lte and nlte models in the uv / optical / nir part of the sed , as only the dust emissivity differs between the models . there is , however , a significant difference between the lte and nlte models in the mir / fir / submm window : in the lte models , all absorbed radiation is re - emitted as modified blackbody emission in the fir / submm region , whereas the nlte models emit part of the absorbed radiation in the mir region . between the dust seds of 2d nlte models for ugc4754 using the brute force ( bf ) approach and the library ( lib ) approach.,scaledwidth=48.0% ] to test the accuracy of our library approach , we ran a second nlte simulation , where we did not use the library approach , but where we calculated the dust emissivity in each cell by an explicit call to the dustem code . the relative difference between the seds of the skirt nlte models using the library approach and the brute force approach is shown in figure [ bruteforce.pdf ] . this figure convincingly shows that the library approach is accurate : the relative error is everywhere below 0.2% and in the region where the dust emission dominates even smaller . the difference in run time between the two approaches is substantial , and this difference is only due to the difference in the number of calls to the dustem routine . in the brute force approach , the number of calls is obviously equal to @xmath81 ( or sometimes a bit less if there are empty dust cells ) . the maximum number of calls to the dustem routine in the library approach is obviously @xmath72 , the number of pixels in the library parameter space . as the intensity of the isrf is the main parameter that drives the shape of the dust emissivity spectrum , it is appropriate to choose @xmath82 somewhat larger than @xmath83 ; the values we propose are @xmath84 and @xmath85 . this infers a maximum of 250 calls to the dustem routine . in most cases , however , not the entire @xmath73 parameter space is covered , such that the number of calls is even further reduced . in the present simulation , we needed to call the dustem routine only 71 times in the library approach , compared to 2500 times in the brute force approach . given that the average dustem run time is about 7 s ( on a typical desktop computer ) and that the overhead of the construction of the library is negligible , this makes a substantial difference . it should be noted that 2500 calls is still a very manageable number , but the brute force approach becomes impossible when moving to 3d grids with several million cells . for example , for a simulation with ten million dust cells , we would need a dustem computation time of more than 2 years on a single core computer , compared to several minutes using the library approach . between the dust seds using the mw isrf template ( tem ) approach and our library ( lib ) approach.,title="fig:",scaledwidth=48.0% ] between the dust seds using the mw isrf template ( tem ) approach and our library ( lib ) approach.,title="fig:",scaledwidth=48.0% ] as a final test and a demonstration of the necessity of our library approach , we ran our 2d simulation again , but now using a set of precomputed template emissivity profiles . in a sense this mimics the approach taken by e.g. @xcite , although their approach is somewhat different as they calculate the emission by big grains using the lte approximation and only use a template for the small grains . the basic idea of our template approach is that the dust emissivity from a cell depends only on a single parameter @xmath86 , being the integrated mean intensity of the isrf in the considered cell expressed in terms of the integrated mean intensity of the isrf in the milky way , @xmath87 for the isrf of the mw , the standard parameterization of is adopted . we constructed a library of 501 dust emissivity profiles , corresponding to values of @xmath86 distributed logarithmically between @xmath88 and @xmath89 . these dust emissivity profiles can be computed once and for all ( using the dustem routine ) and saved in a file . in the dust emission phase , we simply calculate @xmath86 in every dust cell and determine the dust emissivity profile using logarithmic interpolation between the precomputed library profiles . this approach definitely has a strong appeal : it is straightforward to implement and very fast , as it requires only the calculation of @xmath86 and a simple interpolation of precomputed values . in this sense it is more attractive than our library approach , which requires the construction of a library for every simulation . the disadvantage of the template approach , however , is that only the strength and not the shape or hardness of the isrf is taken into account to calculate the dust emissivity . as also argued by @xcite , the shape of the isrf can have a significant importance , both because of the differing cross - sections of grains of different sizes and because high - energy photons will excite larger thermal fluctuations than low - energy photons for a given value of @xmath86 . in figure [ milkywaylibrary.pdf ] we show the comparison of the fixed template approach and our dynamic library approach ( dynamic in the sense that the library is tailored to the specific simulation ) . the top panel shows the total sed and the dust emission sed for our 2d model for ugc4754 obtained using both approaches . at first sight , the seds agree fairly well ( definitely when plotted in log - log scale ) . looking at the bottom panel , where we plot the relative difference of the dust emission sed corresponding to both approaches , we do see a significant difference , with relative deviations up to more than 40% . here we have to give an important side note , in the sense that we believe that this 40% difference is actually a underestimate of the error one can make . the shape of the intrinsic sed of the stellar population in our model is independent of position ; as a result , the variations in the shape of the isrf in different cells in the simulation are only the result of varying levels of absorption and scattering . since the stellar population model we adopted for ugc4754 ( an 8 gyr old population with an exponentially decaying star formation rate and an initial burst duration of 150 myr ) is not very different from the average stellar population in the milky way , this implies that the shape of the isrf in the different cells in our model will on average be quite close to the shape of the milky way isrf . as far as the comparison between our library approach and the milky way template approach concerns , we must hence conclude that ugc4754 simulations are not the strongest test . for systems with isrfs which deviate much more from the average milky way isrf , such as starburst galaxies or circumstellar discs around young hot stars , we expect much larger differences than the 40% we obtained here . having tested the accuracy of our library approach , we are ready to run full - scale 3d nlte radiative transfer models with skirt . the first 3d model we consider has exactly the same set - up as the 2d nlte model , except that we now consider a uniform 3d cartesian grid . in the @xmath90 and @xmath91 directions we consider 401 grid points each and a maximum extent of 30 kpc , in the vertical direction we use 61 grid points with a maximum extent of 2 kpc . this results in @xmath92 grid cells , each with a dimension of 150 pc in the @xmath90 and @xmath91 directions and 66.7 pc in the vertical direction . note that we need to store , at each dust grid cell , the entire isrf @xmath66 at each of the wavelength grid points , which basically turns our grid into a 4-dimensional grid structure with @xmath93 grid cells . the memory required to run such a large - scale skirt radiative transfer simulation is about 23 gb . between the seds of 2d and 3d nlte models for ugc4754 . the red curve corresponds to a 3d model with a uniform cartesian dust grid in all three dimensions , the blue curve corresponds to a 3d model with a power - law grid in the vertical direction , similar to the vertical grid structure of the 2d model.,scaledwidth=48.0% ] figure [ compare2d3d.pdf ] compares the sed of the smooth 2d and 3d models for ugc4754 . the relative differences are below 2% in the entire uv - mm domain . the existing small differences are mainly due to the discretization of the grid in the vertical direction : for the 2d model we used a vertical grid with a power - law distribution , with smaller bins close to the equatorial plane , where the dust density has the strongest gradients . the innermost grid cell has a height of only 8.75 pc , compared to the 66.7 pc in the case of the 3d grid . the result is that the discretization of the dust density on the 3d grid is much coarser . to demonstrate that this vertical grid distribution is the origin of this @xmath94 difference , we ran another 3d simulation where we now apply the same power - law distribution for the vertical grid cells as we did for the 2d model . the relative differences between the sed of this model and the sed of the 2d simulation are also indicated in figure [ compare2d3d.pdf ] . the reason why we considered a uniform cartesian dust grid is because such a grid forms the basis for a fully 3d model with a clumpy , two - phase dust distribution . to generate such a model , we followed the strategy outlined by @xcite . the dusty interstellar medium consists of two phases , a smooth inter - clump component and a clumpy component , and is characterized in terms of two parameters , namely the volume filling factor _ ff _ of the dense clumps and the density contrast @xmath95 between the clump and inter - clump medium . the practical construction of the two - phase medium consists of randomly assigning a status ( clump or inter - clump ) to each dust cell in the dust medium . typical values for the parameters @xmath95 and _ ff _ vary widely in the literature . we use the values @xmath96 and @xmath97 in this work , which are within the range of typical parameters used in other studies ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . between the seds of clumpy and smooth 3d nlte models for ugc4754.,scaledwidth=48.0% ] in figure [ clumpysmooth.pdf ] , we show simulated images corresponding to the clumpy and smooth 3d models for the inner region of ugc4754 at three different wavelengths ( 0.55 , 24 and 500 @xmath78 m ) . it is important to note that the pixel - to - pixel variations in the images on the bottom row are not due to poisson noise in the monte carlo routine , but represent real intensity variations due to the clumpy nature of the dust in the models . figure [ compareclumpysmooth.pdf ] compares the total sed of the clumpy and smooth models . the relative differences are below 6% in the entire uv - mm domain . at uv wavelengths , the clumpy model is slightly brighter than the smooth model , or put differently , slightly less uv radiation is absorbed by the dust . this is in agreement with conclusions found by other authors : for a fixed amount of dust , a clumpy dust medium absorbs radiation less efficiently than a smooth dust medium ( e.g. * ? ? ? * ; * ? ? ? when moving to longer wavelengths , the difference between the smooth and clumpy models decreases , as the fraction of absorbed versus unattenuated radiation decreases with increasing wavelength . at nir wavelengths , there is virtually no difference anymore between the seds of the smooth and clumpy models . moving to mir and fir wavelengths , we find that the clumpy model emits significantly less , particularly at wavelengths up to about 100 @xmath78 m . this is no surprise , as the clumpy model was less efficient in absorbing uv and optical radiation . the result is that , for the same total mass , the dust in a clumpy model is on average both cooler and less luminous than the dust in a smooth model . we have presented an updated version of the 3d monte carlo radiative transfer code skirt . the code uses various advanced optimization techniques , both well - known and novel ones , that make the monte carlo process orders of magnitude more efficient than the most basic monte carlo technique . these techniques include an optimized combination of eternal forced scattering and continuous absorption , a multi - gaussian expansion technique and an efficient foam generator to generate random positions from the stellar density , and the use of peeling - off and smart detectors to create high signal - to - noise images and seds . the main novelty of the new skirt code is the possibility to calculate the dust temperature distribution and the associated infrared and submm emission with a full incorporation of the emission of transiently heated grains and pah molecules . to achieve this , we have chosen to link the skirt code to dustem , a publicly available , state - of - the - art numerical tool designed to calculate the nlte emission of arbitrary mixtures of dust grains . the advantages of this approach is that no lte approximation is made , even for large grains , and that new physics ( such as spinning dust emission or a temperature - dependent dust emissivity ) can readily be included . we have implemented a library approach to limit the computational cost of the nlte dust emission calculations inherent in dustem . our approach is inspired by the work by , but uses a slightly different approach that makes maximum use of all information in the simulation to calculate the dust emissivity and avoids the need for additional low - resolution simulations . we have tested the accuracy of the skirt code , in particular of our nlte library approach , through a set of simulations based on the edge - on spiral galaxy ugc 4754 , previously modelled by . the models we ran were gradually refined from a smooth , 2d , lte model to a fully 3d model that includes nlte dust emission and a clumpy structure of the dusty ism . using 2d models , we demonstrated the accuracy of our library approach : the relative differences in the sed between a model that uses the library approach and a model that uses brute force to calculate the dust emission are less than 0.2% at all wavelengths . even for this 2d model with only 2500 dust cells , the difference in run time between both approaches are substantial ; for 3d grids with several million dust cells the brute force approach becomes impossible . we have also explored the possibility to use a fixed set of precomputed dust emission templates instead of a dynamic library as the one we have chosen . while a template approach has the advantage that it is easier to implement and faster to run , we have demonstrated that it leads to significant deviations due to the fact that it does not take into account the shape of the interstellar radiation field . this highlights the need for a more advanced approach such as the library approach we propose . we have subsequently applied the skirt code to calculate full - scale 3d nlte models for ugc 4754 . we found small differences ( @xmath98% ) between 2d and 3d smooth models that are mainly due to differences in the vertical discretization of the internal grid . finally , we have compared 3d models with a smooth and a clumpy interstellar dust medium . we confirm the result found by other authors that , for a fixed amount of interstellar dust , a clumpy dust medium absorbs radiation less efficiently than a smooth dust medium . as a direct consequence , the dust in clumpy models is on average both cooler and less luminous , and the observed infrared emission of clumpy models is less than the emission at these wavelengths of smooth models with the same dust mass . our simulations demonstrate that , given the appropriate use of optimization techniques , it is possible to efficiently and accurately perform monte carlo radiative transfer simulations of arbitrary 3d structures of several million dust cells , including a full calculation of the nlte emission by arbitrary dust mixtures . this significantly increases the number of applications where detailed radiative transfer modeling can be used . for example , we have started an investigation of the energy balance crisis in a set of edge - on spiral galaxies : our intention is to fit detailed radiative transfer models to uv / optical / nir images for a set of edge - on spiral galaxies , predict the resulting mir / fir / submm emission and compare these predictions with the available long wavelength data . many other applications ( agns , circumstellar discs , merging galaxies , ) are possible , and the authors welcome all projects that can make use of skirt .	we present an updated version of skirt , a 3d monte carlo radiative transfer code developed to simulate dusty galaxies . the main novel characteristics of the skirt code are the use of a stellar foam to generate random positions , an efficient combination of eternal forced scattering and continuous absorption , and a new library approach that links the radiative transfer code to the dustem dust emission library . this approach enables a fast , accurate and self - consistent calculation of the dust emission of arbitrary mixtures of transiently heated dust grains and polycyclic aromatic hydrocarbons , even for full 3d models containing millions of dust cells . we have demonstrated the accuracy of the skirt code through a set of simulations based on the edge - on spiral galaxy ugc4754 . the models we ran were gradually refined from a smooth , 2d , lte model to a fully 3d model that includes nlte dust emission and a clumpy structure of the dusty ism . we find that clumpy models absorb uv and optical radiation less efficiently than smooth models with the same amount of dust , and that the dust in clumpy models is on average both cooler and less luminous . our simulations demonstrate that , given the appropriate use of optimization techniques , it is possible to efficiently and accurately run monte carlo radiative transfer simulations of arbitrary 3d structures of several million dust cells , including a full calculation of the nlte emission by arbitrary dust mixtures .	16,424	354
our title clearly alludes to the story of columbus landing in what he called the west indies " , which later on turned out to be part of the new world " . i have substituted antarctica in place of the new world " , following a quip from frank paige after he realized that i was talking all the time about _ penguins_. at the end of the millennium , we are indeed on another discovery voyage . we are at the dawn of observing cp violation in the b system . the stage is the emerging penguins . well , had columbus seen penguins in _ his _ west indies " , he probably would have known he was onto something really new . the em penguin ( emp ) @xmath0 ( and later , @xmath1 ) was first observed by cleo in 1993 . alas , it looked and walked pretty much according to the standard model ( sm ) , and the agreement between theory and experiment on rates are quite good . perhaps the study of cp asymmetries ( @xmath2 ) could reveal whether sm holds fully . the strong penguins ( p ) burst on the scene in 1997 , and by now the cleo collaboration has observed of order 10 exclusive modes @xcite , as well as the surprisingly large inclusive @xmath3 mode . the @xmath4 , @xmath5 and @xmath6 modes are rather robust , but the @xmath7 and @xmath8 rates shifted when cleo ii data were recalibrated in 1998 and part of cleo ii.v data were included . the @xmath9 and @xmath10 modes are still being reanalyzed . the nonobservation , so far , of the @xmath11 , @xmath12 and @xmath13 modes are also rather stringent . the observation of the @xmath14 mode was announced in january this year , while the observation of the @xmath15 and @xmath16 modes were announced in march . cleo ii.v data taking ended in february . with 10 million or so each of charged and neutral b s , new results are expected by summer and certainly by winter . perhaps the first observation of direct cp violation could be reported soon . with belle and babar turning on in may , together with the cleo iii detector upgrade all with @xmath17 separation ( pid ) capability ! we have a three way race for detecting and eventually disentangling _ direct _ cp violation in charmless b decays . we expect that , during 19992002 , the number of observed modes may increase to a few dozen , while the events per mode may increase from 1070 to @xmath18@xmath19 events for some modes , and sensitivity for direct cp asymmetries would go from the present level of order 30% down to 10% or so . it should be realized that _ the modes that are already observed _ ( @xmath20 ) _ should be the most sensitive probes . _ our first theme is therefore : _ is large @xmath2 possible in @xmath20 processes ? _ and , _ if so , whither new physics ? _ however , as an antidote against the rush into the brave new world , we point out that the three observed @xmath21 modes may indicate that the west indies " interpretation is still correct so far . our second subject would hence be _ whither ewp ? now ! ? _ that is , we will argue for the intriguing possibility that perhaps we already have some indication for the electroweak penguin ( ewp ) . it is clear that 1999 would be an exciting landmark year in b physics . so , work hard and come party at the end of the year / century / millennium celebration called third international conference on b physics and cp violation " , held december 3 - 7 in taipei @xcite . we shall motivate the physics and give some results that have not been presented before , but refer to more detailed discussions that can be found elsewhere @xcite . our interests were stirred by a _ rumor _ in 1997 that cleo had a very large @xmath2 in the @xmath6 mode . the question was : _ how to get large @xmath2 ? _ with short distance ( bander - silverman - soni @xcite ) rescattering phase from penguin , the cp asymmetry could reach its maximum of order 10% around the presently preferred @xmath22 . final state @xmath23 rescattering phases could bring this up to 30% or so , and would hence mask new physics . but a 50% asymmetry seems difficult . new physics asymmetries in the @xmath1 process @xcite and @xmath24 process @xcite are typically of order 10% , whereas asymmetries for penguin dominant @xmath20 transitions are expected to be no more than 1% . the answer to the above challenge is to _ hit sm at its weakest ! _ * _ weak spot of penguin _ : dipole transition + -0.3 cm 0.8 cm 1.3 cm + note that these two terms are at same order in @xmath25 and @xmath26 expansion . the effective charge " is @xmath27 which vanishes when the @xmath28 or @xmath29 goes on - shell , hence , only the @xmath30 dipole enters @xmath1 and @xmath31 transitions . it is an sm quirk due to the gim mechanism that @xmath32 ( the former becoming @xmath33 coefficients in usual operator formalism for gluonic penguin ) . hence one usually does not pay attention to the subdominant @xmath34 which goes into the variously called @xmath35 , @xmath36 , or @xmath37 coefficients . in particular , @xmath31 rate in sm is only of order 0.2% . but if new physics is present , having @xmath38 is natural , hence the gluonic dipole could get greatly enhanced . while subject to @xmath1 constraint , this could have great impact on @xmath39 process . * _ blind spot of detector ! _ + because @xmath31 leads to _ jetty , high multiplicity _ @xmath20 transitions + -0.3 cm 0.8 cm 0.9 cm + at present , 510% could still easily be allowed . the semileptonic branching ratio and charm counting deficits , and the strength of @xmath40 rate provide circumstantial _ hints _ that @xmath31 could be more than a few percent . * _ unconstrained new cp phase _ via @xmath41 + if enhanced by new physics , @xmath34 is likely to carry a new phase + -0.27 cm 0.8 cm 0.9 cm + however , one faces a severe constraint from @xmath1 . for example it rules out the possibility of @xmath42 as source of enhancement . but as alex kagan @xcite taught me at last dpf meeting in minnesota , the constraint can be evaded if one has sources for radiating @xmath29 but not @xmath28 . * uncharted territory of nonuniversal squark masses + susy provides a natural possibility via gluino loops : + -0.35 cm 0.9 cm 1.3 cm + the simplest being a @xmath43@xmath44 mixing model @xcite . since the first generation down squark is not involved , one evades all low energy constraints . this is a new physics cp model tailor - made for @xmath20 transitions . with the aim of generating huge cp asymmetries , we can now take @xmath45 and study @xmath46 transitions at both inclusive and exclusive level @xcite . in both we have used operator language . one needs to consider the tree diagram , which carries the cp phase @xmath47 ; the standard penguin diagrams , which contain short distance rescattering phases ; the enhanced @xmath48 dipole ( susy loop induced ) diagram ; finally , diagrams containing @xmath49 loop insertions to the gluon self - energy which are needed to maintain unitarity and consistency to order @xmath50 in rate differences @xcite . at the inclusive level , one finds a @xmath31 pole " at low @xmath51 which reflects the jetty @xmath31 process that is experimentally hard to identify . destructive interference is in general needed to allow the @xmath46 rate to be comparable to sm . but this precisely facilitates the generation of large @xmath2s ! more details such as figures can be found in @xcite . dominant rate asymmetry comes from large @xmath51 of the virtual gluon . to illustrate this , table i gives inclusive br ( arbitrarily cutoff at @xmath52 gev@xmath53 ) and @xmath2 for sm and for various new cp phase @xmath54 valus , assuming @xmath31 rate of order 10% . one obtains sm - like branching ratios for @xmath55 , and @xmath2 also seem to peak . this becomes clearer in table ii where we give the results for @xmath56 , where @xmath57 ( perturbative ) rescattering is fully open . we see that 2030% asymmetries are achieveable . this provides support for findings in exclusive processes . exclusive two body modes are much more problematic . starting from the operator formalism as in inclusive , we set @xmath58 , take @xmath59 and try to _ fit observed brs _ with @xmath60 . we then find the @xmath2 preferred by present rate data . one finds that , analogous to the inclusive case , destructive interference is needed and in fact provides a mechanism to suppress the pure penguin @xmath61 mode to satisfy cleo bound . for the @xmath6 and @xmath7 modes which are p - dominated , one utilizes the fact that the matrix element @xmath62 could be enhanced by low @xmath63 values ( of order 100120 mev ) to raise @xmath64 , which at same time leads to near degeneracy of @xmath6 and @xmath7 rates . the upshot is that one finds rather large cp asymmetries , i.e. @xmath65 35% , 45% and 55% for @xmath7 , @xmath6 and @xmath13 modes , respectively , and all of the same sign . such pattern can not be generated by sm , with or without rescattering . we expect such pattern to hold true for many @xmath20 modes . .inclusive br ( in @xmath66)/@xmath2 ( in % ) for sm and for @xmath67 . [ cols="^,^,^,^,^,^,^,^",options="header " , ] we have left out the prominent @xmath68 modes from our discussion largely because the anomaly contribution -0.35 cm 0.9 cm 1.3 cm to compute such diagrams , one needs to know the @xmath69 fock component of the @xmath70 meson ! this may be at the root of the rather large size of @xmath68 mode . before we get carried away by the possibility of large cp asymmetries from new physics , there is one flaw ( or two ? ) that emerged after summer 1998 . because of p - dominance which is certainly true in case of enhanced @xmath31 , @xmath8 is only half of @xmath71 . the factor of 1/2 comes from @xmath72 , which is just an isospin clebsch factor that originates from the @xmath73 wave function . although this seemed quite reasonable from 1997 data where @xmath8 mode was not reported , a crisis emerged in summer 1998 when cleo updated their results for the three @xmath21 modes . they found @xcite @xmath74 instead ! curiously , @xmath75 also , which can not change the situation . in any case the expectation that @xmath76 can not make a factor of 2 change by interference . miraculously , however , this could be the first indication of the last type of penguin , the ewp . the yet to be observed ewp ( electroweak penguin ) , namely @xmath77 , occurs by @xmath78 followed by @xmath79 . the strong penguin oftentimes obscure the @xmath46 case ( or so it is thought ) , and to cleanly identify the ewp one has to search for pure " ewp modes such as @xmath80 , @xmath81 which are clearly rather far away . one usually expects the @xmath82 mode to be the first ewp to be observed , which is still a year or two away , while clean and purely weak penguin @xmath83 is rather far away . with the hint from @xmath74 , however , and putting back on our sm hat , we wish to establish the possibility that ewp may be operating behind the scene already @xcite . it should be emphasized that , unlike the gluon , the @xmath84 coupling depends on isospin , and can in principle break the isospin factor of 1/2 mentioned earlier . 3.2 truein 3.2 truein -.2 cm we first show that simple @xmath23 rescattering can not change drastically the factor of two . from fig . 1(a ) , where we have adopted @xmath85 from current best fit " to ckm matrix @xcite , one clearly sees the factor of 2 between @xmath6 and @xmath8 . we also not that rescattering , as parametrized by the phase difference @xmath86 between i = 1/2 and 3/2 amplitudes , is only between @xmath87 and @xmath88 . when we put in the ewp contribution , at first sight it seems that the effect is drastic . on closer inspection at @xmath89 , it is clear that the ewp contribution to @xmath90 and @xmath6 modes are small , but is quite visible for @xmath8 and @xmath91 modes . this is because the @xmath8 and @xmath91 modes suffer from @xmath92 suppression in amplitude because of @xmath73 wave function . however , it is precisely these modes which pick up a sizable @xmath93 penguin contribution via the @xmath73 ( the strength of @xmath94 is roughly a quarter of @xmath95 and @xmath96 ) . as one dials @xmath86 , @xmath87 and @xmath88 rescattering redistributes this ewp impact and leads to the rather visible change in fig . 1(b ) . we notice the remarkable result that the ewp reduces @xmath6 rate slightly but raises the @xmath8 rate considerably , such that the two modes become rather close . we have to admit , however , to something that we have sneaked in . to enhance the relative importance of ewp , we had to suppress the strong penguin effect . we have therefore employed a much heavier @xmath97 mev as compared to 100120 mev employed previously in new physics case . otherwise we can not bring @xmath6 and @xmath8 rates close to each other . 3.2 truein 3.2 truein -.2 cm having brought @xmath6 and @xmath8 modes closer , the problem now is that @xmath7 lies above them , and the situation becomes worse for large rescattering . to remedy this , we play with the phase angle @xmath28 which tunes the weak phase of the tree contribution t. setting now @xmath89 , again we start without ewp in fig . the factor of two between @xmath6 and @xmath8 is again apparent . dialing @xmath28 clearly changes t - p interference . for @xmath28 in first quadrant one has destructive interference , which becomes constructive in second quadrant . this allows the @xmath6 mode to become larger than the pure penguin @xmath7 mode , which is insensitive to @xmath28 . however , nowhere do we find a solution where @xmath74 is approximately true . there is always one mode that is split away from the other two . putting in ewp , as shown in fig . 2(b ) , the impact is again quite visible . as anticipated , the @xmath6 and @xmath8 modes come close to each other . since their @xmath28 dependence is quite similar , one finds that for @xmath98@xmath99 , the three observed @xmath21 modes come together as close as one can get , and are basically consistent with errors allowed by data . note that @xmath8 is never larger than @xmath6 . we emphasize that a large rescattering phase @xmath86 would destroy this achieved approximate equality , as can be seen from fig . 3 , where we illustrate @xmath86 dependence for @xmath100 . it seems that @xmath86 can not be larger than @xmath101 or so . 3.2 truein -.2 cm 3.2 truein 3.2 truein -.2 cm as a further check of effect of the ewp , we show the results for @xmath89 in fig . 4 . in absence of rescattering , the change in rate ( enhancement ) for @xmath8 mode from adding ewp is reflected in a dilution of the asymmetry , which could serve as a further test . this , however , depends rather crucially on absence of rescattering . once rescattering is included , it would be hard to distinguish the impact of ewp from cp asymmetries . however , even with rescattering phase , the @xmath28 dependence of cp asymmetries can easily distinguish between the two solutions of @xmath102 and @xmath103 , as illustrated in fig . 5 , where ewp effect is included . from our observation that a large @xmath86 phase would destroy the near equality of the three observed @xmath21 modes that we had obtained , we find that @xmath104 even with presence of rescattering phase @xmath86 . 3.2 truein 3.2 truein -.2 cm it should be emphasized that the @xmath28 value we find necessary to have @xmath71 is in a different quadrant than the present best fit " result of @xmath105@xmath106 . in particular , the sign of @xmath107 is preferred to be negative rather than positive . an extended analysis @xcite to @xmath108 , @xmath109 and @xmath110 modes confirm this assertion . intriguingly , the size of @xmath15 and @xmath16 @xcite was anticipated via this @xmath28 value . perhaps hadronic rare b decays can provide information on @xmath28 , and present results seem to be at odds with ckm fits @xcite to @xmath111 , @xmath112 , @xmath113 mixing , and in particular the @xmath114 mixing bound , which rules out @xmath115 . e prepared for violation ! ! we first illustrated the possibility of having @xmath116@xmath117 from new physics in _ already observed modes _ , such as @xmath21 , @xmath118 , and @xmath119 mode when seen . our existence proof " was the possibility of enhanced @xmath31 dipole transition , which from susy model considerations one could have a new cp phase carried by @xmath120 . note that this is just an illustration . we are quite sure that nature is smatter . we then made an about - face and went back to sm , and pointed out that the ewp may have already shone through the special slit " of @xmath74 , where we inferred that @xmath98@xmath99 is preferred , which implies that @xmath115 , contrary to current ckm fit " preference . see talks by f. w " urthwein and y. gao , these proceedings , hep - ex/9904008 . please see the web page http://www.phys.ntu.edu.tw / english / bcp3/. g.w.s . hou , hep - ph/9902382 , expanded version of proceedings for 4th international workshop on particle physics phenomenology , kaohsiung , taiwan , r.o.c . , june 1998 ; and workshop on cp violation , adelaide , australia , july 1998 . hou and k.c . yang , phys . lett . * 81 * , 5738 ( 1998 ) . m. bander , d. silverman and a. soni , phys . lett . * 43 * , 242 ( 1979 ) . l. wolfenstein and y.l . wu , phys . . lett . * 73 * , 2809 ( 1994 ) . w.s . hou and b. tseng , phys . * 80 * , 434 ( 1998 ) . kagan , _ phys . _ d51 , 6196 ( 1995 ) . x.g . he and w.s . hou , phys . * b 445 * , 344 ( 1999 ) . chua , x.g . he and w.s . hou , phys . rev . d60 , 014003 ( 1999 ) . j .- grard and w.s . hou , phys . lett . * 62 * , 855 ( 1989 ) ; phys . rev . d43 , 2909 ( 1991 ) . m. artuso et al . ( cleo collaboration ) , cleo conf 98 - 20 . deshpande et al . , phys . . lett . * 82 * , 2240 ( 1999 ) . f. parodi , p. roudeau and a. stocchi , hep - ph/9802289 . hou and k.c . yang , hep - ph/9902256 .	discovery voyage into the age of :	5,785	9
boutet de monvel s calculus @xcite is a pseudodifferential calculus on manifolds with boundary . it includes the classical differential boundary value problems as well as the parametrices to elliptic elements . many operator - algebraic aspects of this algebra ( spectral invariance , noncommutative residues and traces , composition sequence , @xmath14-theory ) have been studied recently @xcite . the problem of identifying this algebra as the pseudodifferential algebra ( or as an ideal of one ) of a lie groupoid may be the key to an effective application of the methods of noncommutative geometry . if that is acomplished , one could then seek for extensions of the calculus , and for a better understanding of its index theory @xcite . basic definitions and certain facts about boutet de monvel s algebra are recalled in section [ bdmc ] . the groupoid approach to pseudodifferential calculus was developed in noncommutative geometry , following the seminal work of a. connes for foliations @xcite . the guiding principle in that approach is that the central object in global analysis is a groupoid . to study a particular situation , for a class of singular manifolds for instance , one needs to define a groupoid adapted to the context and then use the general pseudodifferential tools for groupoids , as developed in @xcite . this has been applied to the context of manifolds with corners , with the goal of studying melrose s @xmath15-calculus ( see @xcite ) . groupoids were defined whose pseudodifferential calculi recover the @xmath15-calculus and the cusp - calculus . basic definitions and certain facts about pseudodifferential calculus on groupoids are recalled in section [ psi ] . the starting point of this work is the following result ( see @xcite ) : the kernel of the principal symbol map for boutet de monvel s calculus is equal to the norm closure @xmath16 of the ideal of singular green operators . since in the pseudodifferential calculus on a groupoid , the @xmath6-algebra of the groupoid is the kernel of the principal symbol map , this gives a hint that finding a groupoid whose @xmath6-algebra is @xmath17 could give some insight about the relationship between the boutet de monvel algebra and groupoid pseudodifferential algebras . besides , @xmath16 fits into a short exact sequence ( see ( * ? ? ? * section 7 ) ) : @xmath18 which is similar to that for pseudodifferential operators on smooth manifolds : @xmath19 in section [ sgoi ] , we show that @xmath2 is actually morita - equivalent to the norm - closure @xmath7 of the algebra of pseudodifferential operators on the boundary . since @xmath2 is a stable , it is thus isomorphic to @xmath20 . on the other hand , we define in section [ gr - sgo ] a groupoid whose @xmath6-algebra contains an ideal @xmath21 which fits in an extension analoguous as that of @xmath20 . by showing in section [ main ] that the @xmath22-theory elements induced by these extensions coincide , we infer from a theorem by voiculescu that @xmath2 and @xmath21 are isomorphic . let @xmath23 denote a compact manifold of dimension @xmath24 with boundary @xmath25 and interior @xmath26 . given a pseudodifferential operator @xmath27 , defined on an open neighborhood @xmath28 of @xmath23 , and @xmath29 , one defines @xmath30 as equal to the restriction to @xmath26 of @xmath31 , where @xmath32 is the extension of @xmath33 to @xmath28 which vanishes outside @xmath23 . in general , singularities may develop at the boundary , and one gets only a mapping @xmath34 . one says that @xmath27 has the _ transmission property _ if the image of the truncated operator @xmath35 is contained in @xmath36 ( a subspace of @xmath37 ) . this was defined by boutet de monvel in @xcite , where he proved that the transmission property for a classical ( polyhomogoneous ) pseudodifferential operator is equivalent to certain symmetry conditions for the homogeneous components of the asymptotic expansion of its symbol at the boundary . later @xcite , he constructed an algebra whose elements are operators of the form @xmath38 where @xmath27 is a pseudodifferential operator on @xmath23 satisfying a condition that ensures the transmission property , @xmath39 is a pseudodifferential operator on @xmath25 , while @xmath1 , @xmath14 and @xmath40 belong to classes of operators he then defined and named , respectively , singular green , poisson and trace operators . we call an operator as in ( [ bdmo ] ) a boutet de monvel operator . for detailed expositions of his calculus , we refer to @xcite . a boutet de monvel operator has an _ order _ , roughly the usual order of pseudodifferential operators . the entries @xmath40 and @xmath1 have , moreover , an integer _ class _ assigned to them . the class of a trace operator is related to the order of the derivatives that appear in the boundary - value conditions it prescribes . one must assign a class also to singular green operators due to the fact that the composition @xmath41 is a singular green operator . there exist isomorphisms between suitable sobolev spaces such that the composition of a given operator of arbitrary order and class with one of them has order and class zero . for index theory purposes it is therefore sufficient to consider operators of order and class zero . these form an adjoint invariant subalgebra of the algebra @xmath42 of all bounded operators on a suitable hilbert space @xmath43 . adopting the definition of order in @xcite for @xmath14 and @xmath40 , we here choose @xmath44 . if , as does grubb @xcite , one keeps the original definition ( which makes more sense if one needs general @xmath45 estimates ) then one must take a sobolev space of order @xmath46 over the boundary . boutet de monvel operators can also be defined as mappings between smooth sections of vector bundles . if @xmath47 is a bundle of positive rank over @xmath23 , and @xmath48 is an arbitrary bundle over @xmath25 , then the algebra of all boutet de monvel operators of order and class zero acting between sections of @xmath47 and @xmath48 is morita equivalent ( * ? ? ? * section 1.5 ) to the algebra obtained by taking a rank - one trivial bundle over @xmath23 and the zero - bundle over @xmath25 . this partly justifies , again if one is interested in index theory , to consider only the operators appearing in the upper - left corner of the matrix in ( [ bdmo ] ) and to assume , as we did at the beginning , that the bundle over @xmath23 is @xmath49 . the problem of computing the fredholm index of a boutet de monvel operator acting between sections of different bundles over each side can be reduced to the case of equal bundles on both sides by a device developed by boutet de monvel @xcite , recalled in ( * ? ? ? * section 1.1 ) . let us now explain what a singular green operator @xmath1 is , in the case of order and class zero and of a rank - one trivial bundle over @xmath23 . its distribution kernel is smooth outside the boundary diagonal ; i.e , if @xmath50 , and if we denote by @xmath51 the operator of mulitiplication by @xmath52 , then @xmath53 and @xmath54 are integral operators with smooth kernels . the push - forward of @xmath1 by a boundary chart is an operator - valued - symbol pseudodifferential operator on the variables tangential to the boundary , as we describe below . it is perhaps worth stressing , however , that it is in general not a pseudodifferential operator on all variables , because of its particular way of acting on the normal variable . given @xmath55 , @xmath56 , let @xmath57 denote the vector - valued fourier transform of @xmath33 with respect to the @xmath58 first variables , @xmath59 in local coordinates for which the boundary corresponds to @xmath60 and the interior to @xmath61 , @xmath1 is given by @xmath62 the integrals in ( [ ft ] ) or in ( [ defg ] ) should be regarded , for fixed @xmath63 or @xmath64 , respectively , as @xmath65-valued integrals . for each @xmath66 , @xmath67 in ( [ defg ] ) is an integral operator with kernel @xmath68 equal to the restriction to @xmath69 of a function belonging to the schwartz space of rapidly decreasing functions on @xmath70 . the function @xmath71 ( called by grubb the _ symbol - kernel _ of @xmath1 ) is smooth and satisfies the estimates ( * ? ? ? * ( 1.2.38 ) ) . this is invariantly defined ( * ? ? ? * theorem 2.4.11 ) with respect to coordinate changes that preserve the set @xmath72 . we denote by @xmath73 the set of all polyhomogeneous operators @xmath74 of order and class zero on @xmath23 , and by @xmath75 its subset of all singular green operators . it follows from the rules of boutet de monvel s calculus that @xmath73 is an algebra and that @xmath75 is an ideal in @xmath73 . in the sequel , we shall restrict ourselves to coordinate changes which preserve the variable @xmath76 , i.e. , we choose a normal coordinate . then two * -homomorphisms are defined on @xmath73 , the principal symbol and the boundary principal symbol : @xmath77 the principal symbol of a given @xmath74 is , by definition , the usual principal symbol of @xmath27 @xmath78 where @xmath79 is the leading term in the aymptotic expansion of the symbol of @xmath27 . at a point @xmath80 in @xmath13 , the boundary principal symbol of @xmath35 is defined to be the truncated fourier multiplier @xmath81 of symbol @xmath82 . the boundary principal symbol of @xmath83 is the integral operator @xmath84 with the rapidly decreasing kernel @xmath85 , where @xmath86 denotes the leading term in the asymptotic expansion of @xmath87 , cf . * ( 1.2.39 ) ) . then @xmath88 maps @xmath75 into @xmath89 , with the ideal @xmath90 of compact operators on @xmath65 . let @xmath91 and @xmath2 denote the norm closures of @xmath73 and @xmath75 , respectively ; and let @xmath92 denote the set of all compact operators on @xmath93 . it follows from theorem 1 in @xcite that @xmath94 and @xmath88 can be extended to @xmath6-algebra homomorphisms @xmath95 moreover , by corollary 2 in @xcite and ( * ? ? ? * theorems 5 and 6 ) , we have that : @xmath96 and @xmath97 maps @xmath2 onto @xmath98 . in other words , the restriction of the boundary principal symbol to @xmath2 gives rise to the exact sequence of c@xmath99-algebras @xmath100 in section [ sgoi ] we use ( [ pses ] ) to prove that @xmath2 is isomorphic to the tensor product @xmath9 of the @xmath6-closure @xmath7 of the pseudodifferential operators of order zero on @xmath25 by the compacts . for that we need to use trace and poisson operators . similarly as for the singular green operators , the trace operators and the poisson operators ( @xmath40 and @xmath14 in ( [ bdmo ] ) ) are , locally , operator - valued - symbol pseudodifferential operators on the variables tangential to the boundary , given by @xmath101 and @xmath102 the mappings @xmath103 and @xmath104 are defined , for each @xmath105 , each @xmath106 and each @xmath107 , by @xmath108 and @xmath109(x_n)= \alpha \tilde k(x^\prime , x_n,\xi^\prime).\ ] ] for each @xmath66 , @xmath110 and @xmath111 are restrictions to @xmath112 of functions in the schwartz class on @xmath113 . the functions @xmath114 and @xmath115 , called the symbol - kernels of @xmath40 and @xmath14 , are smooth and satisfy certain estimates . in the polyhomogenous case , they have asymptotic expansions in homogeneous components , whose leading terms we denote by @xmath116 and @xmath117 , respectively . the estimates and expansions for @xmath118 and @xmath119 listed or explained in ( * ? ? ? * section 1.2 ) are not the right ones for our definition of order ( and consequent choice of hilbert space ) : we need to shift some of the exponents there by @xmath120 . the boundary - principal symbols of @xmath40 and @xmath14 are @xmath121 defined as in ( [ defst ] ) and ( [ defsk ] ) , except that @xmath116 and @xmath117 replace @xmath118 and @xmath119 . lastly , the boundary principal symbol of a polyhomogeneous pseudodifferential operator on @xmath25 is simply its usual principal symbol , and we get a * -homomorphism @xmath122 where @xmath123 denotes the set of all polyhomogeneous boutet de monvel operators of order and class zero on @xmath23 . it has a continuous extension to the norm - closure of @xmath123 , but we will not use this fact . [ es1 ] there exists a zero - order poisson operator @xmath14 such that @xmath124 is a strictly positive operator on @xmath125 . _ proof _ : it is well - known that the dirichlet problem @xmath126 defines a bounded invertible operator . we denote by @xmath127 an order reduction of order 3/2 on @xmath25 and by @xmath128 and order reduction of order @xmath129 on @xmath23 . this gives us an isomorphism @xmath130 which is an element of order and class 0 in boutet de monvel s calculus . its inverse therefore also is in boutet de monvel s calculus ; it is of the form @xmath131 with suitable @xmath132 , and @xmath14 of order and class zero . in particular , @xmath14 is a right inverse for the trace operator @xmath133 . for @xmath134 we thus have @xmath135 we then get @xmath136 for some @xmath137 , so that @xmath124 is strictly positive . @xmath138 [ es2 ] there exist a trace operator of order and class zero @xmath139 and a poisson operator of order zero @xmath140 such that @xmath141 is equal to the identity operator on @xmath125 , @xmath142 and @xmath143 . _ proof _ : let @xmath144 be a zero - order poisson operator such that @xmath145 is a strictly positive operator on @xmath125 , and let @xmath146 . @xmath147 is a zero - order pseudodifferential operator on @xmath25 . take @xmath148 and @xmath149 . @xmath138 we denote by @xmath7 the norm closure of the algebra of all polyhomogeneous pseudodifferential operators of order zero on @xmath25 , and by @xmath150 the continuous extension of the principal - symbol homomorphism . it is well - known ( this is mentioned in @xcite and follows from ( * ? ? ? * theorem a.4 ) , or from ( * ? ? ? * theorem 3.3 ) ) that @xmath151 induces the short exact sequence of @xmath6-algebras @xmath152 where @xmath153 denotes the ideal of compact operators on @xmath125 . by lemma [ es2 ] a @xmath6-homomorphism @xmath154 can be defined by @xmath155 since @xmath156 is compact if @xmath157 is compact , we can use @xmath158 to couple the sequences ( [ pses ] ) and ( [ as ] ) . together they yield the commutative diagram of exact sequences of @xmath6-algebras @xmath159 [ here ] the homomorphism @xmath158 imbeds @xmath7 as a hereditary subalgebra of @xmath17 . _ proof _ : we have to prove , that if @xmath160 then @xmath1 is again of the form @xmath161 with @xmath162 . since @xmath41 acts as the identity on @xmath163 it also acts as the identity on @xmath1 and we therefore get @xmath164 . @xmath138 [ full ] let @xmath165 be a commutative diagram of short exact sequences , where @xmath166 and @xmath167 are embeddings . then @xmath168 is full provided that @xmath169 and @xmath167 are full . _ proof _ : we have to prove that the two - sided ideal generated by @xmath170 is dense in @xmath171 . we thus have to prove that to a given @xmath172 and a given @xmath173 we can find an element @xmath15 in the twosided ideal generated by the image of @xmath168 such that @xmath174 . since @xmath167 is full we can find an element @xmath175 in the twosided ideal generated by @xmath176 such that @xmath177 . the element @xmath175 can be lifted to an element @xmath178 in the twosided ideal generated by @xmath179 and we can therefore find an element @xmath180 such that @xmath181 . since @xmath169 is full there is an element @xmath182 in the twosided ideal generated by @xmath183 with @xmath184 . as the desired @xmath15 we can therefore choose @xmath185.@xmath138 the algebras @xmath17 and @xmath186 are isomorphic . _ proof _ : by lemma [ here ] , the diagram [ cd ] and lemma [ full ] the imbedding @xmath158 is full and hereditary . it follows from the remark below theorem 8 on page 155 in @xcite that @xmath17 and @xmath7 are strongly morita equivalent . by the results in @xcite and @xcite we have @xmath187 is isomorphic to @xmath186 . however @xmath17 is stable since it is the extension of @xmath188 with a stable algebra , namely @xmath189 , ( see proposition 6.12 in @xcite ) . this gives the isomorphism.@xmath138 groupoids were introduced in the context of global analysis when a. connes showed that in the case of foliations the index takes values in a which is defined as the of the holonomy groupoid of the foliation . he defined a pseudodifferential calculus on a foliation using the groupoid structure . in several papers ( @xcite ) , generalizations of this approach to a larger class of groupoids were achieved . one particular aspect of this theory is that , as a. connes showed in @xcite for smooth manifolds , it is possible to define the analytic index using a groupoid , the _ tangent groupoid_. a groupoid is a small category in which all morphisms are invertible . this means that a groupoid @xmath1 has a set of units , denoted by @xmath190 , and two maps called _ range _ and _ source _ , @xmath191^{r } \ar@<-.5ex>[r]_{s } & \g0}$ ] . two elements @xmath192 are composable if and only if @xmath193 : we recall briefly the main aspects of this theory . let @xmath1 be a lie groupoid , which means that it has a smooth structure . then one can define an algebra of pseudodifferential operators @xmath194 : a pseudodifferential operator on @xmath1 is a @xmath1-equivariant continuous family of pseudodifferential operators on the fibers of @xmath1 . for example , if @xmath195 is a manifold without boundary , and @xmath196 , with set of units @xmath197 , and range and source maps @xmath198 , and composition @xmath199 , then @xmath194 is the algebra of pseudodifferential operators on @xmath195 . if @xmath1 is a lie group , @xmath194 is the algebra of @xmath1-equivariant pseudodifferential operators on @xmath1 . in order to work with singular manifolds , the framework of lie groupoids needs to be extended . that was done in @xcite , where the algebras of pseudodifferential operators on continuous family groupoids , which are groupoids whose fibers are smooth manifolds , were defined . on the algebra of pseudodifferential operators one can define a symbol map , @xmath94 . the algebra of order 0 operators can be completed as a , denoted by @xmath200 , and the symbol map extends to this algebra . the `` regularizing operators '' of the calculus , which are the operators with trivial symbol , are the elements of the of the groupoid , and we have the following atiyah - singer exact sequence : @xmath201 where @xmath202 is the cosphere bundle of the lie algebroid @xmath203 , which can be thought of as a tangent space . we next recall in more detail the construction of the adiabatic groupoid @xmath204 associated with a smooth manifold @xmath25 : @xmath205 with the tangent bundle @xmath206 of @xmath25 . the groupoid structure is given as follows : @xmath207 @xmath208 this groupoid is endowed with a differential structure , through an exponential , in the following way : * on @xmath209 , the structure is that of a product of manifolds . * define a map on an open neighborhood @xmath210 of @xmath211 in @xmath212 , with values in @xmath204 , by @xmath213 in other terms , the topology is such that a sequence of terms @xmath214 of @xmath209 converges to @xmath215 , if and only if we have locally @xmath216 note that a. connes tangent groupoid is just the restriction of @xmath204 to @xmath217 $ ] . the main interest of this groupoid is that it provides a way to define the analytic index . consider indeed the decomposition of the groupoid as an open and a closed subgroupoid , which gives rise to the exact sequence : @xmath218 this simplifies since @xmath219 , and @xmath220 . a. connes proved that the boundary map of the 6-terms exact sequence induced by this extension is nothing but the analytic index @xmath221 suppose we could identify the @xmath6-closure of boutet de monvel s algebra with the @xmath6-algebra @xmath0 of pseudodifferential operators on a lie groupoid @xmath1 . then , as pointed out above , the kernel of the principal symbol map would be isomorphic to @xmath3 . as the kernel of the principal symbol map in boutet de monvel s calculus consists of the singular green operators , we thus wish to identify these with the of a groupoid . we will actually not identify them with a groupoid , but with an ideal in a groupoid . let us consider the following action of the group @xmath222 on @xmath204 : * on @xmath206 , @xmath222 acts by dilations : @xmath223 * on @xmath209 , @xmath222 acts by @xmath224 . this is a continuous action : if @xmath225 converges to @xmath226 ( which means that @xmath227 ) , then @xmath228 since @xmath229 it is thus possible to construct the semi - direct product @xmath230 of the adiabatic groupoid by @xmath231 : as a set , it is @xmath232 , with set of units @xmath233 , such that : * @xmath234 , for @xmath235 ; * @xmath236 , for @xmath237 ; * @xmath238 * @xmath239 note that the action of @xmath240 on the adiabatic groupoid induces an action on its and that j. renault proved in @xcite that for any locally compact groupoid @xmath17 one has @xmath241 the evaluation at @xmath237 provides a map @xmath242 . also , the evaluation at the zero - section @xmath243 induces a map @xmath244 . but since the action of @xmath222 on @xmath25 is trivial , the latter algebra is just the algebra of the ( regular ) product : @xmath245 let @xmath246 and @xmath247 . the kernel of @xmath248 is @xmath249 . but @xmath250 is directly isomorphic to the pair groupoid @xmath251 : to clarify the proof , let us denote @xmath252 and @xmath253 . then let @xmath254 be defined by @xmath255 this a morphism of groupoids : the composition of @xmath256 with @xmath257 gives @xmath258 , and @xmath259 while @xmath260 hence the kernel of @xmath248 is just the algebra of compact operators , @xmath188 . to make this clear , here is the commutative diagram describing this : @xmath261 & 0 \ar[d]&\\ 0 \ar[r ] & \mathcal{k } \ar[r ] \ar@{=}[d ] & \mathcal i \ar[r ] \ar[d ] & c \ar[r ] \ar[d]^{j } & 0\\ 0 \ar[r ] & \mathcal{k } \ar[r ] & c^(g ) \ar[r]^<<<<<{e_0 } \ar[dr ] & c_0(t^y)\rtimes { \mathbb r}^_+ \ar[r ] \ar[d]^{r_0 } & 0\\ & & & c_0(y\times { \mathbb r}^_+ ) & \\ } \ ] ] we will use this diagram and extension theory to prove that @xmath10 is isomorphic to the algebra of singular green operators . @xmath262 is isomorphic to @xmath263 , where @xmath13 is the sphere bundle in @xmath264 . first of all notice that @xmath262 is isomorphic to @xmath265 : indeed , the exact sequence @xmath266 induces the exact sequence @xmath267 but @xmath268 , so that @xmath269 and @xmath270 now @xmath271 where we again used renault s result for the second isomorphism and the isomorphism of @xmath272 with the pair groupoid @xmath273 for the third . this ends the proof . we have just shown that @xmath10 is an extension of @xmath274 by @xmath188 , and this is also the case for the algebra of s. the main result is the following : it remains to show that the extensions give rise to the same element of @xmath293 . but since @xmath294 is separable , @xmath295 is isomorphic to the group of invertibles of @xmath296 , thanks to a result of kasparov ( @xcite ) . the @xmath263 being nuclear , @xmath296 is actually a group , thus it is isomorphic to @xmath295 . it induces the extension @xmath303 whose class is denoted by @xmath304 . the relation between @xmath305 and @xmath306 is made clear by considering the following exact sequence @xmath307 where @xmath308 is the ball bundle over @xmath25 . its class is denoted by @xmath309 , and one has the well - known equality : @xmath310 for the convenience of the reader , we now recall the diagram ( [ diagram ] ) : @xmath311 & \mathcal{k } \ar[r ] \ar@{=}[d ] & \mathcal i \ar[r ] \ar[d ] & c \ar[r ] \ar[d]^{j } & 0 & ( \partial)\\ 0 \ar[r ] & \mathcal{k } \ar[r ] & c^(g ) \ar[r]^<<<<<{e_0 } \ar[dr ] & c_0(t^y)\rtimes { \mathbb r}^_+ \ar[r ] \ar[d]^{r_0 } & 0 & ( \alpha)\\ & & & c_0(y\times { \mathbb r}^_+ ) & & \\ } \ ] ] let us denote the class of the first sequence by @xmath312 ; it is thus given by the kasparov product : @xmath313 consider the following commutative diagram : @xmath311 & c_0(t^y\setminus y ) \ar[r ] \ar[d ] & c_0(b^y\setminus y ) \ar[r ] \ar[d ] & c(s^y ) \ar[r ] \ar@{=}[d ] & 0\\ \label{eq1 } 0 \ar[r ] & c_0(t^y ) \ar[r ] \ar[d ] & c(b^y ) \ar[d ] \ar[r ] & c(s^y ) \ar[r ] & 0\\ & c(y ) \ar@{=}[r ] & c(y ) & & \\ \label{eq2 } } \ ] ] the first exact sequence actually decomposes as @xmath315 so that its @xmath316-class is the identity of @xmath317 . there is an action of @xmath222 on each algebra of the previous diagram , which is trivial on @xmath318 and @xmath319 . this gives the following : @xmath311 & c_0(t^y\!\setminus\ ! y)\!\rtimes { \mathbb r}^_+ \ar[r ] \ar[d]^{j } & c_0(b^y\!\setminus\ ! y)\!\rtimes { \mathbb r}^_+ \ar[r ] \ar[d ] & c(s^y\times { \mathbb r}^_+ ) \ar[r ] \ar@{=}[d ] & 0 & ( \partial_1)\\ \label{eq1a } 0 \ar[r ] & c_0(t^y)\!\rtimes { \mathbb r}^_+ \ar[r ] \ar[d ] & c(b^y)\rtimes { \mathbb r}^_+ \ar[d ] \ar[r ] & c(s^y\times{\mathbb r}^_+ ) \ar[r ] & 0 & ( \varphi)\\ & c(y\times{\mathbb r}^_+ ) \ar@{=}[r ] & c(y\times{\mathbb r}^_+ ) & & & \\ \label{eq2a } } \ ] ] denote by @xmath320 ( resp . @xmath52 ) the class of the first ( resp . second ) exact sequence of this diagram , and by @xmath321 the element induced by @xmath322 . one has thus the equality @xmath323 so that @xmath324 but @xmath325 is the image of @xmath326 under the thom - connes isomorphism , and @xmath320 is also a thom - connes element in @xmath22-theory . hence the classes in @xmath316 of the extensions of @xmath21 and of @xmath186 are the same . voiculescu s theorem implies that these algebras are isomorphic . lars hrmander . pseudo - differential operators and hypoelliptic equations . in _ singular integrals ( proc pure math . x , chicago , ill . , 1966 ) _ , pages 138183 . soc . , providence , r.i . , 1967 .	can boutet de monvel s algebra on a compact manifold with boundary be obtained as the algebra @xmath0 of pseudodifferential operators on some lie groupoid @xmath1 ? if it could , the kernel @xmath2 of the principal symbol homomorphism would be isomorphic to the groupoid @xmath3 . while the answer to the above question remains open , we exhibit in this paper a groupoid @xmath1 such that @xmath3 possesses an ideal @xmath4 isomorphic to @xmath2 . in fact , we prove first that @xmath5 with the @xmath6-algebra @xmath7 generated by the zero order pseudodifferential operators on the boundary and the algebra @xmath8 of compact operators . as both @xmath9 and @xmath10 are extensions of @xmath11 by @xmath12 ( @xmath13 is the co - sphere bundle over the boundary ) we infer from a theorem by voiculescu that both are isomorphic .	9,220	269
the mixed volume is one of the fundamental notions in the theory of convex bodies . it plays a central role in the brunn minkowski theory and in the theory of sparse polynomial systems . the mixed volume is the polarization of the volume form on the space of convex bodies in @xmath2 . more precisely , let @xmath7 be @xmath8 convex bodies in @xmath2 and @xmath9 the euclidean volume of a body @xmath10 . then the mixed volume of @xmath11 is @xmath12 where @xmath13 denotes the minkowski sum of bodies @xmath14 and @xmath15 . it is not hard to see that the mixed volume is symmetric and multilinear with respect to minkowski addition . also it coincides with the volume on the diagonal , i.e. @xmath16 and is invariant under translations . moreover , it satisfies the following _ monotonicity property _ , which is not apparent from the definition , see ( * ? ? ? * ( 5.25 ) ) . if @xmath17 are convex bodies such that @xmath18 for @xmath19 then @xmath20 the main goal of this paper is to give a geometric criterion for strict monotonicity in the class of convex polytopes . we give two equivalent criteria in terms of essential collections of faces and mixed cells in mixed polyhedral subdivisions , see theorem [ t : main2 ] and theorem [ t : main3 ] . the criterion is especially simple when all @xmath21 are equal ( corollary [ c : mv = v ] ) which is the situation in our application to sparse polynomial systems . in the general case of convex bodies this is still an open problem , see @xcite for special cases and conjectures . the role of mixed volumes in algebraic geometry originates in the work of bernstein , kushnirenko , and khovanskii , who gave a vast generalization of the classical bezout formula for the intersection number of hypersurfaces in the projective space , see @xcite . this beautiful result which links algebraic geometry and convex geometry through toric varieties and sparse polynomial systems is commonly known as the bkk bound . in particular , it says that if @xmath22 is an @xmath8-variate laurent polynomial system over an algebraically closed field @xmath23 then the number of its isolated solutions in the algebraic torus @xmath24 is at most @xmath25 , where @xmath26 are the newton polytopes of the @xmath27 . ( here @xmath28 denotes @xmath29 . ) systems that have precisely @xmath30 solutions in @xmath24 must satisfy a _ non - degeneracy condition _ which means that certain subsystems have to be inconsistent , see theorem [ t : bkk ] . let @xmath22 be a laurent polynomial system over @xmath23 with newton polytopes @xmath5 . replacing each @xmath27 with a generic linear combination of @xmath31 over @xmath23 produces an equivalent system with the same number of solutions in @xmath24 . such an operation replaces each individual newton polytope @xmath26 with the convex - hull of their union , @xmath32 . thus , starting with a system for which @xmath33 , one obtains a system with all newton polytopes equal to @xmath34 and which has less than @xmath35 solutions in @xmath24 , i.e. is degenerate . the geometric criterion of corollary [ c : mv = v ] allows us to characterize such systems without checking the non - degeneracy condition , which could be hard . in fact , theorem [ t : ber ] delivers a simple characterization in terms of the coefficient matrix @xmath36 and the augmented exponent matrix @xmath37 of the system ( see section [ s : pol ] for definitions ) . in particular , it says that if @xmath34 has a proper face such that the rank of the corresponding submatrix of @xmath36 is less than the rank of the corresponding submatrix of @xmath37 then the system has less than @xmath35 isolated solutions in @xmath24 . here is another consequence of theorem [ t : ber ] . if no maximal minor of @xmath36 vanishes then the system has the maximal number @xmath35 of isolated solutions in @xmath24 ( corollary [ c : nice ] ) . this can be thought of as a generalization of cramer s rule for linear systems . this project began at the einstein workshop on lattice polytopes at freie universitt berlin in december 2016 . we are grateful to mnica blanco , christian haase , benjamin nill , and francisco santos for organizing this wonderful event and to the harnack haus for their hospitality . in this section we recall necessary definitions and results from convex geometry and set up notation . in addition , we recall the notion of essential collections of polytopes for which we give several equivalent definitions , as well as define mixed polyhedral subdivisions and the combinatorial cayley trick . throughout the paper we use @xmath38 $ ] to denote the set @xmath39 . for a convex body @xmath14 in @xmath2 the function @xmath40 , given by @xmath41 is the _ support function _ of @xmath14 . we sometimes enlarge the domain of @xmath42 to @xmath43 . for every @xmath44 , we write @xmath45 to denote the supporting hyperplane for @xmath14 with outer normal @xmath46 @xmath47 we use @xmath48 to denote the face @xmath49 of @xmath14 . let @xmath50 be the @xmath8-dimensional mixed volume of @xmath8 convex bodies @xmath7 in @xmath2 , see ( [ e : mv ] ) above . we have the following equivalent definition . * theorem 5.1.7)[t : mv ] let @xmath51 be non - negative real numbers . then @xmath52 is a polynomial in @xmath51 whose coefficient of the monomial @xmath53 equals @xmath50 . let @xmath54 be convex bodies in @xmath2 , not necessarily distinct . we say that a multiset @xmath55 is an _ essential collection _ if for any subset @xmath56 $ ] of size at most @xmath8 we have @xmath57 note that every sub - collection of an essential collection is essential . also @xmath58 , where @xmath14 is repeated @xmath59 times , is essential if and only if @xmath60 . the following is a well - known property of essential collections . * theorem 5.1.8)[t : essential ] let @xmath7 be @xmath8 convex bodies in @xmath2 . the following are equivalent : 1 . @xmath61 ; 2 . there exist segments @xmath62 for @xmath19 with linearly independent directions ; 3 . @xmath63 is essential . another useful result is the inductive formula for the mixed volume , see ( * ? ? ? * theorem 5.1.7 , ( 5.19 ) ) . we present a variation of this formula for convex polytopes . let @xmath14 be a convex body and @xmath64 convex polytopes in @xmath2 . given @xmath65 , let @xmath66 denote the @xmath67-dimensional mixed volume of @xmath68 translated to the orthogonal subspace @xmath69 . then we have @xmath70 note that the above sum is finite , since there are only finitely many @xmath65 for which @xmath71 is essential . namely , these @xmath46 are among the outer unit normals to the facets of @xmath72 . let @xmath73 be convex polytopes . the _ cayley polytope _ @xmath74 is the convex hull in @xmath75 of the union of the polytopes @xmath76 for @xmath77 , where @xmath78 is the standard basis for @xmath79 . a finite polyhedral subdivision of @xmath80 is called _ mixed _ if it corresponds to a polyhedral subdivision of @xmath74 with vertices in @xmath81 via the map sending a polytope @xmath82 of the subdivision of @xmath74 to the polytope @xmath83 , where @xmath84 is the image by the projection @xmath85 of @xmath86 . the polytope @xmath82 is determined by @xmath87 ( and vice versa ) and will be denoted @xmath88 . this correspondence is called the _ combinatorial cayley trick _ , see @xcite for instance . a mixed polyhedral subdivision of @xmath80 is called _ pure _ if the corresponding subdivision of @xmath74 is a triangulation . let @xmath83 be a polytope in a pure mixed polyhedral subdivision of @xmath80 . then all @xmath89 are simplices and @xmath90 . the polytope @xmath87 , as well as the corresponding simplex @xmath88 of the triangulation of @xmath74 , is called _ fully mixed _ if @xmath91 for @xmath92 . when @xmath93 , the polytope @xmath87 is fully mixed if and only if @xmath94 and @xmath95 for @xmath92 so that @xmath87 is an @xmath8-dimensional parallelotope . the following result is well - known , see ( * ? ? ? * theorem 2.4 ) or ( * ? ? ? * theorem 6.7 ) . we include a proof for reader s convenience . [ l : key ] for convex polytopes @xmath96 in @xmath2 , the quantity @xmath97 is equal to the sum of the euclidean volumes of the fully mixed polytopes in any pure mixed polyhedral subdivision of @xmath98 . consider a triangulation @xmath99 of @xmath100 and let @xmath101 . to any simplex @xmath102 of @xmath99 , we associate the simplex @xmath103 ) . the image of this map is a triangulation of @xmath104 . then we may compute the euclidean volume of @xmath105 as the sum of the euclidean volumes of the polytopes in the corresponding pure mixed subdivision : @xmath106 the latter equality coming from the fact that @xmath99 is a triangulation . the coefficient of @xmath107 in the last expression is precisely the total euclidean volume of the fully mixed polytopes in our fully mixed subdivision , and by theorem [ t : mv ] coincides with the mixed volume of @xmath5 . in this section we present our first criterion for strict monotonicity of the mixed volume and its corollaries . let @xmath14 be a subset of a convex polytope @xmath108 and let @xmath109 be a facet . @xmath14 touches @xmath110 _ when the intersection @xmath111 is non - empty . we will often make use of the following proposition , which gives a criterion for strict monotonicity in a very special case , see @xcite . [ p : mix ] let @xmath112 be convex polytopes in @xmath2 and @xmath113 . then @xmath114 if and only if @xmath115 touches every face @xmath116 for @xmath46 in the set @xmath117 the above statement easily follows from ( [ e : induct ] ) and the observation @xmath118 with equality if and only if @xmath115 touches @xmath119 . see ( * ? ? ? * sec 5.1 ) for details . here is the first criterion for strict monotonicity . [ t : main2 ] let @xmath0 and @xmath1 be convex polytopes in @xmath2 such that @xmath120 for every @xmath121 $ ] . given @xmath65 consider the set @xmath122 \ \|\ p_i \text { touches } q^u_i \}.\ ] ] then @xmath123 if and only if there exists @xmath65 such that the collection @xmath124\setminus t_u\}$ ] is essential . assume that there exists @xmath65 such that the collection @xmath124\setminus t_u\}$ ] is essential . note that @xmath125 is a proper subset of @xmath38 $ ] , otherwise @xmath126 is a collection of @xmath8 polytopes contained in translates of an @xmath67-dimensional subspace , hence , can not be essential . without loss of generality we may assume that @xmath38\setminus t_u=\{1,\dots , k\}$ ] for some @xmath127 . since @xmath26 does not touch @xmath128 for @xmath92 there is a hyperplane @xmath129 which separates @xmath26 and @xmath128 . let @xmath130 be the half - space containing @xmath26 . then the truncated polytope @xmath131 satisfies @xmath132 . we claim that the collection @xmath133 is essential . indeed , by assumption there exist @xmath8 segments @xmath134 with linearly independent directions such that @xmath135 for @xmath136 . among the first @xmath137 of these segments no more than one can be parallel to @xmath46 , hence , by projecting them along @xmath46 , translating , and possibly reordering , we may assume that @xmath138 for @xmath139 . since the direction vectors of @xmath140 are linearly independent in the orthogonal subspace @xmath69 , the collection in ( [ e : new ] ) is essential . now by proposition [ p : mix ] we obtain @xmath141 finally , by monotonicity we have @xmath142 and @xmath143 . therefore , @xmath144 conversely , assume @xmath123 . then , by monotonicity , for some @xmath145 we have @xmath146 by proposition [ p : mix ] there exists @xmath65 such that @xmath147 is essential and @xmath148 . by choosing a segment in @xmath149 not parallel to @xmath69 ( which exists since @xmath150 , but @xmath151 does not touch @xmath152 ) we see that @xmath153 is essential . it remains to notice that @xmath154 for @xmath155 and , hence , the collection @xmath124\setminus t_u\}$ ] is essential as well . [ r : full - dim ] note that if @xmath1 are @xmath8-dimensional then we can simplify the criterion of theorem [ t : main2 ] as follows : @xmath123 if and only if there exists @xmath65 such that the collection @xmath126 is essential . [ ex:2-dimth ] for @xmath156 theorem [ t : main2 ] says that @xmath157 if and only if , up to reordering , there exists a facet ( i.e. a side ) @xmath119 such that the corresponding face @xmath158 is not touched by @xmath159 . a particular instance of theorem [ t : main2 ] , especially important for applications to polynomial systems , is the case when @xmath0 are arbitrary polytopes and @xmath160 , where @xmath34 is the convex hull of the union @xmath6 . we will assume that @xmath34 is @xmath8-dimensional , otherwise @xmath161 is not essential and , hence , both @xmath162 and @xmath163 are zero . then the strict monotonicity has the following simple geometric interpretation . [ c : mv = v ] let @xmath0 be polytopes in @xmath2 contained in an @xmath8-dimensional polytope @xmath34 . then @xmath164 if and only if for some @xmath165 $ ] there exist @xmath166 polytopes among @xmath0 that do not touch a codimension @xmath166 face of @xmath34 . by theorem [ t : main2 ] and remark [ r : full - dim ] , we have @xmath164 if and only if there exists @xmath65 such that the collection @xmath167 , where @xmath168 is repeated @xmath169 times , is essential . the last condition is equivalent to @xmath170 , which gives @xmath171 where @xmath166 is the codimension of @xmath168 . this precisely means that there exist @xmath166 polytopes among the @xmath26 which do not touch @xmath168 . corollary [ c : mv = v ] can be reformulated as follows . let @xmath0 be polytopes in @xmath2 contained in an @xmath8-dimensional polytope @xmath34 . then @xmath164 if and only if there is a proper face of @xmath34 of dimension @xmath172 which is touched by at most @xmath172 of the polytopes @xmath0 . in particular , if a vertex of @xmath34 does not belong to any of the polytopes @xmath96 , then @xmath164 . [ ex:2-dimcor ] let @xmath173 be convex polytopes in @xmath174 and @xmath34 be the convex hull of their union . then corollary [ c : mv = v ] shows that @xmath175 if and only if either @xmath115 or @xmath159 does not touch some side of @xmath34 . one can obtain a more direct proof of corollary [ c : mv = v ] by modifying the proof of theorem 2.6 in @xcite . [ ex : first ] let @xmath176 be a finite set in @xmath2 with @xmath8-dimensional convex hull @xmath34 and choose a subset @xmath177 . define @xmath178 for @xmath19 and let @xmath26 be the convex hull of @xmath179 . then corollary [ c : mv = v ] leads to @xmath180 . indeed , if @xmath181 $ ] has size @xmath166 and @xmath182 is a face of codimension @xmath166 , then @xmath183 and @xmath184 . therefore , the subsets @xmath185 and @xmath186 can not be disjoint , i.e. the union of any @xmath166 of the @xmath26 touches every codimension face of @xmath34 . going back to the statement of corollary [ c : mv = v ] it is natural to ask : if there is a codimension @xmath166 face of @xmath34 not touched by more than @xmath166 polytopes among @xmath5 , can the inequality @xmath164 be improved ? the answer is clearly no if we do not restrict ourselves to the class of _ lattice polytopes_. recall that for any collection of lattice polytopes @xmath0 the normalized mixed volume @xmath187 is an integer . we have the following improvement of corollary [ c : mv = v ] in this case . [ p : stronger ] let @xmath0 be lattice polytopes in @xmath2 contained in an @xmath8-dimensional lattice polytope @xmath34 . suppose there exists a facet @xmath188 which is not touched by @xmath189 , for some @xmath190 . moreover , suppose that @xmath191 then @xmath192 . by the essentiality of @xmath191 and since @xmath193 it follows that for any @xmath194 the collection @xmath195 , where @xmath168 is repeated @xmath196 times , is essential . by proposition [ p : mix ] we obtain @xmath197 now to obtain @xmath192 we use ( [ e : claim ] ) successively with @xmath198 : @xmath199 multiplying each of the above terms by @xmath200 we get a decreasing sequence of @xmath201 integers , the last being no less than @xmath202 , which provides the required inequality . we remark that the essentiality conditions above can not be removed . indeed , let @xmath203 be the standard @xmath8-simplex , @xmath168 one of its facets , and @xmath204 for @xmath19 . then if @xmath205 equal the vertex of @xmath34 not contained in @xmath168 , then @xmath206 regardless of @xmath59 . it would be interesting to obtain a more general statement than proposition [ p : stronger ] which deals with faces of larger codimension . given a polytope @xmath34 , corollary [ c : mv = v ] provides a characterization of collections @xmath5 such that @xmath204 for @xmath19 and @xmath207 . clearly , when @xmath34 and the @xmath26 are lattice polytopes there are only finitely many such collections . describing them explicitly is a hard combinatorial problem . in the case when @xmath34 is the standard simplex , @xmath203 , this problem was solved by esterov and gusev in @xcite . in this section we obtain another criterion for the strict monotonicity property ( theorem [ t : main3 ] ) based on mixed polyhedral subdivisions and the combinatorial cayley trick . recall that for polytopes @xmath208 in @xmath2 , we have @xmath209 if and only if the collection @xmath210 is essential , which is equivalent to @xmath211 for all @xmath181 $ ] , see theorem [ t : essential ] . we now describe a generalization of this criterion . consider again polytopes @xmath212 for @xmath213 . for any non - zero vector @xmath214 consider the convex polytopes @xmath215 [ t : main3 ] let @xmath0 and @xmath1 be convex polytopes in @xmath2 such that @xmath120 for every @xmath121 $ ] . the following conditions are equivalent : 1 . @xmath216 , 2 . there exists a fully mixed simplex with vertices in @xmath217 which is contained in the closure of @xmath218 , 3 . there exists a non - zero vector @xmath214 such that the collection @xmath219 is essential . consider a fully mixed simplex @xmath220 and let @xmath221 be the corresponding fully mixed polytope in the mixed subdivision of @xmath222 . then @xmath223 is contained in the closure of @xmath218 if and only if there is a supporting hyperplane of @xmath100 which separates the convex sets @xmath100 and @xmath223 . this is equivalent to the existence of @xmath224 with @xmath225 such that @xmath226 we note that is equivalent to @xmath227 , and so @xmath228 , for @xmath213 . since @xmath229 have linearly independent directions , this gives that @xmath219 is essential . this shows the equivalence between @xmath230 and @xmath231 . assuming that @xmath232 is a fully mixed simplex verifying , it is easy to show the existence of a triangulation of @xmath233 with vertices in @xmath217 which contains @xmath223 and restricts to a triangulation of @xmath100 . by the combinatorial cayley trick , this gives a pure mixed subdivision @xmath234 of @xmath235 restricting to a pure mixed subdivision @xmath236 of @xmath98 and a fully mixed polytope @xmath237 contained in @xmath238 . therefore , @xmath239 by lemma [ l : key ] . the implication @xmath240 is proved . assume now that @xmath239 . by proposition [ p : mix ] there exists @xmath65 such that @xmath147 is essential and @xmath148 ( see the proof of theorem [ t : main2 ] ) . thus we may choose a segment @xmath241 , segments @xmath242 for @xmath243 , and segments @xmath244 for @xmath245 such that the corresponding @xmath8 direction vectors are linearly independent . this shows that @xmath219 is essential since @xmath246 and @xmath247 are contained in @xmath248 for @xmath213 . we have proved the implication @xmath249 . note that if @xmath26 touches @xmath128 then @xmath250 and if @xmath26 does not touch @xmath128 then @xmath251 . therefore , the condition ( 3 ) in the above theorem is equivalent to the condition in theorem [ t : main2 ] . consider a finite set @xmath252 where @xmath253 . let @xmath254 be the coordinates of @xmath255 for @xmath256 . consider a laurent polynomial system with coefficients in an algebraically closed field @xmath23 @xmath257 where @xmath258 and as usual @xmath259 stands for the monomial @xmath260 . we shall assume that no polynomial @xmath27 is the null polynomial . call @xmath261 the _ individual support _ of @xmath27 and @xmath262 the _ total support _ of the system . the _ newton polytope _ @xmath26 of @xmath27 is the convex hull of @xmath179 and the _ newton polytope _ @xmath34 of the system is the convex hull of @xmath176 . the matrices @xmath263 are the _ coefficient _ and _ exponent _ matrices of the system , respectively . choose @xmath65 and let @xmath264 . then the _ restricted system _ corresponding to @xmath46 is the system @xmath265 where @xmath266 with @xmath267 if @xmath268 and @xmath269 , otherwise . finally , a system ( [ e : system ] ) is called _ non - degenerate _ if for every @xmath65 the corresponding restricted system is inconsistent . the relation between mixed volumes and polynomial systems originates in the following fundamental result , known as the bkk bound , discovered by bernstein , kushnirenko , and khovanskii , see @xcite . [ t : bkk ] the system has at most @xmath270 isolated solutions in @xmath24 counted with multiplicity . moreover , it has precisely @xmath270 solutions in @xmath24 counted with multiplicity if and only if it is non - degenerate . systems with fixed individual supports and generic coefficients are non - degenerate . moreover , the non - degeneracy condition is not needed if one passes to the toric compactification @xmath271 associated with the polytope @xmath272 . namely , a system has at most @xmath270 isolated solutions in @xmath271 counted with multiplicity , and if it has a finite number of solutions in @xmath271 then this number equals @xmath273 counted with multiplicity . there are two operations on that preserve its number of solutions in the torus @xmath24 . first , left multiplication of @xmath36 by an invertible matrix of @xmath274 produces an equivalent system . the second operation consists in multiplying each equation by a given monomial and making a monomial change of coordinates of @xmath24 . this second operation corresponds to left multiplication of the _ augmented exponent matrix _ @xmath275 ( obtained from @xmath108 by adding a first row of @xmath276 ) by a matrix in @xmath277 . [ r : left ] consider a non - degenerate system with coefficient matrix @xmath36 . left multiplication of @xmath36 by an invertible matrix preserves the total support of the system , but does not preserve the individual supports and newton polytopes in general . however , since this operation produces an equivalent system , theorem [ t : bkk ] shows that the mixed volumes of the corresponding @xmath8-tuples of newton polytopes are equal . assume that has precisely @xmath278 solutions in @xmath24 counted with multiplicity and @xmath36 has a non - zero maximal minor . up to renumbering , we may assume that this minor is given by the first @xmath8 columns of @xmath36 . left multiplication of by the inverse of the corresponding submatrix of @xmath36 gives an equivalent system with invidual supports @xmath279 for @xmath19 . thus this new system has precisely @xmath280 solutions in @xmath24 counted with multiplicity . by theorem [ t : bkk ] this number of solutions is at most @xmath281 , where @xmath282 is the convex hull of @xmath283 . on the other hand , by monotonicity of the mixed volume we have @xmath284 . we conclude that @xmath285 . the second equality is also a consequence of corollary [ c : mv = v ] , see example [ ex : first ] . [ t : ber ] assume @xmath286 . if a system has @xmath287 isolated solutions in @xmath24 counted with multiplicity , then for any proper face @xmath110 of @xmath34 the submatrix @xmath288 of @xmath36 with columns indexed by @xmath289 \ , , \ , a_j \in f \cap { { \mathcal{a}}}\}$ ] satisfies @xmath290 or equivalently , @xmath291 conversely , if is satisfied for all proper faces @xmath110 of @xmath34 and if the system is non - degenerate , then it has precisely @xmath287 isolated solutions in @xmath24 counted with multiplicity . first , note that for any proper face @xmath110 of @xmath34 we have @xmath292 . consider a proper face @xmath110 of @xmath34 of codimension @xmath293 and assume that @xmath294 . then there exist an invertible matrix @xmath15 and @xmath181 $ ] of size @xmath295 such that the submatrix of @xmath296 with rows indexed by @xmath297 and columns indexed by @xmath298 is the null matrix . the matrix @xmath299 is the coefficient matrix of an equivalent system with the same total support , see remark [ r : left ] . denote by @xmath300 the individual newton polytopes of this equivalent system . then the polytopes @xmath282 for @xmath301 do not touch the face @xmath110 of @xmath34 . since @xmath302 and @xmath295 , it follows then from corollary [ c : mv = v ] that @xmath303 . theorem [ t : bkk ] applied to the system with coefficient matrix @xmath299 gives that it has at most @xmath304 isolated solutions in @xmath24 counted with multiplicity . the same conclusion holds for the equivalent system . therefore , if has @xmath35 isolated solutions in @xmath24 counted with multiplicity , then is satisfied for all proper faces @xmath110 of @xmath34 . conversely , assume that is non - degenerate and that is satisfied for all proper faces @xmath110 of @xmath34 . then has @xmath305 isolated solutions in @xmath24 counted with multiplicity by theorem [ t : bkk ] . suppose that @xmath306 . then by corollary [ c : mv = v ] there exists a proper face @xmath110 of @xmath34 of codimension @xmath293 and @xmath181 $ ] of size @xmath295 such that the polytopes @xmath26 for @xmath301 do not touch @xmath110 . but then @xmath307 , which gives a contradiction . thus @xmath308 and has @xmath35 isolated solutions in @xmath24 counted with multiplicity . as an immediate consequence of theorem [ t : ber ] from which we keep the notations , we get the following . consider any laurent polynomial system with @xmath286 . if there exists a proper face @xmath110 of @xmath34 such that @xmath309 , then the system has either infinitely many solutions , or strictly less than @xmath287 solutions in @xmath24 counted with multiplicity . assume the existence of a proper face @xmath110 of @xmath34 such that @xmath310 . if has precisely @xmath287 solutions in @xmath24 counted with multiplicity , then it is non - degenerate by theorem [ t : bkk ] and thus @xmath311 by theorem [ t : ber ] , a contradiction . a very nice consequence of theorem [ t : ber ] is the following result , which can be considered as a generalization of cramer s rule to polynomial systems . [ c : nice ] assume that @xmath286 and that no maximal minor of @xmath36 vanishes . then the system has the maximal number of @xmath312 isolated solutions in @xmath24 counted with multiplicity . note that @xmath313 since @xmath286 ( recall that @xmath314 is the number of columns of @xmath36 ) . thus a maximal minor of @xmath36 has size @xmath8 and the fact that no maximal minor of @xmath36 vanishes implies that for any @xmath315 $ ] the submatrix of @xmath36 with rows indexed by @xmath38 $ ] and columns indexed by @xmath316 has maximal rank . this rank is equal to @xmath8 if @xmath317 or to @xmath318 if @xmath319 . since @xmath320 for any face @xmath110 of @xmath34 , we get that @xmath321 for any proper face @xmath110 of @xmath34 . moreover , no restricted system is consistent for otherwise this would give a non - zero vector in the kernel of the corresponding submatrix of @xmath36 . thus is non - degenerate and the result follows from theorem [ t : ber ] . when the polytope @xmath322 is the standard simplex , the system is linear and it has precisely @xmath323 solution in @xmath24 if and only if no maximal minor of @xmath324 vanishes , in accordance with cramer s rule for linear systems . we conclude with a few examples illustrating the results of the previous section . let @xmath325 and @xmath326 be individual supports , and @xmath327 the total support of a system . the newton polytopes @xmath328 , @xmath329 , and @xmath330 are depicted in figure [ f : ex1 ] , where the vertices of @xmath115 and @xmath159 are labeled by @xmath331 and @xmath332 , respectively . we use the labeling in figure [ f : ex1 ] to order the columns of the augmented matrix @xmath333 a general system with these supports has the following coefficient matrix @xmath334 where @xmath335 are non - zero . one can see that each side of @xmath34 is touched by at least one of the @xmath26 and , hence , @xmath336 , see example [ ex:2-dimcor ] . also one can check that the rank conditions @xmath337 are satisfied for every face @xmath110 of @xmath34 . ( in fact , both ranks equal 2 when @xmath110 is a side and 1 when @xmath110 is a vertex . ) the augmented exponent matrix and the coefficient matrix are as follows . @xmath341 this time the side of @xmath34 labeled by @xmath342 is not touched by @xmath159 and , hence , @xmath343 . also , the rank condition for @xmath342 fails : @xmath344 and @xmath345 . the submatrix of @xmath36 corresponding to the edge @xmath110 labeled @xmath349 has rank @xmath276 which is less than @xmath350 . therefore the associated system has less than @xmath351 isolated solutions in @xmath352 . ( in fact , it has two solutions . ) in particular , this is a degenerate system . assume that @xmath353 with @xmath286 and any proper face @xmath110 of @xmath34 is a simplex which intersects @xmath176 only at its vertices . assume furthermore that @xmath321 for any proper face @xmath110 of @xmath34 . then is non - degenerate and thus has precisely @xmath312 solutions in @xmath24 counted with multiplicity according to theorem [ t : bkk ] . indeed , if @xmath110 is a proper face of @xmath34 , then the corresponding restricted system has total support @xmath354 . if this restricted system is consistent , then there is a non - zero vector in the kernel of @xmath355 and thus @xmath356 which gives a contradiction .	let @xmath0 and @xmath1 be convex polytopes in @xmath2 such that @xmath3 . we give criteria describing when the mixed volume is strictly increasing @xmath4 this geometric result allows us to characterize sparse polynomial systems with newton polytopes @xmath5 whose number of isolated solutions equals the normalized volume of the convex hull of @xmath6 . in addition , we obtain an analog of cramer s rule for sparse polynomial systems .	10,382	135
ultraluminous infrared galaxies ( ulirgs ) have quasar - like bolometric luminosities ( @xmath5 ) dominated by the far - infrared ( 81000@xmath6 m ) part of the spectrum ( sanders & mirabel , 1996 ) . almost all ulirgs are interacting or merging galaxies ( clements et al . 1996 ) , possibly linking them to the transformation of disk galaxies into ellipticals ( eg . wright et al , 1990 ; baker & clements , 1997 ) . the prodigious luminosity of ulirgs is thought to be powered by a massive starburst , a dust buried agn or some combination of the two . despite a decade of work we still have not been able to decide between these paradigms . various scenarios have also been suggested linking the evolution of quasars with ulirgs ( eg . sanders et al . , 1988 ) . these suggest that part of the luminosity we see from some ulirgs originates in a dust obscured agn which later destroys or expels the enshrouding material . meanwhile , studies of the x - ray background ( mushotzky et al , 2000 ) suggest that dust enshrouded agn make a substantial contribution to its hard component . such objects may also be linked ( trentham & blain , 2001 ; almaini et al . , 1999 ) to the recently discovered cosmic infrared background ( puget et al . 1996 ; fixsen et al . , 1998 ) and the objects that contribute to it ( puget et al . 1999 ; sanders 2000 and references therein ) . as the most obscured objects in the local universe , and as strong candidates for making the cib , ulirgs are ideal local laboratories for studying many of these issues . arp 220 is the nearest ulirg , having an 8 - 1000@xmath6 m luminosity of @xmath7 and a redshift of @xmath8 . as such it is an ideal target for ulirg studies . the consensus since iso is that arp 220 is powered by a massive burst of star formation rather than an agn ( sturm et al 1996 ) , but the possibility of a heavily obscured agn powering the bulk of its emission remains ( haas et al 2001 ) . the evolutionary scenario linking ulirgs to agn also allows the possibility that a weak , but growing , agn may lie at the centre of arp 220 . while this may not be energetically significant at the present time , it may grow to prominence at later stages in the object s evolution . the plausibility of such a scenario has been investigated by taniguchi et al . ( 1999 ) , who show that it is quite possible for a massive black hole ( @xmath9 ) to grow to @xmath10 during the course of a galaxy merger , and thus to be capable of powering a quasar . signs of agn activity can be sought with x - ray observations . the current data for arp 220 includes soft x - ray images from rosat ( heckman et al . these show extended x - ray emission associated with the h@xmath11 nebula ( arribas , colina & clements 2001 ) , which are thought to be produced by a superwind . however the overall soft x - ray luminosity is small relative to the far - ir luminosity when compared to other starbursts , and might allow room for some agn contribution ( iwasawa , 1999 ) . at higher energies , where an agn would be more prominent , data is available from heao-1 ( rieke , 1988 ) , cgro ( dermer et al . , 1997 ) , asca ( iwasawa 1999 ) , and bepposax ( iwasawa et al . these rule out the possibility of an unobscured energetically significant agn in arp 220 . the possibility remains , however , of a compton thick agn , with an obscuring column in excess of 10@xmath12@xmath13 , or of a weaker lower luminosity agn that will grow into a quasar . we have thus undertaken chandra x - ray observations of arp 220 aimed at detecting a weak or obscured agn in its nucleus , and to study the extended superwind emission in detail . this paper presents the first results from our study of the nuclear regions . our results on the superwind can be found elsewhere ( mcdowell et al . 2002 , paper ii ) . we assume a distance of 76 mpc ( kim & sanders 1998 ) to arp 220 throughout this paper . chandra observed arp 220 with the acis - s instrument for 58 ks on 2000 jun 24 . the acis - s instrument was chosen for its good soft response to allow us to study the low energy x - ray emission of the superwind , as well as the harder emission expected from any nuclear source . we chose to use the back - illuminated ccd s3 , for maximum soft response and to avoid any charge transfer difficulties arising in the front - illuminated chips . arp 220 is sufficiently faint that no pile - up issues were expected or found in the data . the data were reduced by the standard chandra pipeline through level 1 ( calibrated event list ) and further analysed using the ciao package version 2.1 and 2.2 . the data were taken with the chip at a temperature of -120c and were gain - corrected using acisd2000-01-29gainn0003.fits from the july 2001 recalibration . the observation was relatively unaffected by background flares and only a small amount of exposure was removed , leaving an effective exposure time of 55756s . astrometry was corrected using a revised geometry file ( teld1999-07-23geomn0004.fits ) which is believed to provide positions across the full acis field accurate to about 1 arcsecond . the standard screening ( good time intervals and grade filtering for grades 0,2,4,5,6 ) was applied to generate a cleaned event file . the x - rays from arp 220 extend over 20 kpc ( paper ii ) , but emission above 2 kev is restricted to the central few kpc . figure [ fig1 ] is a true x - ray color image of the arp 220 nuclear region . it was smoothed in separate bands of 0.2 - 1 ( red ) , 1 - 2 ( green ) and 2 - 10 kev ( blue ) using the ciao adaptive smoothing routine _ csmooth_. the image shows that the nuclear region of arp 220 is clearly distinguished from the rest of the object by being the site of much harder emission . the centroid of the soft emission is displaced 1.5 arcseconds to the northwest of the hard emission . the hard emission coincides with a dust lane in the galaxy ( joy et al . 1986 ) , and indeed the soft emission is suppressed there . however , the absence of hard emission away from the nucleus shows that the spectral change is due to a different type of source , and not merely an absorption effect . figure [ fig2 ] shows an image of the hard emission ( @xmath14 kev ) coming from the nuclear regions of arp 220 , together with circles indicating the area within 1 of the well - studied dual radio and ir nuclei ( see eg . scoville et al . the positional match between the radio / ir nuclei and the hard emission is @xmath151 , within the expected pointing accuracy of chandra . ( the surprising lack of detections of usno stars in the field limits our ability to improve the astrometric accuracy , but three galaxies are found within one arcsecond of their published positions . ) previous observations of arp 220 have shown the presence of hard emission , using , for example , beppo / sax ( iwasawa et al . 2001 ) . however , it is only with the angular resolution of chandra that we have been able to localise some of this emission to the region of the nuclei . the mean off - axis angle of the nucleus during the observation was only 38 arcseconds , so the point spread function ( calculated using the standard ciao tools ) is very close to the on - axis value . to estimate source fluxes , we generated monochromatic point spread functions for the midpoints of the three energy bands using the ciao tool mkpsf , and subtracted a point source from the raw images at the location of the hard peak , with the maximum amplitude that did not create a dip in the local diffuse flux when the data was then smoothed . this procedure was repeated at the location of remaining flux peaks . the central region is consistent ( within the 1 `` absolute astrometric accuracy of the chandra data ) with a pair of hard point sources at the positions of radio nuclei a and b ( scoville et al 1998 ) , with the western nucleus dominant , although the best fit separation for two point sources is only 0.7 '' rather than the 1.1 " of the radio positions and the decomposition of this small region into multiple point sources and diffuse emission is not unique . we designate the sources in order of total flux . the hard band image shows emission concentrated around a nuclear source , x-1 , with extended emission around it . point source subtraction suggests the presence of a much weaker second hard nucleus , x-4 , together with a diffuse component which we denote as the x-1 halo ; its centroid is 0.5 arcseconds east of x-1 . x-2 , further out from the nucleus , is a hard source detected out to 5 kev ; only 33 net counts are seen . its luminosity of @xmath16 puts it in the interesting category of non - nuclear ulx ( ultra - luminous x - ray ) sources . x-3 , the soft peak , is not reliably separated from the extended soft circumnuclear emission , and coincides with the peak of the h@xmath11 emission ( arribas , colina & clements 2001 ; mcdowell et al . 2002 ) . the 210kev luminosity we derive for the nuclear source x-1 and surrounding hard emission ( assuming the spectral model derived in the next section ) is 6.9@xmath17 ergs s@xmath18 , measured from a 3 aperture centred on the nucleus . because of the poorly constrained absorption , here and below we quote observed luminosities rather than absorption - corrected ones unless explicitly stating otherwise . the hard luminosity found here compares to the 210kev luminosity found by iwasawa et al . ( 2001 ) , in a 3 arcminute aperture , of 11@xmath19 ergs s@xmath18 , assuming a similar distance to arp 220 . the background in our observation makes us insensitive to hard emission on 3 arcminute scales , and we can not rule out an extended contribution of this magnitude . however , the 3 sigma upper limit to any remaining hard ( 2 - 8 kev ) flux within 1 arcminute of the nucleus is @xmath20 and we speculate that emission from the nearby southern group may be contributing to the iwasawa et al result . deeper observations with xmm / newton would resolve this issue . we considered two extraction regions : a 5.5 `` radius ` circumnuclear ' region , including x-3 but with the x-1/x-4 region omitted ; a ` nuclear ' region with x-1/x-4 and the hard halo , taking photons within a 3 radius circle , but excluding photons within 0.75 '' of the x-3 location to minimize contamination by the soft source . the extraction regions are indicated in figure [ fig3r ] . for background we extracted counts from two 50 " radius circular regions 3 away on either side of the galaxy and lying on the same node of the chip . responses were generated using the ciao tools _ mkwarf _ and _ mkrmf _ ; a pi ( gain - corrected ) spectrum was extracted and grouped by a factor of 10 ( 0.15 kev bins ) . both xspec and sherpa were used to fit models to the data , and gave similar results . the circumnuclear emission is fit with a raymond - smith or mekal model with a two - temperature plasma of 0.14 and 0.9 kev and absorption of @xmath21 . it is probably composed of a mixture of diffuse emission and point sources . there is no sign of hard emission in this spectrum , confirming the impression from the images that hard emission is restricted to the central kiloparsec of arp 220 . the data are shown in figure [ fig3 ] . fitting the region of hard nuclear emission we find a spectrum similar to the soft circumnuclear emission but with an additional power law component . a representative fit has a power law component of photon index 1.4@xmath221 behind @xmath23 . the absorbed 0.3 - 10 kev luminosity of the power law component is @xmath24 , and most of the counts are above 2 kev . the thermal components , assumed to be superimposed circumnuclear emission , have twenty percent of the flux of the power law , so the hard component dominates the energetics of the nuclear emission . the overall fit , with 9 free parameters , has a reduced chi - squared of 0.84 for 59 d.o.f and the c - statistic is 62.4 ; see figure [ fig4 ] . if instead of modelling the soft emission directly , we use the off - nuclear circumnuclear emission as a background , we find a fit to the remaining emission consistent with a simple power law of index 1@xmath220.5 , a lower nh around @xmath25 , and a similar absorbed luminosity of @xmath26 . however , the remaining emission is also consistent with the slope 1.4 power law plus a modest amount of additional thermal emission at 0.9 kev . the thermal models are not good at fitting the 2.0 kev ( rest ) si xiv line ; an extra contribution at 1.95 kev ( rest ) with l=@xmath27 reduces the @xmath28 , suggesting a possible slight gain error . there is a high bin near 6.5 kev which could be a weak fe k line . to evaluate this possibility we inspected the raw gain - corrected pulse height data without binning . there are a mere 10 photons with gain - corrected energies in the 0.4 kev wide bin between 6.3 and 6.7 kev , compared to 7 photons in the 0.8 kev on either side between 5.9 - 6.3 and 6.7 - 7.1 , a 2 sigma ` detection ' . the distribution of the counts is consistent with the instrumental width at that energy , and their spatial distribution suggests a distribution centered towards the eastern source x-4 ( although a tighter spatial selection only improves the detection significance to 2.6 sigma ) . if real , this line would have a luminosity of @xmath29 , an equivalent width of 700 ev and a rest energy of @xmath30 kev ; these may be taken as approximate upper limits to any narrow fe k line . the absorption - corrected luminosity of the nuclear emission , including the hard halo , is @xmath31 . based on the psf subtraction calculations , we assign 30 percent of this to x-1 , giving @xmath32 which is in the range of both ultra - low luminosity agn and ultra - luminous binaries ( fabbiano 1998 , king et al . see table 1 for derived source positions and luminosities . could there be a further hard source at this location , behind a much larger column ? using a canonical photon index of 1.7 , we set limits of l(0.2 - 10 kev ) = @xmath33 for a column of @xmath34 , and @xmath35 for a column of @xmath36 . however , for a column of @xmath37 , which is entirely plausible in the center of arp 220 , no useful limit can be set by the acis data as even the most luminous quasars would have their x - ray flux absorbed ; one must turn to the harder x - ray limits from the bepposax pds of @xmath38 in 13 - 50 kev ( iwasawa et al . 2001 ) to eliminate this possibility . in general , because of the low sensitivity of chandra above 6 kev , the limits we set as a function of absorbing column are weaker than those in figure 5 of iwasawa et al . for the range they study ( log @xmath39=24.3 and above ) . the x - ray emission in arp 220 is clearly divided into two parts the compact , hard nuclear emission , and the diffuse , softer extended emission . this diffuse emission clearly extends right into the nucleus since the thermal component of the nuclear emission appears very similar to the thermal emission found in the off - nuclear spectrum . the hard emission is not extended beyond a region @xmath401kpc in size ( 3 at 76 mpc , our adopted distance for arp 220 ) . this makes an interesting contrast with other starburst and interacting / merging galaxies already studied by chandra . for example , in the antennae ( fabbiano , zezas & murray 2001 ) hard emission comes from numerous point sources extended over a 10kpc region , and in ngc 3256 ( ward et al . 2000 ) , we see similar clumps spread over 3.5kpc . if such sources existed in arp 220 , we would expect to detect all those above @xmath155@xmath41ergs s@xmath18 @xmath13 in luminosity . we only see one such source , x-2 , which lies 7 away from the nucleus . the hard x - ray emission from arp 220 thus appears to be significantly more concentrated in the nucleus than that of other interacting or merging galaxies observed by chandra . indeed , the spatial distribution of hard emission in arp 220 would appear to be more similar to that of mrk273 ( xia et al . 2002 ) , a ulirg containing an agn at its core , than ngc3256 or the antennae , neither of which seem to have a significant agn contribution . the x - ray output from the nuclear regions of arp 220 is energetically dominated by an extended hard kpc - scale component , with a significant point source contribution ( @xmath42pc ) , that has a power law spectrum with @xmath43=1.4@xmath221.0 , with an unabsorbed 2 - 10kev luminosity of 4@xmath19 ergs / s . the central question is the origin of this radiation does it arise from young supernovae , x - ray binaries , or the result of accretion onto a more massive body , possibly a weak agn . we consider each of these possibilities in turn : * * young supernovae * young supernovae can be strong x - ray emitters especially if in a dense circumstellar or interstellar environment ( schlegel , 1995 ) . such objects are also strong radio emitters . colina et al ( 2001 ) detected a luminous radio supernova in the nuclear starburst of the seyfert galaxy ngc 7469 , a marginally ultraluminous infrared galaxy . 12 candidates for such radio supernovae have been detected by vlbi / vlba observations in the nuclei of arp 220 ( smith et al . 1998 ) , so we know such objects are present . the x - ray properties of such objects are highly heterogeneous . sn1986j , for example , had an integrated 0.1 - 2.4kev luminosity of 23@xmath4410@xmath45ergs s@xmath18 @xmath13 , while sn1993j has a luminosity in the same band of 3@xmath4410@xmath46ergs s@xmath18 @xmath13 . the temporal behaviour of these objects are also different , ranging from relatively constant emission over several years for sn1978k to decay times of a few weeks or months for sn1993j ( schlegel , 1995 ) . a number of such objects are capable of powering the emission we see in arp 220 , though some long term variability should be detectable , especially since no new radio supernovae appear to have occurred for some years ( lonsdale et al . 2000 ) . spectrally the emission from young supernovae are best fitted by thermal bremsstrahlung spectra . attempts to fit the nuclear spectrum of arp 220 with a thermal bremsstrahlung model give an implausibly high temperature of 10.5 kev with a 1.7@xmath47@xmath13 absorbing column ( @xmath28=52.4 with 77 degrees of freedom ) ; supernovae typically have t@xmath150.5kev . fixing the emission to this temperature gives a fit with a poorer @xmath28 than other models , with @xmath28 = 83 with 78 degrees of freedom . * * x - ray binaries and ulx sources * typical galactic x - ray binaries have luminosities up to @xmath1510@xmath48ergs s@xmath18 @xmath13 ( white , nagase & parmar 1995 ) , so several thousand would be required to produce the luminosity seen in arp 220 s nucleus , which is implausible . however , galaxies such as the antennae and ngc3256 , with considerably more active star formation than our own galaxy , have been found to contain ultra - luminous x - ray sources ( ulx ) with luminosity comparable to arp 220 . a survey of 11 nearby spirals with chandra finds 15 ulx sources ( kilgard et al , in preparation ) . the ulx sources may be accreting objects with mass @xmath49 , as seen in m82 ( griffiths et al . 2000 , kaaret et al . 2001 ) ; or , the objects may be high - mass x - ray binaries with compact object masses in the stellar range , but with beamed emission ( king et al . 2001 ) . the m82 source appears to have a 10kev thermal bremsstrahlung spectrum , which can fit the data well ( see above ) , so the presence of several ulx sources in the core of arp 220 is feasible . * * inverse compton emission * moran et al . ( 1999 ) suggested that the hard x - ray emission in ngc3256 is produced through inverse compton scattering of the far - ir photons by relativistic electrons produced by supernova remnants in the starburst . conditions suitable for inverse compton emission might also exist in the nuclei of arp220 , and give rise to the hard emission seen here . one test of inverse compton emission is that the radio and x - ray power law indicies should match , and this does indeed prove to be the case . however , as noted by iwasawa et al . ( 2001 ) , the inverse compton process in arp220 must be quite inefficient or the hard x - ray luminosity would be much greater . * * active galactic nuclei * the restriction of much of the hard x - ray emission to just the central regions of arp 220 is naturally explained if the source is an agn rather than an agglomeration of xrbs . however , whilst the nuclear regions of arp 220 have a high luminosity for an individual xrb , they have a low luminosity for an agn , being about 2 orders of magnitude fainter than typical seyfert 1 s ( george et al . 1998 ) . however , luminosities this low are not without precedent for suspected agns . ( 2001 ) have conducted a survey of nearby galaxies with chandra , examining the x - ray emission of objects that might contain low luminosity agns . they find weak agns in a substantial fraction of their sample , with x - ray luminosities ( 2 - 10kev ) as low as @xmath50ergs s@xmath18 @xmath13 . they also find a correlation between x - ray and h@xmath11 luminosity which arp 220 fits on when only the nuclear h@xmath11 luminosity is considered ( armus , heckman & miley 1990 ) . if there is a weak agn such as this in arp 220 , it will contribute only about 10@xmath5110@xmath52l@xmath53 to the bolometric luminosity , ie . less than 1% . however , the presence of even a weak agn in the dense environment of the nucleus of arp 220 raises the possibility that it will grow and significantly increase in luminosity as the system continues to evolve . there is also the possibility that we are only seeing a small fraction of the emission from an agn if the source itself is compton obscured . the tentative detection of an emission line at 6.5kev , with equivalent width @xmath151 kev , raises the possibility that the power - law source seen is just a reflected component from an otherwise obscured source . harder x - ray observations by beppo / sax ( iwasawa et al . 2001 ) set the best limits to date for any obscured agn . a moderately compton thick agn , with 2@xmath54 @xmath13 column , according to these limits , could supply a few percent of the bolometric luminosity , assuming standard l@xmath55/l@xmath56 ratios . absorption by @xmath5710@xmath12 @xmath13 is needed before a substantial fraction of arp 220 s luminosity can come from an agn . such extinction is not impossible . haas et al . ( 2001 ) use the 7.7@xmath6m/850@xmath6 m flux ratio to suggest very high extinctions , while the lws spectrum from fischer et al . ( 1999 ) , indicating @xmath58 at 100@xmath6 m , would imply an hi column of 2.7@xmath59@xmath13 . improved sensitivity spectra for the nuclear regions of arp 220 is clearly needed to confirm whether the 6.5kev line is real and if it is associated with reflected emission . the presence of _ extended _ hard x - ray emission in the nuclear regions of arp220 can not be easily explained by a simple agn model . possible explainations would include a mixed nature for the nuclear regions , with unresolved agn emission combining with extended xrb emission in the halo . such an arrangement would naturally follow from taniguchi et al.s ( 1999 ) model of nuclear black hole growth by accretion of xrbs during a merger . alternatively , an obscured agn might be associated with extended scattered hard emission . the acquisition of deeper chandra images to examine any differences in x - ray properties between unresolved and halo hard x - ray emission would be able to address these possibilities . we have shown that there is a source of hard , power - law - like , x - ray emission in the nuclear regions of arp 220 . this source is extended ew , consistent with the emission coming from both the radio / ir nuclei . the central concentration of hard x - ray emission in arp 220 is in contrast to other interacting galaxies , where hard emission comes from clumps distributed across much larger physical distances . this difference may be associated with the merger in arp 220 being older than in the antennae or ngc3256 , and that compact objects have sunk to its core ( tremaine , ostriker & spitzer , 1975 ) but may also be associated with arp 220 s greater luminosity . the origin of the hard emission is unclear . its spectrum is unlikely to be produced by young supernovae , but inverse compton emission , albeit of very low efficiency , accretion onto ultra - luminous x - ray binaries or onto an agn are all possible . if there is an agn contribution , it has too low a luminosity for it to play a significant role in the energetics of the object , but the presence of even a weak agn in arp 220 would support a connection between ulirgs and quasars ( sanders et al . if it is not an agn , we can not rule out the presence of a true agn behind a compton screen of column @xmath60 . if xrbs are responsible for the emission then a large number of conventional xrbs are needed , or a smaller number of ultraluminous ulx sources . in either case , the concentration of these objects in the very centre of arp 220 might indicate that these objects will later merge together to form a supermassive black hole and agn ( taniguchi et al . if there is a weak agn already in the nuclear regions , then this process may already have begun . partial support for this work was provided by the national aeronautics and space administration through chandra award number go1 - 1166 issued by the chandra x - ray observatory center , which is operated by the smithsonian astrophysical observatory for and on behalf of nasa under contract nas8 - 39073 . s. lamb acknowledges support from the doe , through llnl . dlc and acb were supported by pparc , cm by the royal society . we would also like to acknowledge the contributions of bernie peek and charlie baker to this work .	we present the first results from 60ks of observations of arp 220 using the acis - s instrument on chandra . we report the detection of several sources near the galaxy s nucleus , including a point source with a hard spectrum that is coincident with the western radio nucleus b. this point source is mildly absorbed ( @xmath0 ) and has an estimated luminosity of @xmath1 erg / s . in addition , a fainter source may coincide with the eastern nucleus a. extended hard x - ray emission in the vicinity raises the total estimated nuclear 2 - 10 kev x - ray luminosity to @xmath2 erg / s , but we can not rule out a hidden agn behind columns exceeding @xmath3 we also detect a peak of soft x - ray emission to the west of the nucleus , and a hard point source 2.5 kpc from the nucleus with a luminosity of @xmath4 erg / s .	7,921	265
the multiobjective optimization problem ( moop ) is to optimize two or more objective functions simultaneously , subject to given constraints . the multiobjective optimization can be applied to problems where the final decision should be made considering two or more conflicting objectives . moop occurs in various fields such as industrial design , finance , management and many engineering areas . practical goals in these fields can be generalized in such a way that the cost of a process is minimized while the quality of its product is maximized . the primary goal is to find a set of solutions that any individual objective function can not be improved without deteriorating the other objective functions , and such a set is called a pareto set . for efficient decision making , a set of generated solutions ( @xmath0 ) should meet two conditions : it should be as close to the pareto front as possible and the solutions should be distributed as widely as possible . evolutionary algorithm ( ea ) is one of the most popular and successful approaches to solve moops @xcite . a number of ea - based algorithms have been suggested including the vector evaluated genetic algorithm ( vega ) @xcite , the niched pareto genetic algorithm ( npga ) @xcite , the nondominated sorting genetic algorithm ( nsga2 ) @xcite , the strength pareto evolutionary algorithm ( spea2 ) @xcite , the mimetic pareto archived evolution strategy ( m - paes ) @xcite and micro genetic algorithm ( micro - ga ) @xcite . among them , nsga2 and spea2 are arguably the most widely used methods . other approaches include simulated annealing ( sa ) @xcite , tabu search @xcite , particle swarm optimization ( pso ) @xcite , immune algorithm ( ia ) @xcite , ant system @xcite and cultural algorithm @xcite . conformational space annealing ( csa ) is a highly efficient single - objective global optimization algorithm which incorporates advantages of genetic algorithm and sa . it has been successfully applied to diverse single - objective optimization problems in physics and biology , such as protein structure modeling @xcite , finding the minimum energy solution of a lenard - jones cluster @xcite , multiple sequence alignment @xcite and the community detection problem @xcite on networks . in these studies , csa is shown to perform more efficient sampling using less computational resources than the conventional monte - carlo ( mc ) and sa methods . here , we introduce a new multiobjective optimization algorithm by using csa , mocsa . compared to existing eas , mocsa has the following distinct features : ( a ) the ranking system considers the dominance relationship and the distance between solutions in the objective space , ( b ) solutions are updated by using a dynamically varying distance cutoff measure to control the diversity of the sampling in the decision space , and ( c ) a gradient - based constrained minimizer is utilized for local search . the remainder of this paper is organized as follows . in section 2 , the definition of moop and related terms are described . in section 3 , details of mocsa is presented . numerical results and the comparison between mocsa and nsga2 on various test problems are presented in section 4 . the final section contains the conclusion . the mathematical definition of a moop can be defined as follows , @xmath1 where @xmath2 is the decision vector , @xmath3 the decision space , @xmath4 the objective vector and @xmath5 the objective space . due to the presence of multiple objective functions , a final solution of moop consists of a set of non - dominated solutions instead of a single point . the notion of _ dominance _ and related terms are defined below . a decision vector @xmath6 is said to dominate another solution @xmath7 ( denoted by @xmath8 ) , if and only if @xmath9 [ paretodominance]definition a solution @xmath2 is said to be non - dominated by any other solutions ( a pareto optimal solution ) if and only if @xmath10 [ paretodominance]definition for a given moop , a pareto optimal set in the decision space , @xmath11 , is defined as @xmath12 [ paretodominance]definition for a given moop , a pareto optimal set in the objective space , @xmath13 , is defined as @xmath14 since the size of pareto optimal front , @xmath13 is infinite in general , which is impossible to obtain in practice , practical algorithms for moop yield a set of non - dominated solutions of a finite size . it should be noted that @xmath13 is always a non - dominated set by definition while a non - dominated set of solutions generated by an algorithm , which is denoted as a @xmath0 , may not be a subset of @xmath13 . here , a new multiobjective optimization algorithm based on csa is described . the csa was initially developed to obtain the protein structure with the minimum potential energy , _ i.e. _ , to solve a single objective optimization problem . csa has been successfully applied to various kinds of optimization problems with modification . the general framework of csa is shown in figure [ csa_flow_chart ] , and the description of mocsa is given in algorithm [ csa ] . initialize the bank , @xmath15 , with @xmath16 random individuals minimize(@xmath15 ) using a constrained local minimizer initialize seed flags of all individuals to zeros : @xmath17 get average distance , @xmath18 , between all pairs of individuals and set @xmath19 as @xmath20 : @xmath21 initialize generation counter to zero : @xmath22 initialize the reserve bank , @xmath23 , to an empty set generate @xmath16 random individuals , @xmath24 minimize(@xmath24 ) @xmath25expand search space evaluate fitness of @xmath15 select @xmath26 seeds among individuals with @xmath27 and set @xmath28 to 1 @xmath29 generate @xmath30 trial solutions by crossover @xmath31 generate @xmath32 trial solutions by mutation @xmath33trial solutions minimize(@xmath34 ) update(@xmath35 ) @xmath36 reduce @xmath19 @xmath37 [ csaendwhile ] csa is a global optimization method which combines essential ingredients of three methods : monte carlo with minimization ( mcm ) @xcite , genetic algorithm ( ga ) @xcite , and sa @xcite . as in mcm , we consider only the solution / conformational space of local minima ; in general , all solutions are minimized by a local minimizer . as in ga , we use a set of @xmath16 solutions ( called _ bank _ in csa , denoted as @xmath15 ) collectively , and we generate offsprings from the bank solutions by cross - over and mutation . finally , as in sa , we introduce a distance parameter @xmath19 , which plays the role of the temperature in sa . in csa , each solution is assumed to represent a hyper - sphere of radius @xmath38 in the decision space . diversity of sampling is directly controlled by introducing a distance measure between two solutions and comparing it with @xmath19 , to prevent two solutions from approaching too close to each other in the decision space . similar to the manipulation of temperature in sa , the value of @xmath19 is initially set to a large value and is slowly reduced to a smaller value in csa ; hence the name conformational space annealing . compared to the conventional ea for multiobjective problems , mocsa has three distinct features ; ( a ) a ranking algorithm which considers the dominance relationship as well as the distance between solutions in the objective space , ( b ) an update rule with a dynamically varying distance cutoff measure to control the size of search space and to keep the diversity of sampling in the decision space and ( c ) the usage of a gradient - based constrained minimizer , feasible sequential quadratic programming ( fsqp ) , for local search . in csa , we first initialize the _ bank _ , @xmath15 , with @xmath39 random solutions which are subsequently minimized by fsqp constrained minimizer . the solutions in the bank are updated using subsequent solutions found during the course of optimization . the initial value of @xmath19 is set as @xmath40 , where @xmath41 is the average distance in the _ decision _ space between two solutions at the initial stage . a number of solutions ( 20 in this study ) in the bank are selected as _ seeds_. for each seed , 30 trial solutions are generated by cross - over between the seed and randomly chosen solutions from the bank . additional 5 are generated by mutation of the seed . it should be noted that if a solution is used as a seed and not replaced by a offspring , it is excluded from the subsequent seed selection . the generated offsprings are locally minimized by fsqp which guarantees to improve a subset of objective functions without deteriorating the others and without violating given constraints . to limit the computational usage , the minimization is performed only once per every five generation steps ( see algorithm [ csa ] ) . offsprings are used to update the bank , and detailed description on the updating rule is provided in section [ sec : update ] . once all solutions in the bank are used as seeds without generating better solutions , implying that the procedure might have reached a deadlock , we reset all bank solutions to be eligible for seeds again and repeat another round of the search procedure . after this additional search reaches a deadlock again , we expand our search space by adding additional 50 randomly generated and minimized solutions to the bank ( @xmath42 ) , and repeat the whole procedure until a termination criterion is satisfied . the maximum number of generation is set to @xmath43 . typically , with @xmath44 in this study , mocsa is terminated before a deadlock occurs with the final bank size of @xmath39 . for a given set of generated solutions , @xmath0 , the fitness of solution @xmath45 is evaluated in terms of @xmath46 , @xmath47 and @xmath48 . @xmath46 is the number of solutions in @xmath0 which dominate @xmath45 . @xmath47 is the number of solutions in @xmath0 dominated by @xmath45 . @xmath48 is the sum of distances from @xmath45 to its nearest and second nearest neighbors in @xmath0 in the objective space . the relative fitness between two solutions , @xmath45 and @xmath49 , is determined by the comparing function shown in algorithm [ fitness ] . with a set of non - dominated solutions all values of @xmath46 and @xmath47 become zeros and the solution with the least value of @xmath50 is considered as the worst . the solutions generated by crossover and mutation are locally minimized by fsqp constrained minimizer and we call them trial solutions . each trial solution @xmath51 , is compared with the bank @xmath15 for update procedure as shown in algorithm [ update ] . first , @xmath52 , the closest solution in @xmath15 from @xmath51 in the _ decision _ space is identified . if there exist dominated solutions in @xmath15 , the closest conformation search is performed only among them . otherwise , @xmath15 is a set of non - dominated solutions , and all in @xmath15 are considered . once @xmath52 is found , the distance @xmath38 in the decision space between @xmath51 and @xmath52 is calculated . if @xmath53 , the current cutoff distance , which indicates that @xmath51 lies in a newly sampled region in the decision space , remote from the existing solutions in @xmath15 , the dominance relationship between @xmath51 and the worst solution in @xmath15 , @xmath2 , is compared . if @xmath54 , @xmath51 is compared with @xmath52 . the selection procedure , described in section [ sec : select ] , is performed to determine which solution should be kept in @xmath15 . at each iteration step , @xmath19 is reduced with a pre - determined ratio , @xmath55 . after @xmath19 reaches to its final value , @xmath56 , it is kept constant . in algorithm [ update ] , for a given trial solution @xmath51 and a solution @xmath57 in @xmath15 , @xmath15 is updated as follows . if @xmath51 dominates @xmath57 , @xmath51 replaces @xmath57 . if @xmath51 is dominated by @xmath57 , @xmath51 is discarded and @xmath57 stays in @xmath15 . if there is no dominance relationship between @xmath51 and @xmath57 and if @xmath57 is better than @xmath51 by algorithm [ fitness ] , @xmath51 is discarded and @xmath57 stays in @xmath15 . finally , when @xmath51 is better than @xmath57 without dominance relationship between them , algorithm [ select ] is used . for the selection procedure , we introduce an additional set of non - dominated solutions , the reserve bank , @xmath23 . due to the limited size of the bank , we may encounter a situation where a solution exists in @xmath15 which is not dominated by the current bank , but dominated by a solution eliminated from the bank in an earlier generation . to solve this problem , non - dominated solutions eliminated from @xmath15 are stored up to 500 in @xmath23 , which is conceptually similar to an _ archive _ in other eas @xcite . the difference is that @xmath23 is used only when more than half of the solutions in @xmath15 are non - dominated solutions because csa focuses more on diverse sampling rather than optimization at the early stage of the optimization . note that @xmath23 keeps only non - dominated solutions . @xmath58 number of solutions in @xmath15 which dominate @xmath59(@xmath60 ) @xmath61 number of solutions in @xmath15 which are dominated by @xmath59(@xmath60 ) @xmath62 the sum of distances from @xmath45 to the nearest and second nearest neighbors in @xmath15 in the _ objective _ space @xmath59 is better @xmath60 is better @xmath63 @xmath59 is better @xmath60 is better @xmath64 @xmath59 is better @xmath60 is better @xmath65 nearest solution to @xmath51 in @xmath15 in the _ decision _ space @xmath65 nearest solution to @xmath51 among _ dominated _ solutions in @xmath15 in the _ decision _ space distance in the decision space @xmath66 worst solution in @xmath15 @xmath67 @xmath68 @xmath51 dominates @xmath57 @xmath51 replaces @xmath57 @xmath57 dominates @xmath51 @xmath57 stays in @xmath15 and @xmath51 is discarded @xmath57 stays in @xmath15 and @xmath51 is discarded select(@xmath69 ) @xmath70 , @xmath71 , @xmath51 is better than @xmath57 @xmath51 replaces @xmath57 in @xmath15 @xmath57 is non - dominated in @xmath15 and @xmath23 is used @xmath51 replaces @xmath57 in @xmath15 @xmath51 is dominated by @xmath23 @xmath66 nearest dominating solution to @xmath51 in @xmath23 in the _ objective _ space move @xmath2 from @xmath23 to @xmath15 move @xmath57 from @xmath15 to @xmath23 @xmath72 @xmath73 @xmath74 \\ & z_1 , \dotsc , z_k \in [ -5,5 ] \end{aligned}\ ] ] @xmath75 @xmath76 \\ \end{aligned}\ ] ] @xmath77 @xmath78\right ) \\ g & = 1 + 9\sum_{i=1}^{k}z_i / k \\ \end{aligned}\ ] ] @xmath79 for the benchmark test of mocsa , we have selected 12 widely used test problems in the field . they consist of zdt @xcite and dtlz @xcite . each test suite contains several functional forms and can feature various aspects of optimization algorithms . comprehensive analysis on the characteristics of the two test suites are well documented by huband _ et al . _ @xcite . in both suites , the input vector , @xmath57 , is divided into two sets @xmath80 and @xmath81 to construct test problems as follows , @xmath82 , where @xmath83 and @xmath49 are the dimensions of decision and objective spaces respectively and @xmath84 . the zdt problem suite consists of six test problems and is probably the most popular test suite to access multiobjective optimization algorithms . the explicit functional forms of five zdt problems are presented in table [ tab : zdt ] . the zdt test suite has two main advantages : ( a ) the pareto fronts of the problems are known in exact forms and ( b ) benchmark results of many existing studies are available . however , there are shortcomings : ( a ) the problems have only two objectives , ( b ) none of the problems contain flat regions and ( c ) none of the problems have degenerate pareto optimal front @xcite . .five real - valued zdt problems are described . the first objective depends only on the first decision variable as @xmath85 and the second objective is given as @xmath86 , where @xmath87 and @xmath88 , where @xmath83 and @xmath49 are the dimensions of the decision space and the objective space . unless the functional forms of @xmath89 and @xmath90 are separately given , they are identical to those of zdt1 . [ cols="^,<,^",options="header " , ] in this paper , we have introduced a novel multiobjective optimization algorithm by using the conformational space annealing ( csa ) algorithm , mocsa . benchmark results on 12 test problems show that mocsa finds better solutions than nsga2 , in terms of four criteria tested . solutions by mocsa are closer to the pareto front , a higher fraction of them are on the pareto front , they cover a wider objective space , and they are more evenly distributed on average . we note that the efficiency of mocsa arises from the fact that it controls the diversity of solutions in the decision space as well as in the objective space . the authors acknowledge support from creative research initiatives ( center for in silico protein science , 20110000040 ) of mest / kosef . we thank korea institute for advanced study for providing computing resources ( kias center for advanced computation linux cluster ) for this work . we also would like to acknowledge the support from the kisti supercomputing center ( ksc-2012-c3 - 02 ) . coello coello , c. , corts , n. , 2002 . an approach to solve multiobjective optimization problems based on an artificial immune system . in : proceedings of first international conference on artificial immune systems 2002 ; canterbury , uk . . 212221 . coello coello , c. , lechuga , m. , 2002 . mopso : a proposal for multiple objective particle swarm optimization . in : evolutionary computation , 2002 . proceedings of the 2002 congress on . vol . 2 . ieee , pp . 10511056 . deb , k. , agrawal , s. , pratap , a. , meyarivan , t. , 2000 . a fast elitist non - dominated sorting genetic algorithm for multi - objective optimization : nsga - ii . lecture notes in computer science 1917 , 849858 . deb , k. , thiele , l. , laumanns , m. , zitzler , e. , 2002 . scalable multi - objective optimization test problems . in : proceedings of the congress on evolutionary computation ( cec-2002),(honolulu , usa ) . proceedings of the congress on evolutionary computation ( cec-2002),(honolulu , usa ) , pp . 825830 . doerner , k. , gutjahr , w. , hartl , r. , strauss , c. , stummer , c. , 2004 . pareto ant colony optimization : a metaheuristic approach to multiobjective portfolio selection . annals of operations research 131 ( 1 ) , 7999 . hansen , m. , 1997 . tabu search for multiobjective optimization : mots . in : proceedings of the 13th international conference on multiple criteria decision making ( mcdm97 ) , cape town , south africa . citeseer , pp . 574586 . horn , j. , nafpliotis , n. , goldberg , d. , 1994 . a niched pareto genetic algorithm for multiobjective optimization . in : evolutionary computation , 1994 . ieee world congress on computational intelligence . , proceedings of the first ieee conference on . ieee , pp . 8287 . joo , k. , lee , j. , seo , j. , lee , k. , kim , b. , lee , j. , 2009 . all - atom chain - building by optimizing modeller energy function using conformational space annealing . proteins : structure , function , and bioinformatics 75 ( 4 ) , 10101023 . lee , j. , liwo , a. , ripoll , d. , pillardy , j. , scheraga , h. , jan . calculation of protein conformation by global optimization of a potential energy function . proteins structure function and genetics 37 ( s 3 ) , 204208 . liwo , a. , lee , j. , ripoll , d. r. , pillardy , j. , scheraga , h. , may 1999 . protein structure prediction by global optimization of a potential energy function . proceedings of the national academy of sciences of the united states of america 96 ( 10 ) , 54825 . pillardy , j. , czaplewski , c. , liwo , a. , lee , j. , ripoll , d. r. , kamierkiewicz , r. , oldziej , s. , wedemeyer , w. j. , gibson , k. d. , arnautova , y. a. , saunders , j. , ye , y. j. , scheraga , h. , feb . 2001 . recent improvements in prediction of protein structure by global optimization of a potential energy function . proceedings of the national academy of sciences of the united states of america 98 ( 5 ) , 232933 . schaffer , j. , 1985 . multiple objective optimization with vector evaluated genetic algorithms . in : proceedings of the 1st international conference on genetic algorithms . l. erlbaum associates inc . , pp . 93100 . van veldhuizen , d. , lamont , g. , 2000 . on measuring multiobjective evolutionary algorithm performance . in : evolutionary computation , 2000 . proceedings of the 2000 congress on . vol . 1 . ieee , pp .	we introduce a novel multiobjective optimization algorithm based on the conformational space annealing ( csa ) algorithm , mocsa . it has three characteristic features : ( a ) dominance relationship and distance between solutions in the objective space are used as the fitness measure , ( b ) update rules are based on the fitness as well as the distance between solutions in the decision space and ( c ) it uses a constrained local minimizer . we have tested mocsa on 12 test problems , consisting of zdt and dtlz test suites . benchmark results show that solutions obtained by mocsa are closer to the pareto front and covers a wider range of the objective space than those by the elitist non - dominated sorting genetic system ( nsga2 ) . conformational space annealing , multiobjective optimization , genetic algorithm , evolutionary algorithm , pareto front	6,341	225
multipole radiofrequency ion traps @xcite , in particular the 22-pole ion trap @xcite , are versatile devices used in laser spectroscopy @xcite and investigations of chemical reaction processes @xcite of atomic and molecular ions . high order multipole traps offer a large field free region in the trap center , and therefore provide a reduced interaction time of the ions with the oscillating electric field compared to a quadrupole trap @xcite . buffer gas cooling down to cryogenic temperatures of the translational @xcite , rotational @xcite , and vibrational degrees of freedom @xcite of trapped molecular ions has been demonstrated . this enables applications with all the advantages of low temperature experiments , such as a reduced doppler width in spectroscopy studies and a well defined population of internal states . stable confinement of a single ion in the oscillating quadrupole field of a paul trap is precisely predicted , because the mathieu equations of motion can be solved analytically . this is different for oscillating high order multipole fields , where the equations of motion have no analytical solution . however , the movement of the ions in a fast oscillating rf - field justifies the assumption of an effective trapping potential , based on the separation of the ion motion into a smooth drift and a rapid oscillation , called micromotion @xcite . this effective potential can be expressed as @xmath1 where @xmath2 denotes the electric field at the point @xmath3 , @xmath4 is the mass of a test particle with charge @xmath5 in an electric field oscillating on frequency @xmath6 and @xmath7 is a non - oscillating dc potential . for an ideal cylindrical multipole of the order @xmath8 this effective potential can be expressed as @xmath9 where @xmath10 denotes the amplitude of the oscillating rf - field . for a high order multipole field ( @xmath8=11 for the 22-pole trap ) this creates an almost box - like trapping volume with steep walls and a large field free region in the center . this is important to suppress rf heating when cooling ions in neutral buffer gas @xcite . the number of trapped ions that can be excited with radiation depends on the local density of ions in the interaction volume with the light field . in experiments with trapped ba@xmath11 ions in an octupole trap the density has been imaged by spatially resolving the fluorescence signal @xcite . in this article we report on a method based on the photodetachment of trapped anions that allows us to image the density distribution of the trapped ions . in this way we measure the experimental trapping potential of a 22-pole trap with unprecedented accuracy and study its dependence on the radiofrequency field . in the next section the photodetachment tomography is explained , followed by a presentation of measured density distributions at different temperatures and rf amplitudes . we find ten unexpected maxima in the density distribution and analyze their origin with the help of numerical simulations in section [ model ] . the experimental radiofrequency ion trap setup has been described in detail in ref . its central element is a 22-pole trap @xcite with good optical access along the trap axis . in the present tomography experiments , we have used an ensemble of stored oh@xmath0 anions cooled to a translational temperature of either 170 or 300 k. the use of negative ions allows us to perform photodetachment measurements , were the anions are depleted by a laser beam propagating parallel to the symmetry axis of the trap @xcite . the photodetachment process @xmath12 with @xmath13ev yields a depletion rate proportional to the photodetachment cross section and to the overlap of the ion column density with the photon flux . by scanning the position of the laser beam and measuring the depletion rate at each position the relative ion column density distribution is obtained . for small ion densities , where the coulomb interaction between the ions can be neglected , the depletion rate at the transverse position @xmath14 can be expressed as @xmath15 it depends only on the total photon flux @xmath16 , the photodetachment cross section @xmath17 and the single particle column density @xmath18 , reflecting the spatial overlap of the laser beam with the ion distribution @xcite . as the latter is normalized to unity , we can write it as @xmath19 thus , a full two - dimensional tomography scan of the trapped ions can be used to map the entire ion column density in the trap . if @xmath20 is measured for the whole ion distribution , also the absolute photodetachment cross section can be obtained @xcite . ions are produced in a pulsed supersonic expansion of a suitable precursor gas , crossed by a 1kev electron beam . for the oh@xmath0 production we use a mixture of ar / nh@xmath21/h@xmath22o ( 88% , 10% , 2% ) to create nh@xmath23 . a rapid chemical conversion by water forms the oh@xmath0 anions in the source . a bunch of @xmath24500 mass - selected ions is loaded into the trap , which is enclosed by a copper housing that is temperature - variable between 8 and 300k . a typical he buffer gas density of @xmath25@xmath26 , employed for all temperatures , is enhanced by a buffer gas pulse during the ion injection . the trap is operated with @xmath27mhz radiofrequency and different rf amplitudes . along the axial direction ions are confined by end electrodes biased to -2v . to allow good thermalization of the ions , a storage period of 200ms is inserted before the laser beam is switched on . after a given storage and laser interaction time the current signal , proportional to the number of ions that survived the interaction with the photodetachment laser , is detected with a microchannel plate . the two - dimensional tomography scans are performed under the same experimental conditions as in ref . @xcite . a free - running continuous wave diode laser at 661.9 nm ( mitsubishi ml1j27 , 100mw , spectral width 0.7 nm fwhm ) has been employed . it is imaged into the trap and scanned in the @xmath14 plane , perpendicular to the symmetry axis of the trap , by moving the imaging lense on a mesh with 0.25 mm point spacing with a computer - controlled two - dimensional translation stage . the mesh spacing is comparable to the @xmath28 radius of the laser in the trap of 350@xmath29 m . for each laser position the photodetachment depletion rate is obtained in a storage time interval of 0.2 - 2s by fitting an exponential decay to the ion current signal , reduced by typically one percent due to the background loss rate . the data are averaged over typically 4 - 8 scans . within each scan the mesh points are accessed in random order to avoid systematic drifts . fig . [ fig1]a shows a tomography scan of oh@xmath0 anions in the 22-pole trap at 300k with the rf amplitude set to 160v . the figure also contains the sketched arrangement of the trap s copper housing mounted on the coldhead of the cryostat , the position of the 22 rf electrodes and the axial end electrodes . [ fig1]b is a zoom of the scan which more clearly shows the measured ion density distribution . every pixel of the histogram here represents a fitted photodetachment depletion rate @xmath20 ( see previous section ) and is proportional to the single - particle probability density @xmath18 along a column parallel to the @xmath30-axis . as can be seen the ion distribution as a first approximation can be considered circularly symmetric and constant in the center region of the trap , whereas it drops to zero when the ions reach the outer regions of the trapping volume . this distribution directly visualizes the overall storage properties of a 22-pole ion trap with a flat potential in the center and steep walls . note that the ion density drops to zero already for smaller radii than the end electrode ( solid line in fig . [ fig1]b ) , indicating that clipping of the laser at the end electrode is not affecting the measured density distribution . for smaller rf amplitudes the ion density distribution would extend to larger radii and could not be fully probed by the photodetachment tomography . for this reason we have restricted ourselves to large enough rf amplitudes in this study . in fig . [ fig1]c a horizontal cut through the ion distribution is shown . while in the center the distribution is relatively uniform , the population is locally enhanced by up to 40% near the edge of the ion distribution . such a behavior has already been observed in previous measurements @xcite . to study this here in more detail , the effective potential @xmath31 is extracted from the local ion density @xmath18 assuming a boltzmann distribution for the ions in the trap @xmath32 where @xmath33 is the absolute temperature , @xmath34 is boltzmann s constant , and @xmath35 is the partition function . since only the column density is measured , the resulting potential @xmath31 is an average over the @xmath30-direction . fig.[fig1]d shows a cut through the obtained effective potential for the distribution of fig . overall this potential compares well with the calculated potential of an ideal 22-pole potential ( solid line ) , obtained without any free parameters from eq.([multipole_effective_potential ] ) . closer inspection reveals interesting features in the potential that deviate from the ideal multipole . while the potential is relatively flat in the center , it shows a distinct minimum of about 12mev near the left edge of the ion distribution and a weaker minimum of about 5mev near its right edge . it can be excluded that this change of the distribution is caused by space charge effects , because the experiments are performed with only a few hundred ions in a trap volume of about 1@xmath36 . the same features of the effective potential are observed in measurements at a lower trap temperature . [ fig2]a shows a tomography scan at 170k and the same rf amplitude as above . the ion distribution again looks circular symmetric with a distinct cutoff when the ions reach the steep walls of the trapping potential . a horizontal cut through the effective potential , obtained in the same fashion as fig . [ fig1]d , is shown in fig . [ fig2]b . the same minima as for 300k are observed in the effective potential . at this lower temperature the two minima are better resolved and appear similar in depth on the left side and slightly deeper on the right side of the potential as compared to the 300k tomography . further substructure becomes visible in the 170k density distribution . ten clearly separated maxima in the density distribution appear almost equally spaced in angle at a radial position of about 3 mm . according to eq.([density : eq ] ) they correspond to ten localized minima in the trapping potential at this radius with a typical depth of 10mev . these minima have not been significant in the 300k ion distribution at 160v rf amplitude . they become visible , however , for larger amplitudes . fig.[fig3]a shows a 300k ion distribution for an rf amplitude of 270v . it reveals the same ten density maxima and respective potential minima that could only be resolved at lower temperature at 160v . we have studied the dependence of the depth of the ten potential minima on the rf amplitude at 300k . the depth of the deepest minimum is plotted in fig . [ fig3]b . since the effective potential is expected to depend quadratically on the rf amplitude , a fit with only a constant and a quadratic term is applied to the data ( solid line in fig . [ fig3]b ) . it yields an rf - independent offset of about 11mev , which is attributed to the static potential of the end electrodes of the ion trap . these end electrodes produce a radially repulsive potential inside the trap , as discussed in ref.@xcite . it compares well with simulations , as shown in the next section . the ten `` pockets '' in the potential , however , reveal a more complex deviation from the ideal multipole description of eq.([multipole_effective_potential ] ) . an explanation for them will also be discussed in the next section . the effective potential of a @xmath37-pole has a @xmath37-fold rotational symmetry , when averaging over one radiofrequency period . the appearance of the ten observed potential minima is therefore a clear indication for a breaking of the ideal symmetry . to investigate this effect further , the effective potential of the employed 22-pole trap has been modeled using a numerical simulation package based on a fast multipole solver @xcite for the boundary element problem in combination with accurate field evaluation in free space . with this method the electric field @xmath38 can be calculated at any location inside the trap . it is converted into the effective trapping potential using eq . ( [ effective_potential ] ) . we have verified that the simulation of the trapping potential of an ideal 22-pole structure reproduces the effective potential of eq . ( [ multipole_effective_potential ] ) on the numerical level of accuracy . different assumptions have been tested as origin of the observed ten potential minima , such as the influence of the shape and position of the end electrodes and of the copper housing around the trap electrodes , without showing a measurable effect on the potential . this suggests that imperfections of the trap geometry itself may be responsible . to simulate these imperfections a breaking of the ideal symmetry is introduced by displacing one half of the 22 radiofrequency electrodes by a small angle ( see inset in fig.[fig4]a ) . such a small tilt of one set of electrodes against the others occurs to be the most likely displacement during the trap assembly . upon tilting one set of electrodes by only a few tenths of a degree the calculated effective potential of a 22-pole trap at 160v rf amplitude and -2v on the end electrodes immediately shows ten potential minima . in fig . [ fig4]a the dependence of the maximum pocket depth on the tilt angle , as obtained from a series of simulations , is plotted . these simulations have been carried out for 160v rf amplitude . here , an imperfection in the parallelity of only @xmath39 causes a pocket depth of 5mev . the pocket depth is calculated at each angle with and without a potential of -2v applied to the static end electrodes . the end electrode voltage produces an overall quadrupole potential that pushes the ion ensemble towards larger radii in the trap in addition to the tilt - induced pockets . both data sets with and without end electrode potential are described by the same quadratic increase . for the simulations with end electrode potential a constant offset of about 9mev is obtained , in fair agreement with the experiment value of about 11mev . from the measured depths of the potential minima ( fig . [ fig3]b ) a value of between 3 and 5mev is extracted for an rf amplitude of about 160v , after subtracting the influence of the static end electrode ( see fig.[fig3]b ) . such an rf - induced pocket depth is obtained in the simulation for a tilt angle of between @xmath40 and @xmath39 ( see fig.[fig4]a ) . [ fig4]b shows a simulated density distribution for a tilt angle of @xmath40 . following eq . ( [ density : eq ] ) , the simulation has been performed for oh@xmath0 ions that are stored at 300k in the 22-pole trap with 270v rf amplitude and -2v potential on the end electrodes . this simulated density distribution agrees well with the measured distribution of fig . [ fig3]a , which has been obtained with the same trap parameters . a larger tilt angle was found to already overestimate the ten potential minima . note that it is preferable to compare graphs of the density distributions of simulation and experiment instead of effective potentials , because the experimental potential is obtained by a logarithm of the density distribution which suppresses the fine details in the images . when the 22-pole trap was assembled the strong influence of small displacements of the rf electrodes on the effective potential was not known . a tilt of one set of rf electrodes by a tenth of a degree can therefore not be excluded for our trap . such small tilt angles already come close to the mechanical tolerances for the assembly of a 22-pole trap in the presently used design . this shows the need to significantly improve the precision in the rf electrode geometry when a potential with pocket depths in the @xmath29ev range is desired . in searching for an explanation for the observed ten minima , we have extended the electric field simulations to multipoles of different order @xmath8 . these calculations have shown that the number of minima observed in the effective potential of a distorted multipole ion trap is directly connected to the multipole order as @xmath41 . besides the above discussed tilt of one set of electrodes , other distortions of the ion trap , such as a parallel displacement of one set of rf electrodes , also introduces @xmath42 minima . the number of minima therefore seems to be a general consequence of breaking the symmetry of a multipole ion trap . we expect the minima to be related to the points in space where the superimposed time - dependent electric field of the individual multipole electrodes cancels , because according to eq . ( [ effective_potential ] ) these are the global minima of the effective potential . for a perfectly symmetric multipole trap cancellation is expected only in the trap center , but for a distorted symmetry several such points are found at larger radii . in this article we report on a method to directly measure the column density distribution of ions in a 22-pole ion trap using photodetachment of stored oh@xmath0 . the two - dimensional tomography scans yield the effective potential averaged over the length of the trap . the measurements quantitatively confirm the overall validity of the effective potential of a 22-pole ion trap , which scales as @xmath43 . for large rf amplitudes , however , new features in the potential have been observed in the form of ten almost equally spaced potential minima . these minima arise from the breaking of the 22-fold symmetry of the trap . they can be quantitatively explained by a slight tilt of half of the multipole rf electrodes within their mechanical tolerances . also for other multipole ion traps @xmath42 minima in the effective trapping potential have been found as a consequence of a broken symmetry of the trap . this observation , which has become possible due to the high sensitivity of our photodetachment tomography scans , has implications for other spectroscopic experiments in 22-pole ion traps . in particular at cryogenic temperatures , trapped ions will reside predominantly in the ten pockets of the potential . correspondingly , the ion density along the symmetry axis of the trap would become very small and only a small spectroscopic signal would be detected for a laser beam pointing along the trap axis . further studies are needed to find out if ions that reside in the ten pockets may be subject to enhanced radiofrequency heating , similar to the influence of the static end electrode potential , which can increase the translational temperature by a few kelvin @xcite . generally , it is therefore advisable to operate the ion trap at rf amplitudes far below 100v to significantly suppress the pockets . to overcome the pockets a significantly enhanced precision in the manufacturing and assembly of 22-pole traps is required . an interesting alternative for precisely controllable multipole ion traps are planar , chip - based traps @xcite . we thank ferdinand schmidt - kaler for helpful discussions . this work is supported by the deutsche forschungsgemeinschaft under contract no . we 2592/2 - 1 . p.h . acknowledges support by the alexander von humboldt foundation . k.s . acknowledges support by the european commission within emali ( contract no . mrtn - ct-2006 - 035369 ) and the landesstiftung baden - wrttemberg in the framework atomics ( contract no . pn 63.14 ) and the eliteprogramm postdoktorandinnen und postdoktoranden. 10 url # 1#1urlprefix[2][]#2 gerlich d 1992 _ adv . phys . _ * 82 * 1 ions at 300k buffer gas temperature with the rf amplitude set to 160v . the sketched geometry shows the layout of the ion trap , viewed along its symmetry axis . it includes the copper housing , the 22 rf electrodes ( end - on ) , a surrounding shaping electrode , and the end electrodes . b ) zoom into the measured ion density distribution , each pixel represents an individual decay rate measurement . c ) one - dimensional cut through the density distribution along the horizontal axis . d ) effective potential derived from the density distribution by assuming a boltzmanm distribution of the trapped ions at 300k . ] a ) . the right wall of the trap together with the 11 implanted rf electrodes has been tilted by an angle of @xmath44 . b ) calculated ion density distribution for 300k in the effective potential of 270v rf amplitude and -2v static end electrode potential . a tilt angle of 0.15@xmath45 is chosen , which leads to a good agreement with the measured density distribution shown in fig . [ fig3]a . ]	the column density distribution of trapped oh@xmath0 ions in a 22-pole ion trap is measured for different trap parameters . the density is obtained from position - dependent photodetachment rate measurements . overall , agreement is found with the effective potential of an ideal 22-pole . however , in addition we observe 10 distinct minima in the trapping potential , which indicate a breaking of the 22-fold symmetry . numerical simulations show that a displacement of a subset of the radiofrequency electrodes can serve as an explanation for this symmetry breaking .	5,577	134
experimental investigations of the rheology of concentrated suspensions often involve a vane - in - cup geometry ( see @xcite for a review ) . the vane tool offers two main advantages over other geometries . first , it allows the study of the properties of structured materials with minimal disturbance of the material structure during the insertion of the tool [ @xcite ] . it is thus widely used to study the properties of gels and thixotropic materials [ @xcite ] and for in situ study of materials as e.g. in the context of soil mechanics [ @xcite ] . second , it is supposed to avoid wall slip [ @xcite ] , which is a critical feature in concentrated suspensions [ @xcite ] ; the reason for this belief is that the material sheared in the gap of the geometry is sheared by the ( same ) material that is trapped between the blades . consequently , it is widely used to study the behavior of pasty materials containing large particles , such as fresh concrete [ @xcite ] and foodstuff [ @xcite ] . the constitutive law of materials can be obtained from a rheological study with the vane - in - cup geometry provided one knows the coefficients called `` geometry factors '' that allow the conversion of the raw macroscopic data ( torque , rotational angle or velocity ) into local data ( shear stress , shear strain or shear rate ) . however , in contrast with other classical geometries , even the _ a priori _ simple linear problem ( for hookean or newtonian materials ) is complex to solve with a vane tool . this linear problem was studied theoretically by @xcite and @xcite in the general case of a @xmath2-bladed vane tool embedded in an infinite linear medium . the analytical expression found for the torque vs. rotational velocity is in rather good agreement with macroscopic experimental data [ @xcite ] . note however two possible shortcomings of this theoretical approach for its use in practice : the blades are infinitely thin and there is no external cylinder . there is no such approach in the case of nonlinear media ( _ i.e. _ complex fluids ) . a practical method used to study the flow properties of non - linear materials , known as the couette analogy [ @xcite ] , consists in calibrating the geometry factors with hookean or newtonian materials . one defines the equivalent inner radius @xmath3 of the vane - in - cup geometry as the radius of the inner cylinder of a couette geometry that would have the same geometry factors for a linear material . for any material , all macroscopic data are then analyzed as if the material was sheared in a couette geometry of inner cylinder radius @xmath3 . the nonlinearity ( that affects the flow field ) is sometimes accounted for as it is in a standard couette geometry [ @xcite ] . this approach may finally provide constitutive law measurements within a good approximation [ @xcite ] . however , simulations and observations show that @xmath3 is not a universal parameter of the vane tool independent of the properties of the studied material . while the streamlines go into the virtual cylinder delimited by the blades in the case of newtonian media [ @xcite ] , yielding an equivalent radius lower than the vane radius [ @xcite ] , it was found from simulations [ @xcite ] that the streamlines are nearly cylindrical everywhere for shear - thinning fluids if their index @xmath4 is of order 0.5 or less , and thus that @xmath5 in these cases . moreover , for yield stress fluids , simulations and photographs of the shearing zone around a four - bladed vane rotating in bingham fluids [ @xcite ] , simulations of herschel - bulkley and casson fluids flows in a four - bladed vane - in - cup geometry [ @xcite ] , and simulations of bingham fluids flows in a six - bladed vane - in - cup geometry [ @xcite ] , all show that at yield ( _ i.e. _ at low shear rates ) , the material contained in the virtual cylinder delimited by the blades rotates as a rigid body , and that it flows uniformly in a thin cylindrical layer near the blades . this is now widely accepted [ @xcite ] and used to perform a couette analogy with @xmath5 ; the yield stress @xmath6 is then simply extracted from torque @xmath7 measurements at low velocity thanks to @xmath8 , where @xmath9 is the vane tool height ( neglecting end effects ) [ @xcite ] . the flow field in a vane - in - cup geometry and its consequences on the geometry factors have thus led to many studies . however , only theoretical calculations , macroscopic measurements and simulation data exist in the literature : there are no experimental local measurements of the flow properties of newtonian and non - newtonian materials induced by a vane tool except the qualitative visualization of streamlines made by @xcite for newtonian media , and the photographs of @xcite for yield stress fluids . moreover , while the main advantage of the vane tool is the postulated absence of wall slip , as far as we know , this widely accepted hypothesis has been neither investigated in depth nor criticized . in order to provide such local data , we have performed velocity measurements during the flows of a newtonian medium and of a yield stress fluid in both a coaxial cylinder geometry and a vane - in - cup geometry . we have also performed particle concentration measurements in a concentrated suspension of noncolloidal particles in a yield stress fluid , which is a good model system for complex pastes such as fresh concrete [ @xcite ] . our main results are that : 1 . in the newtonian fluid , the @xmath0-averaged strain rate component @xmath1 decreases as the inverse squared radius in the gap , as in a couette geometry , which allows direct determination ( without end - effect corrections ) of the value of @xmath3 : it is here found to be lower than @xmath10 , but slightly higher than for a vane in an infinite medium ; the flow enters deeply the region between the blades , leading to a significant extensional flow ; 2 . in the yield stress fluid , in contrast with results from the literature , the layer of material that is sheared near the blades at low velocity does not have a cylindrical shape ; 3 . in the suspension of noncolloidal particles in a yield stress fluid , the noncolloidal particles are quickly expelled from a thin zone near the blades , leading to the development of a thin slip layer made of the pure interstitial yield stress fluid , in sharp contradiction with the common belief that the vane tool prevents slippage . in sec . [ section_display ] , we present the materials employed and the experimental setup . we present the experimental results in sec . [ section_results ] : velocity profiles obtained with a newtonian oil and with a yield stress fluid are presented in sec . [ section_results][section_oil ] and sec . [ section_results][section_emulsion ] , while sec . [ section_results][section_suspension ] is devoted to the case of suspensions , with a focus on the slip layer created by a shear - induced migration phenomenon specific to the vane tool . throughout this paper , we use cylindrical coordinates @xmath11 . all flows are supposed to be @xmath12 invariants ( _ i.e. _ there are no flow instabilities ) . we define the @xmath0-average @xmath13 of a function @xmath14 as @xmath15 . we study three materials : a newtonian fluid , a yield stress fluid , and a concentrated suspension of noncolloidal particles in this yield stress fluid . + the newtonian fluid is a silicon oil of 20 mpa.s viscosity . + the yield stress fluid is a concentrated water in oil emulsion . the continuous phase is dodecane oil in which span 80 emulsifier is dispersed at a 7% concentration . a 100 g / l cacl@xmath16 solution is then dispersed in the oil phase at 6000 rpm during 1 hour with a sliverson l4rt mixer . the droplets have a size of order 1 @xmath17 m from microscope observations . the droplet concentration is 75% , and the emulsion density is @xmath18 g@xmath19 . the emulsion behavior , measured through coupled rheological and mri techniques described in @xcite ( see fig . [ figure_emulsion_behavior ] ) , is well fitted to a herschel - bulkley behavior @xmath20 with yield stress @xmath21 pa , consistency @xmath22=6.8 pas@xmath23 , and index @xmath24 . + . the empty squares are local data ; the solid line is a herschel - bulkley fit to the data @xmath20 with @xmath21 pa , @xmath22=6.8 pas@xmath23 , and @xmath24.,width=302 ] + the suspension is a suspension of monodisperse polystyrene beads ( density @xmath25 g@xmath19 , diameter @xmath26 m ) suspended at a 40@xmath27 volume fraction in the concentrated emulsion described above . the density matching between the particles and the yield stress fluid is sufficient to prevent shear - induced sedimentation of the particles in the yield stress fluid [ @xcite ] ; in all experiments , we check that the material remains homogeneous in the vertical direction by means of mri density measurements . the rheometric experiments are mainly performed within a six - bladed vane - in - cup geometry ( vane radius @xmath28 cm , outer cylinder radius @xmath29 cm , height @xmath3011 cm ) . the shaft radius is 1.1 cm and the blade thickness is 6 mm . other experiments are performed with a wide - gap couette geometry of slightly different inner cylinder radius @xmath31 cm , due to the presence of sandpaper ( the other dimensions were identical ) . the inner cylinder of the couette geometry , and the outer cylinder of both geometries are covered with sandpaper of roughness equivalent to the size of the largest elements of the materials studied in order to avoid wall slip . + in the rheometric experiments presented here , we control the rotational velocity of the inner cylinder , with values ranging from 0.1 to 100 rpm . proton mri [ @xcite ] was chosen as a non - intrusive technique in order to get measurements of the local velocity and of the local bead concentration inside the sample . experiments are performed on a bruker 24/80 dbx spectrometer equipped with a 0.5 t vertical superconductive magnet with 40 cm bore diameter and operating at 21 mhz ( proton frequency ) . we perform our experiments with a home made nmr - compliant rheometer , equipped with the geometries described in the previous section . this device was already used in a number of previous rheo - nmr studies [ @xcite ] , and is fully described in @xcite . the volume imaged is a ( virtual ) rectangular portion of 4 cm in the axial ( vertical ) direction with a width ( in the tangential direction ) of 1 cm and a length of 7 cm ( in the radial direction , starting from the central axis ) . velocity and concentration profiles @xmath32 and @xmath33 , averaged over the vertical and tangential directions in this volume , are obtained with a resolution of 270 @xmath17 m in the radial direction . this volume is situated at the magnet center ( so as to minimize the effects of field heterogeneities ) and sufficiently far from the bottom and the free surface of the rheometer so that flow perturbations due to edge effects are negligible . we checked that the velocity and concentration profiles are homogeneous along the vertical direction in this volume , which justifies averaging data over this direction . details on the sequence used to obtain velocity profiles can be found in [ @xcite ] . while it is possible to get 2d or 3d maps of 2d or 3d velocity vectors [ @xcite ] , such measurements may actually take minutes and require complex synchronization of the mri sequences and of the geometry position . however , we will show in the following that the azimuthal velocity alone provides a valuable information that can be sufficient for most analyses ; in particular it allows computation of the @xmath0-averaged strain rate component @xmath34 and thus the @xmath0-averaged shear stress @xmath35 ( and the torque @xmath7 ) in the case of the newtonian oil . that is why we have chosen to limit ourselves to 1d profiles of 1d velocity measurements , namely the azimuthal velocity @xmath36 as a function of the radius @xmath37 and time @xmath38 , for which a single measurement may take as little as 1 s ; this has allowed us to perform a sufficient number of experiments , with various materials , geometries , and rotational velocities . depending on the time over which this measurement is averaged as compared to @xmath39 where @xmath40 is the number of blades and @xmath41 is the rotational velocity , this measurement may provide either a time ( or @xmath42)-averaged azimuthal velocity @xmath43 or an instantaneous ( transient ) azimuthal velocity @xmath36 . in this latter case , the @xmath42 dependence of the azimuthal velocity @xmath44 at a given radius @xmath37 can then be easily reconstructed by simply replacing the time @xmath38 dependence by an angular @xmath42 dependence with @xmath45 . it should also be noted that due to incompressibility of the materials we study , the @xmath46 field can be reconstructed thanks to @xmath47 with @xmath48 ; however , this derivation from the experimentally measured values of @xmath44 can not be very accurate . finally , from @xmath46 and @xmath44 , we are also able to evaluate the strain rate components @xmath49 and @xmath50 $ ] . note that the derivative @xmath51 with respect to coordinate @xmath52 of experimental data @xmath53 measured at regularly spaced positions @xmath54 was computed as : @xmath55/[x_{i+1}-x_{i-1}]$ ] . the nmr sequence used in this work to measure the local bead concentration is a modified version of the sequence aiming at measuring velocity profiles along one diameter in couette geometry [ @xcite ] , and is described in full detail in @xcite . the basic idea is that during measurements , only nmr signal originating from those hydrogen nuclei belonging to the liquid phase of the sample ( _ i.e. _ both the oil and water phase of the emulsions ) is recorded : the local nmr signal that is measured is thus proportional to @xmath56 , where @xmath57 is the local particle volume fraction . a rather low absolute uncertainty of @xmath58 on the concentration measurements values was estimated in @xcite . all volume fraction profiles are measured at rest , after a given flow history . this is possible because the particles do not settle in the yield stress fluid at rest : the volume fraction profile induced by shear is gelled by the interstitial yield stress fluid . in this section , we study successively the flow properties of the newtonian oil , the yield stress fluid , and the suspension . in this section , we study the flows observed with a newtonian fluid . we first present a basic theoretical analysis of the flows in a vane - in - cup geometry as compared to flows in a standard couette geometry , which provides the basis for a couette analogy . the @xmath42-averaged azimuthal profiles @xmath59 are then shown , and are compared to predictions of the couette analogy . the full velocity field @xmath60 , @xmath46 is finally presented and analyzed . the stress balance equation projected along the azimuthal axis is : ( 1/r)_r(r^2)+_()-_p=0 [ equation_balance]where @xmath61 is the deviatoric stress tensor and @xmath62 the pressure . the strain rate tensor component @xmath1 is given by : ( r,)=((1/r)_(v_r)+r_r ( v_/r))[equation_strain_rate ] we recall that the constitutive law of a newtonian fluid of viscosity @xmath63 is : = 2[equation_newtonian ] . in all the following analysis , we assume a no - slip boundary condition at the walls of the inner tool and of the cup . in a standard coaxial cylinders couette geometry , due to cylindrical symmetry , eq . [ equation_balance ] becomes @xmath64 which means that the whole shear stress distribution @xmath65 in the gap is known whatever the constitutive law of the material is . if a torque @xmath66 is exerted on the inner cylinder driven at a rotational velocity @xmath41 , @xmath65 is given by : ( r)=-[equation_tau_couette]for a newtonian fluid of viscosity @xmath63 , it follows that the strain rate component @xmath67 is given by : ( r)=-as eq . [ equation_strain_rate ] becomes @xmath68 with cylindrical symmetry , due to the boundary conditions @xmath69 and @xmath70 , from @xmath71 , one gets alternatively @xmath72 $ ] . this yields the following azimuthal velocity profile : v_(r)=[equation_velocity_couette]finally , the viscosity @xmath63 of a newtonian fluid is obtained from the measured torque / rotational velocity relationship @xmath66 through = [ equation_visco_couette ] these equations will be used for the comparison with the flows observed in a vane - in - cup geometry , in particular to determine the radius @xmath3 of the equivalent couette geometry . in a vane - in - cup geometry , there is _ a priori _ no cylindrical symmetry and all quantities _ a priori _ depend on @xmath0 . however , averaging eq . [ equation_balance ] over @xmath42 yields @xmath73 . this means that what is true in a couette geometry , @xmath74 , is still true on average with a vane - in - cup geometry : @xmath75 independently of the material s constitutive law . note that this derivation is true only between @xmath10 and @xmath76 ; this is not true for the material between the blades as the unknown @xmath61 distribution in the blades contributes to the @xmath0-average . the link between this stress distribution and the torque @xmath66 exerted on the vane tool may then seem difficult to build . however , it can be equivalently computed on the outer cylinder as @xmath77 . this means that eq . [ equation_tau_couette ] is still valid for the @xmath42-averaged shear stress in the vane - in - cup geometry , for @xmath78 : _ r(r)=-[equation_tau_vane ] from the @xmath0-averaged eq . [ equation_newtonian ] , this means that the @xmath0-averaged strain rate distribution in a newtonian fluid , for @xmath78 , is : \|d_r(r)=-from the @xmath42-averaged eq . [ equation_strain_rate ] @xmath79 , this means that the @xmath0-averaged azimuthal velocity profile of a newtonian fluid of viscosity @xmath63 in a vane - in - cup geometry for @xmath78 , with a boundary condition @xmath80 is given by : ( r)=[equation_velocity_vane ] finally , the only difference with a standard couette flow , as regards these @xmath42-averaged quantities , is that we do not know the value of @xmath81 ; we only know that @xmath82 , for @xmath83 integer , where @xmath2 is the number of blades . this means that @xmath84 and @xmath59 follow the same scaling with @xmath37 and @xmath41 as in the standard couette geometry , but with a different prefactor . nevertheless , these equations provide a new insight in the couette analogy . the usual way of performing the couette analogy consists in defining the radius of the equivalent couette geometry @xmath3 as the radius that allows measuring the viscosity @xmath63 of a newtonian fluid with the standard couette formula . from eq . [ equation_visco_couette ] , @xmath63 should then be correctly obtained from the torque / rotational velocity relationship @xmath66 measured in a vane - in - cup geometry with : = [ equation_visco_vane]from eqs . [ equation_visco_couette ] and [ equation_visco_vane ] , it means in particular that the torque @xmath85 exerted by the vane tool is decreased by a factor = [ equation_torque_rieq]as compared to the torque @xmath86 exerted by the inner cylinder of a couette geometry of same radius @xmath10 at a same rotational velocity . here , from eqs . [ equation_velocity_vane ] and [ equation_velocity_couette ] , we see that from the local flow perspective , there is a couette analogy in the sense that the @xmath0-averaged azimuthal velocity ( and shear ) profiles will be exactly the same as in a couette geometry . this defines a radius @xmath3 of the equivalent couette geometry , such that @xmath59 and @xmath84 for @xmath78 are given by : ( r)=[equation_velocity_vane_eq ] + @xmath3 are equivalent : combining eqs . [ equation_velocity_vane ] and [ equation_visco_vane ] yields eq . [ equation_velocity_vane_eq ] . this point of view provides an additional meaning to the couette analogy , namely the similarity of the average flows , and offers a new experimental mean to determine @xmath3 , which is more accurate than calibration . in rheological measurements , the @xmath66 relationship has to be corrected for end effects [ @xcite ] and the couette analogy has to be calibrated on a reference material of known viscosity . here , the @xmath59 or @xmath87 measurements provide the value of @xmath3 directly without any correction , as only shear in the gap is involved in the analysis , and independent of the viscosity of the material . this will be illustrated in the following . the only theoretical prediction of the stress field associated with a vane tool is that of @xcite for an infinite @xmath2-bladed vane embedded in an infinite linear medium . in this case , it is shown that the torque @xmath88 exerted on the vane is well approximated by = 1-[equation_prediction_atkinson ] where @xmath89 is the torque exerted on a cylinder of same radius @xmath10 as the vane in an infinite medium ( _ i.e. _ with @xmath90 ) . [ equation_prediction_atkinson ] is in agreement with experimental results [ @xcite ] . @xcite argue that , as the stress distribution varies as @xmath91 in a couette geometry , @xmath92 should be of the order of 1% or less in order to nullify the influence of the outer boundary ; this is clearly the case in their experiments and in field experiments where the vane is embedded e.g. in a soil ; this is clearly not the case in our experiments and in most rheological experiments that make use of a vane - in - cup geometry . however , when the cup to vane radius ratio @xmath93 is not large , no generic theoretical expression exists in the literature . nevertheless , bounds of the value of the torque @xmath94 can be derived using classical results of linear elasticity [ @xcite ] . our starting points are the variational approaches to the solution of the stokes equations describing the flow of an incompressible linear material induced by the rotation of an inner tool ( of any shape ) at a rotational velocity @xmath41 within a cup . in this framework , it can be shown that [ @xcite ] : @xmath95 where @xmath96 denotes the inner tool - fluid interface and @xmath97 the domain occupied by the fluid . in the first inequality , @xmath98 is any stress tensor complying with the stress balance equations , @xmath99 is the azimuthal component of the surface forces applied by the tool on the fluid and @xmath100 is the deviatoric stress tensor associated to @xmath98 . in the second inequality , @xmath101 is the strain rate tensor associated with any velocity field @xmath102 complying with the incompressibility constraint and the boundary conditions prescribed on the tool - fluid and cup - fluid interfaces . [ eq : bounds ] leads in particular to the expected inequalities : @xmath103 the lower bound is obtained by using the velocity field defined by @xmath104 for @xmath105 and @xmath106 for @xmath107 , where @xmath108 is the solution for the @xmath2-bladed vane of radius @xmath10 in a cup of radius @xmath76 . @xmath102 complies with the boundary conditions for the @xmath2-bladed vane of radius @xmath10 in an infinite domain problem . then , putting this test velocity field within the second inequality ( [ eq : bounds ] ) with @xmath109 and using @xmath110 yields the lower bound of the inequality ( [ eq : expectedbounds ] ) for the quantity @xmath111 . in eq . [ eq : clapeyron ] , @xmath112 denotes the strain rate tensor associated with @xmath113 while @xmath114 is the domain occupied by the fluid . the upper bound is obtained using the test velocity field defined by @xmath115 with @xmath116 defined by eq . [ equation_velocity_couette ] for @xmath117 and by @xmath118 for @xmath119 . it is easily checked that @xmath102 complies with the velocity boundary conditions for any vane - in - cup geometry with vane radius @xmath10 and cup radius @xmath76 . putting this test velocity field into the second inequality ( [ eq : bounds ] ) then yields the upper bound of inequality ( [ eq : expectedbounds ] ) . finally , combining inequalities ( [ eq : expectedbounds ] ) , eq . [ equation_prediction_atkinson ] and eq . [ equation_visco_couette ] yields @xmath120 of course , it is possible to improve the lower bound by determining admissible test stress fields for the problem under consideration and the inequalities ( [ eq : bounds ] ) . for example , let us consider the stress field defined between the two blades positioned at @xmath121 by @xmath122 with @xmath123 for @xmath119 and by @xmath124 for @xmath117 . this stress field complies with the balance equations within the fluid domain . let us recall that a stress field does not need to be continuous to comply with the balance equations ( of course , this stress field is not the solution of the problem ) . putting this stress field into the first inequality ( [ eq : bounds ] ) and using a numerical optimization tool to choose the optimal value of the parameter @xmath125 yields a new lower bound for the @xmath2-bladed vane in cup problem , which depends on @xmath2 and @xmath126 . in some cases , this test stress field improves the lower bound of inequality ( [ eq:17dego ] ) : e.g. , for the geometry we use in this study ( @xmath127 ) , the new lower bound is @xmath128 while the lower bound given by inequality ( [ eq:17dego ] ) is @xmath129 . nevertheless , such an improvement is not obtained for all parameter sets ( @xmath2,@xmath126 ) . it is thus necessary to compute the two lower bounds for each value of ( @xmath2,@xmath126 ) in order to obtain the more accurate lower estimate of the torque . although application of variational approaches to the derivation of estimates of the applied torque of a vane - in - cup problem is not classical , it is believed that such a strategy is able to provide useful results when no theoretical prediction of the solution is available for particular geometries . lower bounds of @xmath130 computed using the approach presented above are displayed in tab . [ tab_torque_reduction ] and are compared below to our results and to data in the literature . we first study the @xmath0-averaged azimuthal velocity profiles @xmath59 observed during the flows of a newtonian oil ( fig . [ figure_mean_velocity_vane ] ) . as shown above , these profiles can be used to check the validity of the couette analogy and to determine the couette equivalent radius @xmath3 . the azimuthal dependence of the velocity profiles between two adjacent blades of the vane tool will then be considered . of a newtonian oil in a couette geometry ( @xmath31 cm ) , at various rotational velocities @xmath41 ranging from 2 to 20 rpm ; the solid line is the theoretical profile for a newtonian fluid . b ) dimensionless @xmath0-averaged velocity profile @xmath131 of a newtonian oil in a six - bladed vane - in - cup geometry ( @xmath28 cm ) for @xmath41 ranging from 1 to 9 rpm ; the vertical dashed line shows the radius of the vane ; the dotted line is the theoretical profile for a rigid body rotation ( for @xmath132 ) ; the solid lines are the theoretical profiles for a newtonian fluid in couette geometries of radii , from right to left : ( i ) @xmath28 cm , ( ii ) @xmath133 cm , and ( iii ) @xmath134 cm corresponding to the @xcite theory in an infinite medium.,title="fig:",width=298 ] of a newtonian oil in a couette geometry ( @xmath31 cm ) , at various rotational velocities @xmath41 ranging from 2 to 20 rpm ; the solid line is the theoretical profile for a newtonian fluid . b ) dimensionless @xmath0-averaged velocity profile @xmath131 of a newtonian oil in a six - bladed vane - in - cup geometry ( @xmath28 cm ) for @xmath41 ranging from 1 to 9 rpm ; the vertical dashed line shows the radius of the vane ; the dotted line is the theoretical profile for a rigid body rotation ( for @xmath132 ) ; the solid lines are the theoretical profiles for a newtonian fluid in couette geometries of radii , from right to left : ( i ) @xmath28 cm , ( ii ) @xmath133 cm , and ( iii ) @xmath134 cm corresponding to the @xcite theory in an infinite medium.,title="fig:",width=298 ] in fig . [ figure_mean_velocity_vane]a we observe that the velocity profiles in the gap of a couette geometry are , as expected , in perfect agreement with the theory for a newtonian flow ( eq . [ equation_velocity_couette ] ) . this first observation can be seen as a validation of the measurement technique . in the vane - in - cup geometry ( fig . [ figure_mean_velocity_vane]b ) , we first note that the @xmath0-averaged dimensionless azimuthal velocity profiles @xmath131 measured for several rotational velocities @xmath41 are superposed , as expected from the linear behavior of the material . we also remark that the material between the blades rotates as a rigid body only up to @xmath135 cm , indicating that the shear flow enters deeply the region between the blades ( the vane radius is 4.02 cm ) . the whole limit between the sheared and the unsheared material in the ( @xmath136 ) plane will be determined in sec . [ section_results][section_oil][section_theta_profiles ] ( fig . [ figure_temporal_profiles]b ) . we finally observe that the theoretical velocity profile for a newtonian fluid in a couette geometry of radius equal to that of the vane lies above the data , as expected from the literature . this is also consistent with the observation that the shear flow enters the region between the vane blades . in order to test the couette analogy , we have chosen to plot the @xmath0-averaged strain rate @xmath137 vs. the radius @xmath37 in fig . [ figure_mean_gradient_vane ] . this allows us to distinguish more clearly the difference between the experimental and theoretical flow properties than would the velocity profiles , because the velocity profile always tends to the same limit ( @xmath138 ) at the outer cylinder whereas the strain rate profile does not . note that velocity measurements could not be performed close to the blades , which explains why strain rate data are missing from 4 to 4.2 cm . -averaged strain rate @xmath137 vs. radius @xmath37 for a newtonian oil sheared at 1 rpm in a six - bladed vane - in - cup geometry . the vertical dashed line shows the radius of the vane . the solid lines are the theoretical strain rate profiles for a newtonian fluid in couette geometries of radii : ( i ) @xmath28 cm ( light grey ) , ( ii ) @xmath133 cm ( black ) , and ( iii ) @xmath134 cm ( dark grey ) corresponding to the @xcite theory in a infinite medium.,width=340 ] in fig . [ figure_mean_gradient_vane ] , we first note that @xmath139 is zero up to @xmath140 cm , which corresponds to the limit of the rigid motion of the material ; @xmath137 then increases when @xmath37 tends towards @xmath10 as the material is more and more sheared between the blades . in the gap of the geometry , @xmath139 decreases when @xmath37 increases . as expected , the theoretical strain rate profile for a newtonian material in a couette geometry of radius equal to that of the vane falls well above the data _ at any radius @xmath37 _ ( this was less obvious on the velocity profiles ) . we then observe that the data are well fitted to the theoretical strain rate profile ( eq . [ equation_strainrate_vane ] ) for a newtonian material flowing in an equivalent couette geometry of inner cylinder radius @xmath133 cm ( @xmath141 cm was obtained from a fit of the velocity profile to eq . [ equation_velocity_vane_eq ] ) . this confirms that the @xmath0-averaged strain rate @xmath137 decreases as the inverse squared radius in the gap , in agreement with the couette analogy . from eq . [ equation_torque_rieq ] , we find @xmath142 ( let us recall that we do not need to consider end effects here because we determine the shear rate within the gap , and hence only the contribution to the torque from the material sheared in the gap ) . this value can now be compared to data from the literature . for a six - bladed vane tool in an infinite medium , the @xcite theory would imply a theoretical @xmath143 , which is 8% lower than what we measure , and corresponds to a theoretical `` equivalent radius '' @xmath134 cm when the vane is embedded in a cup of radius @xmath29 cm . [ figure_mean_velocity_vane]b and [ figure_mean_gradient_vane ] show that the flow characteristics predicted with this value of the equivalent radius can be distinguished from our experimental data and fall slightly below the data ( a discrepancy could be expected as @xmath92 is not small in our experiment ) . .ratio @xmath130 between the torque measured when straining a linear medium ( viscous or elastic ) in a vane - in - cup geometry and that measured in a coaxial cylinder geometry of similar dimensions , obtained in various theoretical , numerical and experimental studies of the literature ; only data corrected for ( or free from ) end effects are shown . the number @xmath2 of blades , the cup to vane radius ratio @xmath93 , the shaft radius to vane radius ratio @xmath144 , and the blade thickness to vane radius ratio @xmath145 , are displayed when provided in the manuscripts . the theoretical lower bound computed using variational approaches in sec . [ section_results][section_oil][section_analogy ] is also provided . [ cols="^,^,^,^,^,^,^ " , ] we have gathered experimental and numerical data from the literature where the cup to vane radius ratio @xmath93 is not large in tab . [ tab_torque_reduction ] ; only data corrected for ( or free from ) end effects are shown . first , it should be noted that all torque data obey the theoretical inequalities computed using variational approaches in sec . [ section_results][section_oil][section_analogy ] . however , no clear trends emerge from the comparison of the data . the relative impact of the various geometrical parameters that may affect the flow field , namely the cup to vane radius ratio @xmath93 , the shaft radius to vane radius ratio @xmath144 , and the blade thickness to vane radius ratio @xmath145 , can not be determined at this stage . for example , in very similar geometries , @xcite find a torque ratio @xmath130 close to that of @xcite whereas @xcite find a much lower torque ratio . the only noticeable difference between these two studies ( apart from the numerical method ) is that @xmath144 is higher in @xcite , but our data , with a rather large value of @xmath144 , show different features . we actually note that our study is the only one to report a torque ratio higher than in an infinite medium ; all other data report torque ratios up to 19% lower than expected in an infinite medium . in the general case of a finite vane - in - cup geometry , it thus seems that numerical investigations are still needed , and that , at this stage , a calibration has to be performed to get the geometry factors . we also expect that the bounds obtained using variational approaches in sec . [ section_results][section_oil][section_analogy ] can be improved . to better characterize the flow field , we now study the dependence of the velocity profiles on the angular position @xmath0 . we have performed experiments in which we measure one azimuthal velocity profile per second while the vane tool is rotated at 1 rpm , yielding 10 profiles between two adjacent blades . of a newtonian oil sheared at 1 rpm in a six - bladed vane - in - cup geometry , for various angular positions , @xmath0 , between one blade ( @xmath146 ) and midway between adjacent blades ( @xmath147 ) . the vertical dashed line shows the radius of the vane . the dotted line is the profile for a rigid body rotation ( for @xmath132 ) and the theoretical profile for a newtonian fluid in a couette geometry of radius @xmath10 ( for @xmath148 ) . b ) two - dimensional plot of the limit between rigid motion and shear ( empty circles ) for a newtonian material in the six - bladed vane - in - cup geometry ; the grey rectangles correspond to the blades.,title="fig:",width=328 ] of a newtonian oil sheared at 1 rpm in a six - bladed vane - in - cup geometry , for various angular positions , @xmath0 , between one blade ( @xmath146 ) and midway between adjacent blades ( @xmath147 ) . the vertical dashed line shows the radius of the vane . the dotted line is the profile for a rigid body rotation ( for @xmath132 ) and the theoretical profile for a newtonian fluid in a couette geometry of radius @xmath10 ( for @xmath148 ) . b ) two - dimensional plot of the limit between rigid motion and shear ( empty circles ) for a newtonian material in the six - bladed vane - in - cup geometry ; the grey rectangles correspond to the blades.,title="fig:",width=268 ] in fig . [ figure_temporal_profiles]a , we plot the velocity profiles @xmath149 measured at different angles @xmath0 . we first observe that the velocity profile which starts near from a blade tip ( corresponding to @xmath146by definition ) is very different from the velocity profile in a couette geometry of same radius : it starts with a much steeper slope , which means that the blades tip neighborhoods are regions of important shear as already observed by @xcite . we then observe that , as expected from the @xmath0-averaged velocity profiles , the shear flow enters more and more deeply the region between the blades as @xmath0 tends towards @xmath150(corresponding to midway between two adjacent blades ) ; at this angular position , the rigid rotation stops at @xmath151 cm . from all the velocity profiles , we finally extract a 2d map of the limit @xmath152 between rigid rotation and shear , which is depicted in fig . [ figure_temporal_profiles]b . this provides an idea of the deviation from cylindrical symmetry , and will be compared in the following to the case of yield stress fluids . note that eddies are likely to be present in the `` rigid '' region [ @xcite ] ; however , we did not observe any signature of their existence : they can thus be considered as second - order phenomena . of a newtonian oil sheared at 1 rpm in a six - bladed vane - in - cup geometry , for various angular positions @xmath0 between two adjacent blades ( from @xmath146to @xmath153).,width=340 ] as explained in sec . [ section_display ] , from the @xmath149 measurement and from the material incompressibility , we are able to reconstruct the radial velocity profile @xmath154 ( see fig . [ figure_radial_velocity ] ) . this also allows us to compute the strain rate components @xmath155 and @xmath156 , which are plotted in fig . [ figure_temporal_gradients ] . of course , due to the limited number of profiles between two adjacent blades , this method provides only a rough estimate of these quantities . in addition to their interest for future comparison with models and simulations , these data allow us to evaluate the contribution of the extensional flow to dissipation ; here , in a newtonian medium , the local power density is given by : @xmath157 . ( left ) and @xmath158 ( right ) vs. radius @xmath37 for various angular positions @xmath0 between two adjacent blades ( from @xmath146to @xmath153).,title="fig:",width=274 ] ( left ) and @xmath158 ( right ) vs. radius @xmath37 for various angular positions @xmath0 between two adjacent blades ( from @xmath146to @xmath153).,title="fig:",width=323 ] in fig . [ figure_radial_velocity ] , we first observe that @xmath159 for @xmath146and @xmath147 ; there is thus no extensional flow in these regions of space , as seen in fig . [ figure_temporal_gradients ] . this is actually expected from the fore - aft symmetry of the flow around these angular positions . @xmath160 and its spatial variations ( @xmath161 ) are maximal at @xmath162 . meanwhile , we observe that @xmath1 is maximal near the blades : at @xmath163 it is more than 4 times larger at @xmath146than at @xmath147 . we then find that @xmath1 ( and thus the shear stress @xmath164 ) decreases more rapidly from the blades ( at @xmath146 ) than the @xmath91 scaling of the couette geometry , whereas it does not vary much with @xmath37 midway between adjacent blades ( it even seems to slightly increase with @xmath37 as already observed in simulations by @xcite ) . it is also worth nothing that at @xmath165 , in contrast with what is observed at @xmath163 , the shear stress value is of order two times lower at @xmath146than at @xmath147 . from the whole set of @xmath1 and @xmath161 measurements ( fig . [ figure_temporal_gradients ] ) , we finally find that in regions where @xmath161 is maximal , the contribution of the extensional flow to dissipation is of order 25% . over the whole gap , we then evaluate its average contribution to dissipation to be rather important , of order 5 to 10% . this significant value may be a reason why the torque that has to be exerted to enforce flow is higher than that predicted by @xcite in an infinite medium . the confinement effect induced by a close boundary at a radius @xmath166 likely increases the contribution of the extensional flow to dissipation as compared to the case of an infinite medium ( although other effects may exist , as appears from the comparison of the data of tab . [ tab_torque_reduction ] ) . in this section , we study the flows induced by the vane tool with a yield stress fluid ( a concentrated emulsion ) . we focus on the behavior near the yielding transition , _ i.e. _ on low rotational velocities @xmath41 . -averaged velocity profile @xmath131 of a yield stress fluid ( concentrated emulsion ) in a six - bladed vane - in - cup geometry for @xmath41 ranging from 0.1 to 9 rpm ; the vertical dashed line shows the radius of the vane ; the dotted line is the theoretical profile for a rigid body rotation ( for @xmath132).,width=340 ] in fig . [ figure_average_velocity_yield ] , we plot the @xmath0-averaged azimuthal velocity profiles @xmath59 measured at several @xmath41 values ranging from 0.1 to 9 rpm , corresponding to macroscopic shear rates varying between 0.02 and 2 s@xmath167 . we first observe that flow is localized : the material is sheared only up to a radius @xmath168 . @xmath169 is found to increase as @xmath41 increases . this is a classical feature of flows of yield stress fluids in heterogeneous stress fields . it has been observed in couette geometries [ @xcite ] , where it is attributed to the @xmath91 decrease of the shear stress @xmath164 , which passes below @xmath6 at some @xmath170 at low @xmath41 . in this case , when @xmath41 tends to 0 , @xmath169 tends to @xmath10 and the torque @xmath7 at the inner cylinder tends to @xmath171 . in the vane - in - cup geometry , the same argument holds qualitatively thanks to eq . [ equation_tau_vane ] . it implies that the flow has to stop inside the gap at low @xmath41 . however , in contrast with the case of the couette geometry , as the whole stress field _ a priori _ depends on @xmath0 , this @xmath0-averaged equation does not provide the position of the limit between the sheared and the unsheared material ( which will determined at the end of this section ) . we then observe that , although this effect is less pronounced than with a newtonian material , the shear flow still enters the region between the blades , even at the lowest studied @xmath41 . close examination of the profiles shows that the material trapped between the blades rotates as a rigid body only up to @xmath172 cm at @xmath173 rpm , @xmath174 cm at @xmath175 rpm , and @xmath176 cm at @xmath177 rpm . we recall that @xmath151 cm with a newtonian fluid in the same geometry . of a yield stress fluid ( concentrated emulsion ) sheared at 0.1 rpm in a six - bladed vane - in - cup geometry , for various angular positions @xmath0 between two adjacent blades ( from @xmath146to @xmath153 ) . the vertical dashed line shows the radius of the vane.,width=340 ] as in sec . [ section_results][section_oil][section_theta_profiles ] , to better characterize the flow field , we have performed experiments in which we have measured 10 azimuthal profiles between two adjacent blades at 0.1 rpm . in fig . [ figure_temporal_velocity_yield ] , as for a newtonian fluid , we observe that there is a strong @xmath0-dependence of the velocity profiles . the velocity profile that starts near from a blade tip ( at @xmath146 ) has a much steeper slope than the profile measured midway between adjacent blades ( at @xmath147 ) ; again , this shows that the blade tip neighborhoods are regions of high shear . meanwhile the flow stops at a radius @xmath169 which is larger at @xmath147(4.5 cm ) than at @xmath146(4.3 cm ) . note also that there may be slight fore - aft asymmetry , as sometimes observed with yield stress fluids flows [ @xcite ] , but we did not study this point further . from these velocity profiles , we have reconstructed a 2d map of the flow field ( fig . [ fig_rigid_limit_yield ] ) , indicating both the boundary between the region of rigid body rotation ( between the blades ) and the sheared region , and the boundary between the sheared region and the outer region of fluid at rest ( _ i.e. _ the position where the yield criterion is satisfied ) . flow is found to occur in a layer of complex shape which is far from being cylindrical even at this very low velocity . these observations are in contradiction with the usually accepted picture for yield stress fluid flows at low rates [ @xcite ] , namely that the material contained in the virtual cylinder delimited by the blades rotates as a rigid body , and that it flows uniformly in a thin cylindrical layer near the blades . our results contrast in particular with previous numerical works which showed that the yield surface is cylindrical at low rates for bingham fluids , casson fluids , and herschel - bulkley materials with @xmath178 [ @xcite ] . with apparently similar conditions to those in some of the @xcite simulations , we find an important departure from cylindrical symmetry . this means that further investigation on the exact conditions under which this symmetry can be recovered is still needed . possible difference between our work and that of @xcite is that the blade thickness is zero in this last study . it is particularly striking and counterintuitive that @xmath169 is largest at the angular position ( @xmath147 ) where shear at @xmath10 is smallest ( similar observation was made by @xcite ) . as in sec . [ section_results][section_oil][section_theta_profiles ] , this points out the importance of the extensional flow in this geometry , with @xmath0-dependent normal stress differences which have to be taken into account in the yield criterion , and which thus impact the yield surface . it thus seems that the link between the yield stress @xmath6 and the torque @xmath7 measured at yield with a vane - in - cup geometry is still an open question , although the classical formula probably provides a sufficiently accurate determination of @xmath6 in practice . the same 2d map as above is plotted for @xmath175 rpm in fig . [ fig_rigid_limit_yield ] ; the same phenomena are observed , with enhanced departure from cylindrical symmetry , consistent with the observation that @xmath179 decreases when @xmath41 increases . this result was also unexpected , as simulations find uniform flows for shear - thinning material of index @xmath180 [ @xcite ] ; we would have expected the same phenomenology in a herschel - bulkley material of index @xmath178 ( and thus @xmath179 to tend to @xmath10 when increasing @xmath41 ) . this observation also shows that a couette analogy can hardly be defined for studying the flow properties of such materials in a vane - in - cup geometry because the equivalent couette geometry radius @xmath3 would probably depend also on @xmath41 ( as recently shown by @xcite ) . let us finally note that this departure from cylindrical symmetry has important impact on the migration of particles in a yield stress fluid ( see below ) . in this section , we investigate the behavior of a concentrated suspension of noncolloidal particles in a yield stress fluid ( at a 40% volume fraction ) . a detailed study of their velocity profiles would _ a priori _ present here limited interest : such materials present the same nonlinear macroscopic behavior as the interstitial yield stress fluid , and their rheological properties ( yield stress , consistency ) depend moderately on the particle volume fraction [ @xcite ] . on the other hand , noncolloidal particles in suspensions are known to be prone to shear - induced migration , which leads to volume fraction heterogeneities . this phenomenon is well documented in the case of suspensions in newtonian fluids [ @xcite ] but is still badly known in yield stress fluids ( some studies exist however in viscoelastic fluids [ @xcite ] ) . in the model of @xcite and @xcite , migration is related to shear - induced diffusion of the particles [ @xcite ] . in a wide gap couette geometry , the shear stress heterogeneity is important ( eq . [ equation_tau_couette ] ) ; the shear rate gradients then generate a particle flux towards the low shear zones , which is counterbalanced by a particle flux due to viscosity gradients . a steady state , which results from competition between both fluxes , may then be reached , and is characterized by an excess of particles in the low shear zones of the flow geometry ( near the outer cylinder in a wide - gap couette geometry [ @xcite ] ) . note that there are other models [ @xcite ] in which particle fluxes counterbalance the gradients in the particle normal stresses , and which can be used directly for non - newtonian media . as the development of migration depends on the spatial variations of shear , one may wonder how the azimuthal heterogeneities of shear introduced by the vane tool affect migration ; a related question is that of the relevance of the couette analogy for this phenomenon . in the following , we thus focus on the particle volume fraction distribution evolution when the material is sheared . we first study the behavior at high shear rate , in the absence of shear localization . we shear the suspension in both the standard couette geometry and the vane - in - cup geometry at a rotational velocity @xmath181 rpm . in this first set of experiments , we only study the steady - state of migration . at @xmath181 rpm , this steady - state is reached in less than 30 min ( which corresponds to a macroscopic strain of order 50000 , consistently with strainscale evaluations from data of the literature [ @xcite ] ) . in fig . [ fig_concentration_profile_omega100 ] we plot the steady state volume fraction profiles observed after shearing the suspension at @xmath181 rpm during 1h . rpm in both the couette geometry ( empty circles ) and the vane - in - cup geometry ( squares ) . in the vane - in - cup geometry , the volume fraction profile is determined in a 1 cm thick slice situated exactly between two adjacent blades ( see fig . [ fig_image_vane_omega100]b ) . the inset is a zoom ; the line is a fit of the data measured in the couette geometry to the @xcite model with @xmath182 . b ) @xmath0-averaged azimuthal velocity profile @xmath59 ; the dotted line is the theoretical rigid motion induced by the rotation of the vane tool ; the vertical dashed line shows the radius of the vane.,title="fig:",width=308 ] rpm in both the couette geometry ( empty circles ) and the vane - in - cup geometry ( squares ) . in the vane - in - cup geometry , the volume fraction profile is determined in a 1 cm thick slice situated exactly between two adjacent blades ( see fig . [ fig_image_vane_omega100]b ) . the inset is a zoom ; the line is a fit of the data measured in the couette geometry to the @xcite model with @xmath182 . b ) @xmath0-averaged azimuthal velocity profile @xmath59 ; the dotted line is the theoretical rigid motion induced by the rotation of the vane tool ; the vertical dashed line shows the radius of the vane.,title="fig:",width=289 ] as expected , we first observe that the material is strongly heterogeneous in the couette geometry : the volume fraction varies between 37% near the inner cylinder and 43% near the outer cylinder ( note that the nmr technique we use do not allow quantitative measurements near the walls ) . this heterogeneity is quantitatively similar to that observed in couette flows of newtonian suspensions at a same 40% particle volume fraction [ @xcite ] ; the profiles are actually well fitted to the @xcite model ( see eq . 16 of @xcite ) with a dimensionless diffusion constant @xmath182 which is close to that found by @xcite ( @xmath183 ) , although this model is not expected to hold in non - newtonian suspensions . in the vane - in - cup geometry , the volume fraction profile shows very different features ; note that the profile is measured in a 1 cm thick ( in the azimuthal direction ) slice situated exactly between two adjacent blades ( see fig . [ fig_image_vane_omega100 ] ) . in fig . [ fig_concentration_profile_omega100 ] , we first observe that there is a strong particle depletion in a wide zone between the blades . a homogeneous volume fraction of 40% is observed for radii inferior to 3.1 cm . at a radius @xmath184 cm , there is a strong drop in the volume fraction down to 5% within 1 mm ( corresponding to 4 particle diameters ) . close inspection of the velocity profile fig . [ fig_concentration_profile_omega100 ] shows that this radius @xmath179 corresponds to the transition between rigid motion and shear between the blades . the volume fraction then increases basically linearly with the radius up to a 40.5% volume fraction at a radius @xmath185 cm which is close to the vane radius . the volume fraction then increases only slightly ( between 40.5% and 42.5% ) in the gap of the geometry : the heterogeneity is here much less important than in a standard couette geometry . to get further insight into the new strong depletion phenomenon we have evidenced , we have performed 2d magnetic resonance images of the material . such images provide a qualitative view of the spatial variations of the particle volume fraction as only the liquid phase is imaged . images are coded in grey scales ; a brighter zone contains less particles . in fig . [ fig_image_vane_omega100]b , we first see an image of the homogeneous material . before any shear , as expected , the light intensity is homogeneous in the sample ( intensity variations correspond to noise ) . after a 1h shear at @xmath181 rpm , we observe very bright and thin curves on the , @xmath186 cm ) to the edge of another blade ( at @xmath153 , @xmath186 cm ) , and describes a concave @xmath187 curve whose minimum is @xmath188 cm at @xmath147 . note that as the volume fraction profile is averaged over a slice which is 1 cm thick in the azimuthal direction ( see fig . [ fig_image_vane_omega100]b ) , the fact that we measure a minimum of 5% at @xmath188 cm in the slice probably means that the volume fraction minimum is actually equal to zero in the depletion zone . rpm ( corresponding to a macroscopic strain of order 75000 ) , and ( b ) before any shear . the crosses in fig . [ fig_image_vane_omega100]a correspond to the limit between rigid motion and shear for the newtonian oil of fig . [ figure_temporal_profiles]b . the white rectangle in fig . [ fig_image_vane_omega100]b shows the slice in which the volume fraction profiles of figs . [ fig_concentration_profile_omega100]a and [ fig_concentration_profile_omega1 ] are measured . ( c ) is a zoom of image ( a ) near the edges of a blade . the images are taken in the horizontal plane of the geometry , at middle height of the vane tool , and correspond to vertical averages over 2 cm . the vane tool rotates counterclockwise.,title="fig:",width=453 ] + ( a ) + + rpm ( corresponding to a macroscopic strain of order 75000 ) , and ( b ) before any shear . the crosses in fig . [ fig_image_vane_omega100]a correspond to the limit between rigid motion and shear for the newtonian oil of fig . [ figure_temporal_profiles]b . the white rectangle in fig . [ fig_image_vane_omega100]b shows the slice in which the volume fraction profiles of figs . [ fig_concentration_profile_omega100]a and [ fig_concentration_profile_omega1 ] are measured . ( c ) is a zoom of image ( a ) near the edges of a blade . the images are taken in the horizontal plane of the geometry , at middle height of the vane tool , and correspond to vertical averages over 2 cm . the vane tool rotates counterclockwise.,title="fig:",width=222 ] rpm ( corresponding to a macroscopic strain of order 75000 ) , and ( b ) before any shear . the crosses in fig . [ fig_image_vane_omega100]a correspond to the limit between rigid motion and shear for the newtonian oil of fig . [ figure_temporal_profiles]b . the white rectangle in fig . [ fig_image_vane_omega100]b shows the slice in which the volume fraction profiles of figs . [ fig_concentration_profile_omega100]a and [ fig_concentration_profile_omega1 ] are measured . ( c ) is a zoom of image ( a ) near the edges of a blade . the images are taken in the horizontal plane of the geometry , at middle height of the vane tool , and correspond to vertical averages over 2 cm . the vane tool rotates counterclockwise.,title="fig:",width=223 ] + ( b)(c ) as pointed out above , this curve also likely marks the transition between the unsheared material ( which rotates as a rigid body ) and the sheared material . note in particular the similarity with fig . [ figure_temporal_profiles]b , the data of which are reported in fig . [ fig_image_vane_omega100]a for illustration . a first interpretation of the phenomenon would then simply be that migration is caused by shear and naturally stops at this transition zone . indeed , as shear is maximum near the blades , particles tends to migrate out of this zone ; moreover , there is no source of particle flux from the unsheared zone between the blades to balance the migration towards the outer cylinder . however , this does not explain why the volume fraction drops down to zero : heterogeneities observed at steady - state in the literature are usually moderate and do not lead to zones free of particles . a better understanding of the phenomenon can be gained by zooming on the previous image ( fig.[fig_image_vane_omega100]c ) . we now see that while the depletion phenomenon seems symmetric around both sides of the blades at a macroscopic scale , it is clearly asymmetric at a local scale near the blades and depends on the direction of rotation : depletion is more pronounced at the back of the blade ( note that the vane tool rotates counterclockwise ) . this would mean that the noncolloidal particle trajectories are asymmetric around the blade : a particle that is found at a radius @xmath189 cm just before the blade is necessarily found at a radius slightly higher than 4.02 cm after the blade as there are no particles at @xmath186 cm . this feature is reminiscent of the fore - aft asymmetry that is observed in the bulk of noncolloidal suspensions [ @xcite ] and that leads to their non - newtonian properties [ @xcite ] . it thus seems that , in addition to the shear - induced migration mechanism intrinsic to suspensions , the vane tool induces a specific migration mechanism which has its origin in the direct interactions between the particles and the blades ; this effects leads to the full depletion that is observed at the transition between the sheared and the unsheared material . such direct effect of a flow geometry on migration has also been observed in microchannel flows of colloidal suspensions [ @xcite ] , and also led to full particle depletion . see also @xcite . the kinetics of the phenomenon will be briefly discussed below . the rest of the volume fraction profile results from a complex interplay between shear - induced migration and the fore - aft asymmetry around the blades ; this leads to the rapid increase of the volume fraction between 3.1 cm and 4.02 cm . after 4.02 cm the flow lines do not meet the blade edges , and the phenomenon evidenced above should have basically no effect on the heterogeneity that develops in the gap of the geometry . on the other hand , the mean volume fraction should be slightly higher due to mass conservation ; it is indeed observed to be equal to around 42% . nevertheless , as the mean radial shear rate heterogeneity is basically similar to that observed in a standard couette geometry ( see previous sections ) , we would _ a priori _ expect the heterogeneity to be somehow similar . however , we observe that the volume fraction profile is only slightly heterogeneous : there is less than 5% variation of the volume fraction in the gap , to be compared to the 15% variation observed in the couette geometry . clearly , this means that the couette analogy is irrelevant as regards this phenomenon , and that the details of shear matter . here , the extensional flow that adds to shear may be at the origin of this diminution of migration . a more detailed analysis is out of the scope of this paper . let us now study the behavior at low shear rate . low shear rates are typically imposed with the aim of measuring the yield stress of such materials . starting from a homogeneous suspension at rest , we apply a rotational velocity @xmath175 rpm ( without any preshear ) , and we measure the evolution of the particle volume fraction in time . the corresponding volume fraction profiles are depicted in fig . [ fig_concentration_profile_omega1 ] . rpm measured in the vane - in - cup geometry after different times of shear : 5min , 1h , 14h . the material is homogeneous at the beginning of shear . the inset presents the @xmath0-averaged azimuthal velocity profile @xmath59 measured in the first stages of shear ; the dotted line is the theoretical rigid motion induced by the rotation of the vane tool ; the vertical dashed line shows the radius of the vane.,width=340 ] in fig . [ fig_concentration_profile_omega1 ] , we observe that , although shear is much less important than in the previous experiments , particle depletion also appears between the blades . comparison of the velocity profile and the volume fraction profile shows that depletion also appears between the blades at the transition zone between the sheared and the unsheared materials . this phenomenon appears with a very fast kinetics : the lower volume fraction value in the measurement zone is 36% after only a 5 minute shear ( corresponding to a macroscopic strain of order 50 ) . afterwards , it continues evolving slowly : the minimum observed volume fraction is of order 33% after a 1h shear and of order 32% after a 14h shear ( corresponding to a 10000 macroscopic strain ) . note that the radial position of the minimum value of the volume fraction slightly decreases in time ; it likely corresponds to progressive erosion of the material between the blades ( we did not measure the velocity profiles to check this hypothesis ) . we also note that migration is negligible in the rest of the sheared material as expected from the theory of migration briefly described above ( a larger strain would be needed to observe significant migration ) . nevertheless , we note some particle accumulation ( with a volume fraction value of 43% ) at @xmath169=4.7 cm after a very long time . this corresponds to the yield surface as flow is localized at low velocity ( see velocity profile fig . [ fig_concentration_profile_omega1 ] ) . migration profiles usually result from an equilibrium between various sources of fluxes . on the other hand , the unsheared material does not produce any particle flux while it receives particles from the sheared region . this particle accumulation is thus the signature that the migration phenomenon is indeed active , although not observable on the profile measured in the sheared zone . it is probable that this accumulation process would stop only ( after a very long time ) when there are no more particles in the sheared region . ( b ) is a zoom of image ( a ) between two adjacent blades . the image is taken in the horizontal plane of the geometry , at middle height of the vane tool , and corresponds to a vertical average over 2 cm . the vane tool rotates counterclockwise.,title="fig:",width=298 ] + ( a)(b ) as above , 2d magnetic resonance images of the material provide an insight in the phenomenon . in fig . [ fig_image_vane_omega1 ] , we observe again that particle depletion is enhanced at the rear of the blades ; this confirms that this phenomenon is likely due to direct interactions between the blades and the particles , leading to the asymmetry of the particles trajectory around the blades . this _ a priori _ occurs with any particle whose trajectory is close to the blades , explaining why particle depletion appears so rapidly . there is probably no way to avoid it . note that the images are here much brighter very close to the blades than midway between two adjacent blades ; this would mean that the particle volume fraction is probably close to @xmath190 near the blades , although we observe volume fraction of order 32% between two blades . + finally , let us note that the bright line provides a good idea of the boundary between the sheared material and the material that moves as a rigid body . we see as in sec . [ section_results][section_emulsion ] that this is far from being cylindrical even at this low velocity . we finally present some consequences of this phenomenon . from the above observations , our conclusion is that depletion sets up quickly and is probably unavoidable . then two situations have to be distinguished . if linear viscoelastic properties of a suspension of large particles are measured at rest on the homogeneous material in its solid regime , without any preshear , then these measurements pose no other problem than that of the relevant couette analogy to be used ( see sec . [ section_results][section_oil ] ) . if a yield stress measurement is performed at low imposed rotational velocity , starting from the homogeneous material at rest , then this measurement is likely valid as long as only the peak value or the plateau value at low strain ( of order 1 ) is recorded . on the other hand , any subsequent analysis of the material behavior will _ a priori _ be misleading : irreversible changes have occurred and the material can not be studied anymore . more generally , any measurement performed after a preshear will be incorrect and any flow curve measurement will lead to wrong evaluation of the material properties . in these last cases , the consequence of the new particle depletion phenomenon we have evidenced is a kind of wall slip near the blades , whereas there are no walls . here the `` slip layer '' is made of the ( pure ) interstitial yield stress fluid in a zone close to the blades , as would be observed near a smooth inner cylinder . this contrasts with the common belief that the vane tool prevents slippage . in order to illustrate this feature , we present some results of @xcite : mahaut _ et al . _ performed classical upward / downward shear rate sweeps with a six - bladed vane - in - cup geometry ( @xmath191 cm , @xmath192 cm , @xmath304.5 cm , blade thickness=0.8 mm ) in a pure concentrated emulsion , and in the same emulsion filled with 20% of 140 @xmath17 m ps beads . in these experiments , constant macroscopic shear rates steps increasing from 0.01 to 10 s@xmath167 and then decreasing from 10 to 0.01 s@xmath167 were applied during 30s , and the stationary shear stress was measured for each shear rate value . the results are shown in fig . [ fig_sweep_suspension ] . m ps beads ( filled / open circles ) . figure from @xcite.,width=302 ] while the same curve is observed for the upward / downward shear rate sweeps in the case of the pure emulsion ( as expected for a simple non - thixotropic yield stress fluid ) , the shear stress during the upward shear rate sweep differs from the shear stress during the downward shear rate sweep in the case of the suspension . moreover , any measurement performed on the suspension after this experiment gives a static yield stress equal to the dynamic yield stress observed during the downward sweep . this means that there has been some irreversible change . this irreversible change is actually the particle depletion near the blades we have observed in this paper . as the flow of the suspension is localized near the inner tool at low shear rate , it means that after the first upward sweep that has induced the particle depletion , during the downward shear rate sweep only the pure emulsion created by migration near the blades remains in the sheared layer at sufficiently low rotational velocity . this explains why the same apparent value of the yield stress is found in the suspension during the downward sweep as in the pure emulsion with this experiment . on the other hand , the yield stress at the beginning of the very first upward sweep is that of the suspension as migration has not occurred yet . the conclusion is that the vane tool is probably not suitable to the study of flows of suspensions of large particles . as a conclusion , let us summarize our main findings : * in the case of newtonian fluid flows , our measurements support the couette equivalence approach : the @xmath0-averaged strain rate component @xmath1 decreases as the inverse squared radius in the gap . interestingly , the velocity profiles allow determining the couette equivalent radius without end - effect correction and independently of the viscosity of the material . the torque exerted by the vane in our display is found to be higher ( by 8% ) than the theoretical prediction of @xcite for a vane embedded in an infinite medium , and is thus much closer to the torque exerted by a couette geometry of same radius as the vane than expected from the literature . a key observation may be that there is a significant flow between the blades which adds an important extensional component to shear , thus increasing dissipation . from a short review of the literature , it clearly appears that numerical investigations are still needed in the case of finite geometries . variational approaches are also promising , although they do not yet provide tight bounds . * in the case of yield stress fluid flows , we find that the thin layer of material which flows around the vane tool at low velocity is not cylindrical , in contrast with what is usually supposed in the literature from simulation results . consequently , a non negligible extensional component of shear has probably to be taken into account in the analysis . at this stage , there are too few experimental and simulation data to understand the origin of this discrepancy . it thus seems that progress still has to be made , in particular through simulations , which allow a wide range of parameters to be studied . this may help understanding how the torque is linked to the yield stress of a material at low velocity , depending in particular on the geometry . * an important and surprising result is the observation of particle depletion near the blades when the yield stress fluid contains noncolloidal particles . this phenomenon is thus likely to occur when studying polydisperse pastes like coal slurries , mortars and fresh concrete . it has to be noted that the phenomenon is very rapid , irreversible , and thus probably unavoidable when studying flows of suspensions of large particles . it results in the creation of a pure interstitial yield stress fluid layer and thus in a kind of wall slip near the blades . it contrasts with the classical assumption that is made in the field of concentrated suspension rheology where the vane tool is mainly used to avoid this phenomenon . consequently , we would say that , in the case of pasty materials with large particles , if accurate measurements are needed , the vane tool may finally be suitable only for the study of the solid ( elastic ) properties of materials and for the static yield stress measurements ; as the yield stress measurement may induce irreversible particle depletion near the blades , any new measurement then requires a new sample preparation . furthermore , the vane can be used as a very accurate tool without any hypothesis nor any calibration to measure the relative increase of the elastic modulus of materials as a function of their composition [ @xcite ] . in order to study accurately the flows of pasty materials with large particles , our results suggest that a coaxial cylinders geometry with properly roughened surfaces is preferable when possible . if the use of a vane tool can not be avoided , one should keep in mind our observations in order to carefully interpret any result . 99 abbott , j. r. , n. tetlow , a. l. graham , s. a. altobelli , e. fukushima , l. a. mondy , and t. s. stephens , `` experimental observations of particle migration in concentrated suspensions : couette flow , '' j. rheol . * 35 * , 773 - 795 ( 1991 ) . at - kadi , a. , p. marchal , l. choplin , a .- chrissemant , and m.bousmina , `` quantitative analysis of mixer - type rheometers using the couette analogy , '' the canadian journal of chemical engineering * 80 * , 1166 - 1174 ( 2002 ) . atkinson , c. , and j. d. sherwood , `` the torque on a rotating n - bladed vane in a newtonian fluid or linear elastic medium , '' proceedings of the royal society of london series a - mathematical physical and engineering sciences * 438 * , 183 - 196 ( 1992 ) . corbett , a. m. , r. j. phillips , r. j. kauten , and k. l. mccarthy , `` magnetic resonance imaging of concentration and velocity profiles of pure fluids and solid suspensions in rotating geometries , '' j. rheol . * 39 * , 907 - 924 ( 1995 ) . hanlon , a. d. , s. j. gibbs , l. d. hall , d. e. haycock , w. j. frith , s. ablett , and c. marriott , `` a concentric cylinder couette flow system for use in magnetic resonance imaging experiments , '' measurement science and technology * 9 * , 631 - 637 ( 1998 ) . huang , p. y. , and d. d. joseph , `` effects of shear thinning on migration of neutrally buoyant particles in pressure driven flow of newtonian and viscoelastic fluids , '' j. non - newtonian fluid mech . * 90 * , 159 - 185 ( 2000 ) . jau , w .- c . , and c .- t . yang , `` development of a modified concrete rheometer to measure the rheological behavior of fresh concrete , '' cement and concrete composites ( 2010 ) , in press doi:10.1016/j.cemconcomp.2010.01.001 mahaut , f. , s. mokddem , x. chateau , n. roussel , and g. ovarlez , `` effect of coarse particle volume fraction on the yield stress and thixotropy of cementitious materials , '' cem . concr . res . * 38 * , 1276 - 1285 ( 2008 ) . ovarlez , g. , f. bertrand , and s. rodts , `` local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging , '' j. rheol . * 50 * , 259 - 292 ( 2006 ) . ovarlez , g. , s. rodts , a. ragouilliaux , p. coussot , j. goyon , and a. colin , `` wide - gap couette flows of dense emulsions : local concentration measurements , and comparison between macroscopic and local constitutive law measurements through magnetic resonance imaging , '' phys . e * 78 * , 036307 ( 2008 ) . phillips , r. j. , r. c. armstrong , r. a. brown , a. l. graham , and j. r. abbott , `` a constitutive equation for concentrated suspensions that accounts for shear - induced particle migration , '' phys . fluids * 4 * , 30 - 40 ( 1992 ) . raynaud , j. s. , p. moucheront , j. c. baudez , f. bertrand , j. p. guilbaud , and p. coussot , `` direct determination by nmr of the thixotropic and yielding behavior of suspensions , '' j. rheol . * 46 * , 709 - 732 ( 2002 ) . rodts , s. , f. bertrand , s. jarny , p. poullain , and p. moucheront , `` dveloppements rcents dans lapplication de lirm la rhologie et la mcanique des fluides , '' comptes rendus chimie * 7 * , 275 - 282 ( 2004 ) . shapley , n. c. , r. a. brown , and r. c. armstrong , `` evaluation of particle migration models based on laser doppler velocimetry measurements in concentrated suspensions , '' j. rheol . * 48 * , 255 - 279 ( 2004 ) . zhu , h. , n. s. martys , c. ferraris , and d. de kee , `` a numerical study of the flow of bingham - like fluids in two - dimensional vane and cylinder rheometers using a smoothed particle hydrodynamics ( sph ) based method , '' j. non - newtonian fluid mech . * 165 * , 362 - 375 ( 2010 ) .	we study the local flow properties of various materials in a vane - in - cup geometry . we use mri techniques to measure velocities and particle concentrations in flowing newtonian fluid , yield stress fluid , and in a concentrated suspension of noncolloidal particles in a yield stress fluid . in the newtonian fluid , we observe that the @xmath0-averaged strain rate component @xmath1 decreases as the inverse squared radius in the gap , in agreement with a couette analogy . this allows direct comparison ( without end - effect corrections ) of the resistances to shear in vane and couette geometries . here , the mean shear stress in the vane - in - cup geometry is slightly lower than in a couette cell of same dimensions , and a little higher than when the vane is embedded in an infinite medium . we also observe that the flow enters deeply the region between the blades , leading to significant extensional flow . in the yield stress fluid , in contrast with the usually accepted picture based on simulation results from the literature , we find that the layer of material that is sheared near the blades at low velocity is not cylindrical . there is thus a significant extensional component of shear that should be taken into account in the analysis . finally and surprisingly , in the suspension , we observe that a thin non - cylindrical slip layer made of the pure interstitial yield stress fluid appears quickly at the interface between the sheared material and the material that moves as a rigid body between the blades . this feature can be attributed to the non - symmetric trajectories of the noncolloidal particles around the edges of the blades . this new important observation is in sharp contradiction with the common belief that the vane tool prevents slippage , and may preclude the use of the vane tool for studying the flows of pasty materials with large particles .	22,097	451
currently a number of intense mid - infrared light sources are being developed @xcite , spurred on by their uses in sub - attosecond pulse generation @xcite , strong - field holography @xcite and laser - induced electron diffraction @xcite . the low frequency and high intensity of these new sources mean that the tunneling picture is an appropriate framework for describing how these light sources interact with atoms and molecules . in this work we deal with the process of dissociative tunneling ionization in molecules , where a static electric field tunnel ionizes an electron , after which the nuclei dissociate . to our knowledge this is the first work on the theory of dissociative tunneling ionization . in the theory we treat the nuclear and electronic degrees freedom on an equal footing and fully quantum mechanically . the reflection principle @xcite is often used to describe the process of dissociative ionization . this principle can be applied within the framework of the born - oppenheimer ( bo ) approximation to relate the nuclear kinetic energy release ( ker ) spectrum to the nuclear wave function . it was formulated as early as 1928 @xcite , and later put on a more rigorous foundation @xcite ( see also references therein for a list of early uses ) . in time - dependent cases where the time - scale of the electric field is shorter than that of nuclear motion , one assumes that the electrons make an instantaneous frank - condon transition to a dissociative electronic state . the probability distribution in the new electronic state is then the absolute value squared of the initial nuclear wave function times some dipole coupling factor . in case this dipole coupling factor is almost constant , the wave packet that enters the dissociative state is practically identical to the initial nuclear wave function . classical energy conservation then dictates that the nuclear ker spectrum can be obtained by reflecting the nuclear wave function in the dissociative potential curve . this is the regime considered mostly in the literature . in the time - independent tunneling case , the electronic ionization rate takes the same role as the dipole coupling factor in the time - dependent case , that is , it multiplies the nuclear wave function before it enters the continuum . however , this electronic rate has an exponential dependence on the internuclear coordinate and can by no means be considered constant ( as was also pointed out in ref . it is therefore essential to consider the effect of this additional factor on the ker spectrum ; the spectrum can not be found simply by reflection of the nuclear wave function . imaging of the nuclear wave function is made possible through the reflection principle , by applying it in reverse on a measured ker spectrum @xcite . this is often referred to as coulomb explosion imaging . in the tunneling case the exponential dependence of the electronic rate on the internuclear coordinate means that the product of the electronic rate and the nuclear wave function is essentially different from the bare nuclear wave function , and the electronic rate therefore needs to be included to image the nuclear wave function based on the ker spectrum . in ref . @xcite it was demonstrated that the bo approximation breaks down for weak fields . in this case the weak - field asymptotic theory ( wfat ) @xcite provides us with accurate results for the ker spectrum . the paper is organized as follows . in sec . [ sec : theory ] the theory for dissociative tunneling ionization of homonuclear molecules is outlined . we derive an exact expression for the ker spectrum and a corresponding expression in the bo approximation . section [ sec:1d - calculation ] exemplifies the theory with numerical reduced dimensionality calculations . numerically exact ker spectra are compared to ker spectra obtained in the bo approximation using the reflection principle . imaging of the nuclear part of the wave function from the ker spectrum is demonstrated . section [ sec : conclusion - outlook ] concludes the paper . atomic units @xmath0 are used throughout . we consider a three - body system consisting of two heavy nuclei with masses @xmath1 and charges @xmath2 , and one electron with mass @xmath3 and charge @xmath4 . in the center - of - mass frame these have coordinates @xmath5 and @xmath6 fulfilling @xmath7 . let us introduce the reduced masses @xmath8 effective charge @xmath9 and jacobi coordinates @xmath10 we assume that the orientation of the internuclear axis @xmath11 is fixed in space . we also assume that the field is directed along the @xmath12-direction , @xmath13 and choose to consider @xmath14 for definiteness . due to the azimuthal symmetry of the molecule only the polar angle @xmath15 between @xmath11 and @xmath16 matters . this @xmath15 angle takes the role as an external parameter , and we omit explicit reference to it in the following . with these assumptions we can write the time - independent schrdinger equation ( se ) within the single - active - electron approximation as @xmath17 \psi({\mathbf{r}},r ) & = 0,\label{eq : schrodinger}\end{aligned}\ ] ] where the effective @xmath18 potential describes how the nuclei interact with each other and the effective @xmath19 potential describes how the electron interacts with the nuclei . for a system with several electrons the @xmath18 potential represents the bo potential of the system without the active electron . in this work , we assume that @xmath18 is monotonically decreasing , i.e. , it corresponds to a purely dissociative bo curve . we assume that the nuclei can not pass through each other . this gives the boundary condition @xmath20 and we consider eq . ( [ eq : schrodinger ] ) in the interval @xmath21 . we also impose outgoing - wave boundary conditions in the electronic coordinate @xmath22 , the exact form of these will be specified below . with these boundary conditions the wave function we seek as a solution of eq . ( [ eq : schrodinger ] ) is a siegert state @xcite , with a complex energy @xmath23 , where @xmath24 is the ionization rate , and it is normalized by @xmath25 the outgoing - wave boundary condition in the electronic coordinate means that the solution we seek to eq . ( [ eq : schrodinger ] ) describes tunneling of the electron . this tunneling is followed by dissociation of the nuclei for the considered class of strictly dissociative potentials @xmath18 . in the following we will describe the energy distribution of the dissociated nuclei . our aim is to describe the energy distribution of the nuclei , i.e. , the ker spectrum , after the molecule is ionized by tunneling of the electron . to this end we need to consider the problem in the @xmath26 limit , where the electron is far away from the nuclei . in this limit we assume that the electron - nuclear interaction potential takes the form @xmath27 where @xmath28 is the total charge of the remaining core system . this assumption makes our problem separable in electron and nuclear coordinates in this asymptotic region . by seeking the partial solutions in the form @xmath29 , eq . ( [ eq : schrodinger ] ) can be written as the separated equations @xmath30 f({\mathbf{r}},k ) & = 0,\label{eq : as_x_eq}\\ \left[- \frac{1}{2 m } { \frac { d^2 } { d { r } ^2 } } + { u}(r ) - e_{r}\right ] g(r , k ) & = 0,\label{eq : as_r_eq } \end{aligned}\ ] ] with separation constants given by @xmath31 where we assume @xmath32 and @xmath33 is the wave number for the state @xmath34 . equation ( [ eq : zero_bc ] ) amounts to @xmath35 we choose the continuum solutions of eq . ( [ eq : as_r_eq ] ) to be real and normalized by @xmath36 the conditions eqs . ( [ eq : zero_bc_g])-([eq : g_norm ] ) completely specify the nuclear problem eq . ( [ eq : as_r_eq ] ) . the electronic problem eq . ( [ eq : as_x_eq ] ) has a potential consisting of a coulomb term and a linear field term . this problem is separable in parabolic coordinates @xcite , which we will therefore use . first we introduce mass - scaled quantities @xmath37 then the following form of the parabolic coordinates is introduced ( as in ref . @xcite ) [ eq : parab_coord ] @xmath38 with this choice of coordinates a potential barrier forms in the @xmath39 coordinates and therefore @xmath39 takes the role as the tunneling coordinate. in the asymptotic region @xmath40 eq . ( [ eq : as_x_eq ] ) has a solution that is a linear combination of partial solutions of the form @xcite @xmath41 where the outgoing - wave @xmath42 is given by @xmath43\right),\label{eq : f_eta}\end{aligned}\ ] ] @xmath44 is the ionization channel function defined by @xmath45 \phi_\nu(\xi,\varphi ) & = \beta_\nu \phi_\nu(\xi,\varphi),\label{eq : xi_phi_ad_eq}\end{aligned}\ ] ] and @xmath46 is a set of parabolic quantum numbers labeling the different ionization channels , see fig . [ fig : parab_coord ] . with our choice of @xmath14 the potential in eq . ( [ eq : xi_phi_ad_eq ] ) goes to infinity as @xmath47 goes to infinity , so the parabolic channels @xmath48 are purely discrete . . the gray paraboloid is the same for a smaller value of @xmath39 . the electron is ionized in the negative @xmath12 direction due to its negative charge , given that the electric field points in the positive @xmath12-direction . the @xmath44 states [ eq . ( [ eq : xi_phi_ad_eq ] ) ] live in the constant @xmath39 paraboloids . the colors in the blue / red surface illustrates an example of the nodal structure of such a @xmath44 state . the curvature of the paraboloids means that the @xmath44 states are bound . this means that @xmath39 is the only coordinate where we have to consider the wave function at infinity , i.e. , @xmath39 is the tunneling coordinate . for large @xmath39 the polar angle @xmath15 , which specifies the orientation of the molecule , does not matter for the asymptotic form of the wave function in parabolic coordinates , though it matters for the size of the coefficients [ eq . ( [ eq : spectrum_ampl_def ] ) ] . ] the full wave function can be expressed as a linear combination , discrete in @xmath48 , continuous in @xmath49 , of the @xmath50 products , @xmath51 where the asymptotic expansion coefficient @xmath52 can be calculated by @xmath53 @xmath54 indicates integration w.r.t . the coordinates @xmath47 and @xmath55 over their full range . note that the polar angle @xmath15 , which we suppressed in the notation , only enters eq . ( [ eq : spectrum_ampl_def ] ) through the wave function @xmath56 . the ker dissociation spectrum into the channel @xmath48 is defined in terms of these expansion coefficients by @xmath57 this is the main observable of interest . by inserting eq . ( [ eq : f_eta ] ) and eq . ( [ eq : spectrum_ampl_def ] ) and assuming @xmath58 to be real , which is approximately the case for small @xmath59 , we obtain @xmath60 the exact ker spectrum in the channel @xmath48 can thus be obtained by projecting the wave function on the channel state @xmath44 , and further projecting this on the continuum states @xmath34 of the @xmath18 potential . the total ker spectrum can then be obtained by summing over all the channels @xmath61 in the @xmath62 limit the total rate can be obtained by @xmath63 now that we have a recipe for finding the exact ker spectrum , we consider some approximations for ease of predictions and gain in physical insight . we first consider the bo approximation , which appears in the limit @xmath64 . in this limit @xmath65 , and the wave function takes the form @xmath66 . the electronic and nuclear part of bo wave function fulfills the bo equations @xmath67 \psi_e({\mathbf{r}};r ) & = 0,\label{eq : bo_elec_eq}\\ \left[- \frac{1}{2 m } { \frac { d^2 } { d { r } ^2 } } + u(r)+e_e(r;f ) - e_{\text{bo}}(f)\right ] \chi(r ) & = 0.\label{eq : bo_nuc_eq } \end{aligned}\ ] ] we impose zero boundary condition for the nuclear wave function @xmath68 and the following normalizations @xmath69 in the asymptotic limit @xmath40 the electronic eq . ( [ eq : bo_elec_eq ] ) takes the same form as eq . ( [ eq : as_x_eq ] ) , and it can be written in parabolic coordinates in the same manner . the electronic wave function then takes the outgoing - wave form @xmath70 where @xmath42 is from eq . ( [ eq : f_eta ] ) and @xmath44 are solutions of eq . ( [ eq : xi_phi_ad_eq ] ) , with @xmath58 replaced by @xmath71 in both . the asymptotic coefficient @xmath72 defines the ionization amplitude in channel @xmath48 @xcite . the partial electronic ionization rate is given by @xmath73 by considering the flux of the electron probability through a surface at large negative @xmath12 , one can show that in the weak field limit the total electronic rate @xmath74 is given as a sum over @xmath48 of all the partial electronic rates . by inserting the bo wave function into eq . ( [ eq : spec_expr ] ) we obtain @xmath75 the separation of electronic and nuclear coordinates in the bo approximation means that this expression for the ker spectrum does not contain any explicit reference to electronic coordinates , as opposed to the expression for the exact spectrum eq . ( [ eq : spec_expr ] ) . equation ( [ eq : spec_bo ] ) is similar to a result previously put forward in the literature [ eq . ( 1 ) of ref . @xcite ] , except that the correct complex ionization amplitude @xmath72 was taken as @xmath76 . in the cases we have considered , the phase variations of @xmath72 are sufficiently small that they can be safely neglected , explaining the successful use of the aforementioned replacement in ref . @xcite , but this is not generally true . to evaluate the integral in eq . ( [ eq : spec_bo ] ) we will use the reflection principle @xcite . at the heart of the reflection principle lies an important mathematical component which we denote the reflection approximation @xcite . this approximation amounts to setting @xmath77 which is exact in the @xmath78 limit . in eq . ( [ eq : g_as_delt ] ) @xmath79 is the classical turning point potentials , so there is only one classical turning point . ] for the @xmath34 function defined by @xmath80 in order to determine the derivative @xmath81 the form of the dissociative @xmath18 potential must be known . inserting eq . ( [ eq : g_as_delt ] ) in eq . ( [ eq : spec_bo ] ) yields @xmath82 this result shows that using the reflection approximation in conjunction with the bo approximation we obtain a ker spectrum that is expressed as a product of a jacobian factor , the electronic rate and the field - dressed nuclear wave function [ eq . ( [ eq : bo_nuc_eq ] ) ] . this is a lot simpler to calculate than evaluating either integrals in eqs . ( [ eq : spec_expr ] ) or ( [ eq : spec_bo ] ) , and is easily reversed to give a way to image the field - dressed nuclear wave function , and it is applicable to any molecule with a dissociative bo curve . the exact electronic rate @xmath83 is often not available , since finding it requires solving the electronic problem eq . ( [ eq : bo_elec_eq ] ) , which is a highly non - trivial task for many systems . in such cases the weak - field asymptotic theory ( wfat ) @xcite can be employed to obtain the rate . wfat is an analytic theory which expresses the ionization rate in terms of properties of the field - free state . it is applicable in the weak - field limit . let @xmath84 and @xmath85 denote the adiabatic eigenvalues and eigenfunctions solving eq . ( [ eq : xi_phi_ad_eq ] ) for @xmath86 with the field - free electronic energy @xmath87 replacing @xmath58 . ref . @xcite provides analytic expressions for these quantities . in terms of these the asymptotic field - free electronic wave function can be written @xmath88 where @xmath89 the electronic wfat rate is then given by @xcite @xmath90 where the field factor @xmath91 is defined by @xmath92 and the asymptotic coefficients @xmath93 can be found from the electronic wave function by inversion of eq . ( [ eq : field_free_elec_wf ] ) @xmath94 wfat can also be applied for the exact state , and not just in the bo approximation as above . in this section we will give the pertaining formulas . let , as before , @xmath95 and @xmath96 denote the adiabatic eigenvalues and eigenfunctions solving eq . ( [ eq : xi_phi_ad_eq ] ) for @xmath86 now with the field - free energy @xmath97 . in terms of these the asymptotic field - free wave function can be written @xcite @xmath98 where @xmath99 the wfat @xcite yields the following expression for the ker spectrum @xmath100 where the field factor @xmath101 is given by @xmath102 and the field - free asymptotic coefficients @xmath103 can be found by inversion of eq . ( [ eq : field_free_wf ] ) @xmath104 solving eq . ( [ eq : schrodinger ] ) in 3d is a computationally heavy task , so we have used a 1d model to illustrate our central points . in this section we compare exactly calculated ker spectra with those obtained through the bo approximation , eq . ( [ eq : spec_result_bo ] ) , and the wfat , eq . ( [ eq : spec_wfat ] ) , within this 1d model . in the following we will consider a model of h@xmath105 as an example . the potentials we consider are thus @xmath106 with @xmath107 and @xmath108 . the interaction between the nuclei and the electrons @xmath109 is described by a soft - core coulomb potential . the function @xmath110 is chosen in such a way that the bo potential of this potential reproduces the bo potential energy curve of 3d h@xmath105 @xcite . we use the method described in ref . @xcite to solve the 1d equivalent of eq . ( [ eq : schrodinger ] ) given by @xmath111 \psi(z , r ) & = 0.\label{eq : schrodinger_1d}\end{aligned}\ ] ] in the 1d model the index @xmath48 , which describes what happens in the paraboloids of constant @xmath39 transversal to @xmath12 , is of no meaning , and it hence does not appear in any of the 1d equivalents of the 3d equations . the 1d equivalent of the exact ker spectrum eq . ( [ eq : spec_expr ] ) is @xmath112 equations ( [ eq : spec_bo ] ) and ( [ eq : spec_result_bo ] ) apply to the 1d case with appropriately redefined quantities . the wfat expressions eqs . ( [ eq : rate_wfat_elec ] ) , ( [ eq : field_factor_elec ] ) and ( [ eq : spec_wfat ] ) , ( [ eq : full_field_factor ] ) are the same as in the 1d case , but the asymptotic coefficients are now found from @xmath113 and @xmath114 in eq . ( [ eq:1d_asymp_coeff ] ) , @xmath115 denotes the field - free 1d electronic bo wave function . ( light blue shaded area in the lower @xmath116 bo curve ) is multiplied by the electronic rate @xmath117 ( dashed purple line ) and reflected in the dissociative @xmath18 bo curve to give a ker spectrum ( solid blue line in upper right corner , [ eq . ( [ eq : spec_result_bo ] ) ] ) , using the relation @xmath118 to translate @xmath49 into @xmath119 . this is compared to the exact ker spectrum @xmath120 ( red dashed line , [ eq . ( [ eq : spec_expr_1d ] ) ] ) . a field strength of @xmath121 was used for this calculation . the solid gray line in the lower part of the figure shows the field - free nuclear wave function @xmath122 . the surface plot in the upper part of the figure shows the continuum states @xmath34 of the @xmath18 potential , these are solutions of eq . ( [ eq : as_r_eq ] ) . ] figure [ fig : spec_1d_ground ] illustrates how the bo approximation can be used in conjunction with the reflection principle to determine the ker spectrum . the figure shows a calculation for the ground state of the h@xmath105 model at @xmath121 . the field dressed nuclear wave function @xmath123 is multiplied by the electronic rate @xmath117 . the exponential dependence of the electronic rate @xmath117 on the internuclear coordinate means that the product @xmath124 ( see [ eq . ( [ eq : spec_result_bo ] ) ] ) has its maximum at a value of @xmath125 , which is significantly different from the maximum of the bare nuclear wave function at @xmath126 . this in turn means that the transition to the continuum which is determined by the product @xmath124 and not the bare nuclear wave function is far from vertical in @xmath119 with respect to the initial nuclear wave function , and the spectrum peaks at a lower energy around @xmath127 and not at @xmath128 . using wfat within the bo approximation we can make a statement about in which direction the maximum of the spectrum shifts when the field is varied . in these approximations the main dependence of the electronic rate on the field is contained in the exponent @xmath129 , see eq . ( [ eq : field_factor_elec ] ) . the electronic energy @xmath87 , in terms of which @xmath130 is defined , generally depends very much on the system considered . in the case of h@xmath105 it is a monotonically increasing function of @xmath119 , since when the two potential wells around each of the nuclei start to overlap the electron is more tightly bound . this in turn means that the electronic rate is an increasing function of @xmath119 , as can also be seen in fig . [ fig : spec_1d_ground ] . when the strength of the field increases the exponent @xmath129 grows , but at the same time the slope of this exponent with respect to @xmath119 decreases , since @xmath131 is multiplied by a smaller number . the smaller slope means that the location of the maximum of the product @xmath124 is shifted less from the maximum of @xmath123 as the field strength increases , and conversely , as the field strength is decreased the maximum of the product @xmath124 is shifted more towards larger @xmath119 . these shifts are directly reflected in the spectrum , which is given as the reflection of the @xmath124 product in the bo and reflection approximations . figure [ fig : spec_1d ] shows ker spectra obtained using as initial state the first vibrationally exited state of h@xmath105 . we have chosen to show these results as they are for the lowest state with a non - trivial nodal structure in @xmath119 . in the figure two different field strengths are considered . in the top panel we see that the nodal structure of the nuclear wave function is reflected in the ker spectrum , although one peak is a lot larger than the other . this asymmetry can be understood in the bo approximation , see eq . ( [ eq : spec_result_bo ] ) , as due to the fact that the electronic rate @xmath117 has an exponential dependence on @xmath119 . in the wfat it can be understood as resulting from the exponential dependence of the field factor [ eq . ( [ eq : full_field_factor ] ) ] on @xmath49 . for the lower field strength the structures at @xmath132 are not visible as the ker spectrum falls below the numerical precision limit of our calculation . ( [ eq : spec_expr_1d ] ) ] . dashed dotted ( blue ) line : bo combined with reflection principle [ eq . ( [ eq : spec_result_bo ] ) ] . short dashed ( green ) line : wfat [ 1d equivalent of eq . ( [ eq : spec_wfat ] ) ] . the insets show the normalized ker spectra on a linear scale . the critical field for use of bo [ eq . ( [ eq : f_bo ] ) ] is for h@xmath105 : @xmath133 . ( a ) @xmath134 and @xmath135 . ( b ) @xmath136 and @xmath137 . ] for the large field strength [ fig . [ fig : spec_1d](a ) ] we see that the bo ker spectrum has a shape much closer to the exact ker spectrum than for the lower field strength . also the maximum value of the bo ker spectrum is more than an order of magnitude closer to the maximum value of the exact ker spectrum for the larger field strength . this can be understood on the basis of the retardation argument provided in ref . @xcite : the bo approximation is expected to hold as long as the electron is close enough to the nuclei that the time it takes for the electron to go to its present location from the nuclei is shorter than the time it takes for the nuclei to move . a typical electron velocity can be estimated as @xmath138 , where @xmath139 is the equilibrium internuclear distance , which for h@xmath105 is @xmath140 . a typical time scale for the nuclear motion can be estimated as @xmath141 , where @xmath142 is obtained by expanding the bo potential around @xmath139 to second order @xmath143 . using these estimates ref . @xcite defines a critical distance @xmath144 such that for @xmath145 we expect bo to work well , while for @xmath146 we expect it to break down . since the magnitude of the wave function is essentially unchanged after the tunneling , the bo approximation is expected to work well when the outer turning point is within this @xmath147 distance , so a critical field @xmath148 can be estimated , such that the bo approximation is expected to give good results for larger fields , but fail for smaller fields . the two field strengths of fig . [ fig : spec_1d ] lies on either side of this critical field , which for the system under consideration is @xmath133 . as we increase the field strength further the bo gives even better results . for the lower field strength where bo fails we can apply the wfat , see sec . [ sec : full_wfat ] . in fig . [ fig : spec_1d ] we see that the shape of the wfat ker spectrum indeed is closer to the exact ker spectrum than the bo ker spectrum for the weaker field strength , and it is also closer in magnitude to the maximum value . for the larger field strength the wfat ker spectrum is further from the exact ker spectrum in both shape and magnitude . ( upper right corner , [ eq . ( [ eq : spec_expr_1d ] ) ] ) at @xmath121 the magnitude of the asymptotic wave function has been found by reversing the reflection principle , giving @xmath149 , using the relation @xmath118 to translate @xmath49 into @xmath119 . from this , the field - dressed nuclear wave function has been imaged by dividing with the electronic rate @xmath117 and normalizing . in the lowest part of the plot , the short dashed ( purple ) line shows this imaging using the exact electronic rate @xmath74 , the long dashed ( red ) line shows it using the bo wfat approximation @xmath150 [ eq . ( [ eq : rate_wfat_elec ] ) ] . the solid gray line shows the field - free nuclear wave function @xmath122 . the shaded ( light blue ) area shows the field - dressed nuclear wave function @xmath123 . the surface plot in the upper part of the figure shows the continuum states @xmath34 . ] the field dressed nuclear wave function can be imaged from a measurement of the ker spectrum by inverting eq . ( [ eq : spec_result_bo ] ) for fields sufficiently large that the bo approximation applies . to demonstrate this we have taken the exact ker spectrum from our calculation at @xmath121 for the first vibrationally exited state and divided it by the jacobian factor and the electronic rate to obtain an image of the nuclear density . since an experimental ker spectrum is typically not known on an absolute scale , we have then normalized this quantity . in a calculation on a more complicated system than the one considered here the exact electronic rate is often not available , so we also show the result using the wfat approximation for the electronic rate [ eq . ( [ eq : rate_wfat_elec ] ) ] . the results are compared to the nuclear wave function known from the calculation in our model in fig . [ fig : reconstructed_chi ] . they do not agree perfectly , but the nodal structure is correctly reproduced . for smaller field strengths where the bo is not applicable this type of imaging is not possible . the ker spectrum , however , does give us access to the asymptotic wave function , as it is the norm square of the expansion coefficients of this , see eq . ( [ eq : wf_expansion ] ) . for the cases we have looked at , the phase of the asymptotic coefficient @xmath151 varies very little over the range where it has support . in our model we have access to the full wave function , and this we show in fig . [ fig : wf ] . the imaging through the 1d equivalent of eq . ( [ eq : wf_expansion ] ) would only give access to the part at large negative @xmath152 . in the classically allowed region at large negative @xmath152 the maximum of the wave function follows a classical trajectory . this is a prediction of the wkb theory , which applies as long we are not too close to the turning line . the classical trajectories can be found using newton s second law @xmath153 [ eq : clas_traj_newton ] a tempting choice of initial condition for the differential eqs . ( [ eq : clas_traj_newton ] ) would be to choose the @xmath154 values at the intersection of the outer turning line and the maximum ridge of the wave function , with zero velocity in both @xmath152 and @xmath119 direction . however , the wkb fails near the turning line , and therefore we can not expect the wave function to follow a classical trajectory here . instead we have chosen as initial condition some point at the maximum of the wave function at a large negative @xmath152 value away from the turning line . the influence of the @xmath155 potential can be neglected for sufficiently large negative @xmath152 , in this region we can write the separated energy conservation equations @xmath156 [ eq : clas_traj_energy ] the initial velocities have then been determined from eqs . ( [ eq : clas_traj_energy ] ) , using the real part of the total ( quantum ) energy for @xmath157 and the @xmath49 at which the ker spectrum @xmath120 [ eq . ( [ eq : spec_expr_1d ] ) ] peaks . the classical trajectories shown in fig . [ fig : wf ] were found using such initial conditions , and then propagated inwards . from fig . [ fig : wf ] it can be seen that contrary to the exact wave function , the position of the ridge of the bo wave function in @xmath119 does not change with @xmath152 . this is expected as the bo approximation appears in the limit of infinite nuclear mass , so classical motion in the nuclear coordinate is not possible . the asymptotic wave function that we can image using eq . ( [ eq : wf_expansion ] ) is therefore a non - bo wave function . it might seem strange that the bo is able to give the correct ker spectrum when the spectrum is the norm square of the expansion coefficients of the asymptotic wave function , and the bo gives a wrong description of this asymptotic wave function . however , the fact that the bo wave function does not obtain a probability current ( or velocity in the classical picture ) in the @xmath119-direction does not alter its projection on the continuum states . the important point is whether the bo wave function is similar to the exact wave function as it emerges at the outer turning line after tunneling , and this is the case if the turning line is within the critical bo distance @xmath147 [ eq . ( [ eq : z_bo ] ) ] . in fig . [ fig : wf ] we also see that for the larger field strength the tunneling is completed before the critical bo distance is reached , contrary to at the smaller field strength . we see that for the large field strength the electronic and full turning lines agree quite well in the region where most of the wave function is localized , but for the smaller field strength they do not . for @xmath158 . solid purple lines : full turning lines @xmath159 . the long dashed red line shows for each @xmath12 the @xmath119 at which the wave function @xmath160 has its maximum . the solid pink line shows a classical trajectory [ eq . ( [ eq : clas_traj_newton ] ) ] . the black dot at the end of the classical trajectory is the exit point @xmath161 determined from the maximum of the spectrum @xmath162 ( see main text ) . the short dashed lines are the simple straight line estimates for the tunneling and initial classical motion described around eq . ( [ eq : directions ] ) . ] one can notice that a phenomenon reminiscent of light refraction occurs for the wave function around the turning line in fig . [ fig : wf ] . it is evident , that the direction in which the maximum of the wave function moves changes noticeably at the turning line , when the wave function escapes from the classically forbidden tunneling region into the classically allowed region . the change of direction is due to the two different types of motion involved . when the wave function emerges from the tunneling region it has essentially zero average velocity in the @xmath119 direction . this means that we can apply the reflection principle in reverse on the spectrum to find the @xmath163 coordinate at which the maximum of the wave function emerges from the tunneling region by the relation @xmath164 , where @xmath162 is the value of @xmath49 for which the spectrum @xmath120 has its maximum . the @xmath12 value corresponding to this @xmath163 can then by found by considering the turning line @xmath165 . in fig . [ fig : wf_max ] we see that near the turning line the location of the wave function ridge differs from the classical trajectory . this is expected , since the prediction that the wave function ridge should follow a classical trajectory comes from wkb theory , which fails near the turning line . nevertheless , we can roughly describe the dissociative tunneling ionization process in two steps . first the system tunnels from the central region around @xmath166 to the exit point @xmath161 . this motion can roughly be described by a straight line from the maximum of the nuclear wave function @xmath123 that has the largest @xmath119 value , since this is the maximum that will dominate the tunneling , to the exit point . notice that this tunneling is not simply the electron tunneling out , but a correlated process involving both the electronic and nuclear degrees of freedom . in the classically allowed region the initial direction of the wave function from the exit point can be found from the classical trajectory : the initial slope of the classical trajectory that starts at the exit point @xmath161 with zero velocity in both @xmath12 and @xmath119 directions can be found to be @xmath167 this is not exactly the trajectory that describes the motion of the wave function ridge , but it is quite close . these two directions are different as they come from different types of motion , and hence we see the refraction - like phenomenon at the turning line . we have formulated theory for the dissociative tunneling ionization process , and derived exact formulas for the ker spectrum , as well as approximations in the framework of the bo and reflection approximations . we have demonstrated that the reflection principle can be used in conjunction with the bo approximation to image the field - dressed nuclear wave function from the ker spectrum . for weaker fields , where the bo approximation fails , the wfat can be used to find the ker spectrum . we have also demonstrated a qualitative difference between asymptotic bo and exact wave functions , as the latter shows classical motion in the nuclear coordinate , whereas the former does not move at all due to the infinite nuclear mass of the bo approximation . around the turning line the wave function exhibits a behavior similar to refraction of light . this work was supported by the erc - stg ( project no . 277767-tdmet ) , and the vkr center of excellence , quscope . the numerical results presented in this work were performed at the centre for scientific computing , aarhus http://phys.au.dk / forskning / cscaa/. o. i. t. acknowledges support from the ministry of education and science of russia ( state assignment no . 3.679.2014/k ) . 29ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1126/science.1218497 [ * * , ( ) ] \doibase http://dx.doi.org/10.1038/ncomms7611 [ * * , ( ) ] link:\doibase 10.1103/physrevx.5.021034 [ * * , ( ) ] link:\doibase 10.1364/optica.2.000623 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.013901 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.033002 [ * * , ( ) ] link:\doibase 10.1126/science.1198450 [ * * , ( ) ] link:\doibase 10.1103/physreva.84.043420 [ * * , ( ) ] link:\doibase 10.1038/nature10820 [ * * , ( ) ] link:\doibase 10.1103/physrev.32.858 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1679721 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.073202 [ * * , ( ) ] link:\doibase 10.1103/physreva.90.063408 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.92.163004 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.82.3416 [ * * , ( ) ] link:\doibase 10.1103/physreva.58.426 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.153003 [ * * , ( ) ] link:\doibase 10.1103/physreva.84.053423 [ * * , ( ) ] link:\doibase 10.1103/physreva.89.013421 [ * * , ( ) ] link:\doibase 10.1103/physrev.56.750 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.79.2026 [ * * , ( ) ] link:\doibase 10.1103/physreva.82.023416 [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1103/physreva.87.043426 [ * * , ( ) ] link:\doibase 10.1103/physreva.53.2562 [ * * , ( ) ] link:\doibase 10.1103/physreva.67.043405 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.253003 [ * * , ( ) ] link:\doibase 10.1103/physreva.91.013408 [ * * , ( ) ]	we present a theoretical study of the dissociative tunneling ionization process . analytic expressions for the nuclear kinetic energy distribution of the ionization rates are derived . a particularly simple expression for the spectrum is found by using the born - oppenheimer ( bo ) approximation in conjunction with the reflection principle . these spectra are compared to exact non - bo _ ab initio _ spectra obtained through model calculations with a quantum mechanical treatment of both the electronic and nuclear degrees freedom . in the regime where the bo approximation is applicable imaging of the bo nuclear wave function is demonstrated to be possible through reverse use of the reflection principle , when accounting appropriately for the electronic ionization rate . a qualitative difference between the exact and bo wave functions in the asymptotic region of large electronic distances is shown . additionally the behavior of the wave function across the turning line is seen to be reminiscent of light refraction . for weak fields , where the bo approximation does not apply , the weak - field asymptotic theory describes the spectrum accurately .	11,537	258
plasma oscillations in solids are possibly the simplest manifestation of collective effects in condensed matter , and their understanding in terms of _ plasmon _ modes one of the earliest triumphs of quantum many - body theory . @xcite on the experimental side , collective charge - density fluctuations can be probed through electron energy - loss ( eel ) or inelastic x - ray scattering ( ixs ) spectroscopies , two techniques that have been steadily producing a wealth of data since the early 60s and 70s , respectively.@xcite in the present day the engineering of novel materials down to the nanometer scale makes it possible to design devices where electromagnetic fields interact with collective oscillations of structures of sub - wavelength size . the strong dependence of plasmon dynamics on the size and shape of these nanostructures holds the promise of an extraordinary control over the optical response of the resulting devices , with applications to such diverse fields as photovoltaics,@xcite proton beam acceleration,@xcite or biosensing,@xcite to name but a few . this is plasmonics , _ i.e _ photonics based on collective electronic excitations in strongly heterogeneous systems , where surface effects play a fundamental role . plasma oscillations at surfaces have recently aroused a renewed attention by themselves , since it was shown that some metal surfaces unexpectedly exhibit _ acoustic _ plasmons.@xcite these are collective charge excitations localized at the surface , whose frequency vanishes linearly with the wavevector , and are not damped by the bulk electron - hole continuum.@xcite it is thought that these modes may offer the possibility of light confinement at designated locations on the surface , with possible applications in photonics and nano - optics.@xcite most of the theoretical understanding of the optical response in nano - plasmonic systems relies on a classical approach : the nanostructure is usually described as an assembly of components , each characterized by an effective macroscopic dielectric function , and separated from the others by abrupt interfaces . the overall optical response is then computed by solving maxwell s equation for the resulting heterogeneous system.@xcite when distances between the nanoscale components are themselves nanometric , however , quantum effects must be accounted for , and a fully quantum - mechanical description is called for . early quantum - mechanical approaches to the dynamics of charge - density fluctuations@xcite were based on the random - phase approximation as applied to the jellium model that , albeit exceedingly successful in simple metals and semiconductors , is not suitable for more complex materials , nor can it capture the fine , system - specific , features of even simple ones . the effects of crystal inhomogeneities on plasmon resonances in semiconductors ( the so called _ local - field _ effects ) were first addressed in the late 70s,@xcite using the empirical pseudopotential method,@xcite along similar lines as previously followed for the optical spectra.@xcite in the present day the method of choice for describing charge dynamics in real materials ( as opposed to simplified models , such as the jellium one ) is time - dependent ( td ) density - functional theory ( dft).@xcite although some attempts to investigate eel and ixs spectra using many - body perturbation theory have been made,@xcite the vast majority of the studies existing to date relies on tddft , which in fact has been successfully used to study plasmons in a number of bulk@xcite and surface@xcite systems . the conventional tddft approach to plasmon dynamics relies on the calculation of the charge - density susceptibility , @xmath0 ( or , equivalently , inverse dielectric matrix , @xmath1 ) , starting from the independent - electron susceptibility , @xmath2 , via a dyson - like equation.@xcite although successful in ( relatively ) simple systems that can be described by unit cells of moderate size , this methodology can hardly be applied to more complex systems , such as low - index or nano - structured surfaces , because of its intrinsic numerical limitations . in particular : _ i _ ) the calculation of @xmath2 requires the knowledge of a large number of empty states , which is usually avoided in modern electronic - structure methods ; _ ii _ ) the solution of the dyson - like equation requires the manipulation ( multiplication and inversion ) of ( very ) large matrices , and _ iii _ ) all the above calculations have to be repeated independently for each value of the frequency to be sampled . in this paper we introduce a new method , based on td density - functional perturbation theory ( dfpt),@xcite that allows to calculate eel and ixs cross sections avoiding all the above drawbacks , and thus lending itself to numerical simulations in complex systems , potentially as large as several hundred independent atoms . although the new methodology is general in principle , our implementation relies on the pseudopotential approximation , which limits its applicability to valence ( or shallow - core ) loss spectra . inner - core loss spectra are currently addressed using different methods , as explained _ e.g. _ in refs . . the salient features of our method are : _ i _ ) the adoption of a representation from time - independent dfpt@xcite allows to avoid the calculation of kohn - sham ( ks ) virtual orbitals and of any large susceptibility matrices ( @xmath0 or @xmath2 ) altogether ; and _ ii _ ) thanks to the use of a lanczos recursion scheme , the bulk of the calculations can be performed only once for all the frequencies simultaneously . the numerical complexity of the resulting algorithm is comparable , for the _ whole _ spectrum in a wide frequency range , to that of a _ single _ standard ground - state ( or static response ) calculation . the paper is organized as follows . in sec . [ sec : theory ] we describe our basic theoretical and algorithmic frameworks , including the implementation of the newly proposed methodology for the response of a periodic system to a monochromatic perturbation , relevant to the calculation of eel and ixs cross sections ; in sec . [ sec : applications ] we benchmark our technique on the prototypical examples of bulk silicon and aluminum , for which many experimental and well established theoretical results already exist ; finally , our conclusions are presented in sec . [ sec : conclusions ] . electron energy - loss spectroscopy probes the diffusion of a beam of fast electrons through a solid . according to van hove,@xcite the corresponding double - differential cross section for inelastic scattering reads:@xcite @xmath3 where @xmath4 and @xmath5 are the electron charge and mass , @xmath6 , @xmath7 , and @xmath8 are the incoming , outgoing , and transferred momenta , respectively , and @xmath9 is the dynamic structure factor per unit volume . while eel spectroscopy is not suitable for samples enclosed in high - pressure cells , plasmon dynamics under pressure can be probed by ixs spectroscopy.@xcite the double - differential cross - section reads in this case : @xmath10 where @xmath11 and @xmath12 are the incoming and scattered photon polarization directions , and @xmath13 and @xmath14 are the corresponding frequencies . according to the fluctuation - dissipation theorem@xcite @xmath9 is proportional to the imaginary part of the charge - density susceptibility , @xmath15 : @xmath16 in periodic solids the transferred momentum can be split into a component in the first brillouin zone , @xmath17 , and a reciprocal - lattice vector , @xmath18 , as @xmath19 , and @xmath0 is often expressed in terms of the inverse dielectric matrix , defined as:@xcite @xmath20 where @xmath21 . the function @xmath22 $ ] is usually referred to as the _ loss function_. in tddft electron dynamics is described by td one - electron equations for the occupied molecular orbitals . these td ks equations read:@xcite @xmath23 where @xmath24 and @xmath25 are the td ks orbitals and hamiltonian ( quantum mechanical operators are indicated with a caret ) , respectively , the index @xmath26 spans the @xmath27 occupied ( _ valence _ ) states , and atomic units ( @xmath28 are used henceforth . the ks hamiltonian reads : @xmath29 where @xmath30 and @xmath31 are the external and hartree - plus - exchange - correlation ( hxc ) potentials , respectively . let us assume that the external potential can be split into a static term , plus a small td perturbation : @xmath32 where @xmath33 is the td strength of the perturbation . the total ks potential is perturbed accordingly : @xmath34 , @xmath35 being the response hxc potential . the response of the ks orbitals is defined as @xmath36 @xmath37 and @xmath38 being the unperturbed ground - state ks orbitals and energies , respectively . the charge - density susceptibility is the response of the electron charge density , which only depends on the projection of the response of the valence ks orbitals onto the empty - state ( conduction ) manifold . the fourier transforms ( indicated by tilde @xmath39 hereafter ) of such projected response orbitals are obtained from standard first - order perturbation theory via the linear systems : @xmath40 where @xmath41 is the projector over the unperturbed conduction - state manifold . expressing the latter in terms of valence orbitals ( @xmath42 ) allows one to compute response ks orbitals without making any reference to unoccupied states , much in the same way as it is done in time - independent dfpt.@xcite the solution of eq . ( [ eq : lin - resp_w_eq1 ] ) requires one to express the total response potential , @xmath43 , in terms of its own solutions , through the response charge density , which is the diagonal of the response density matrix , @xmath44 , whose fourier transform is defined as : @xmath45 where the factor two accounts for spin degeneracy in a non - polarized system . note that @xmath46 , as a consequence of the reality of @xmath47 . the equation for the complex conjugate of @xmath48 reads : @xmath49 where use has been made of the reality of the perturbing potential ( @xmath50 ) . equations and describe the _ resonant _ and _ anti - resonant _ contributions to charge - density response , respectively . their left - hand sides just differ by the sign of the frequency , while , by using time - reversal symmetry of the unperturbed system ( @xmath51 ) their right - hand side can be made look alike . the equations for the resonant and anti - resonant components of the charge - density response are coupled by the hxc potential , which is determined self - consistently by the density response itself , through the relation : @xmath52 where @xmath53 is the hxc kernel , which we assume to be independent of frequency , consistently with the adiabatic dft approximation.@xcite the td ks equations ( [ eq : td - ks_equation ] ) can be equivalently expressed in terms of a quantum liouville equation for the one - particle density matrix , @xmath54:@xcite @xmath55 . \label{eq : liouville_eq_1}\ ] ] upon linearization and fourier transformation , eq . ( [ eq : liouville_eq_1 ] ) takes the form : @xmath56 , \label{eq : liouville_eq_ft_1_general}\ ] ] where @xmath57 is the unperturbed density matrix and @xmath58 is the liouvillian super - operator , defined by the relation:@xcite @xmath59 + \left [ \hat v'_{\mathrm{hxc}}[\hat{\rho } ' ] , \hat{\rho}^\circ \right ] . \label{eq : liouvillian_def}\ ] ] the response of an arbitrary one - electron hermitian operator , @xmath60 , to an external perturbation , @xmath61 , is described by the generalized susceptibility : @xmath62 \bigl ) , \label{eq : susceptibility_def_2}\end{aligned}\ ] ] where @xmath63 indicates a scalar product in an abstract operator manifold.@xcite equation states that , within tddft , the most general susceptibility can be expressed as an off - diagonal element of the resolvent of the liouvillian . the calculation of susceptibilities from eq . requires the explicit representation of the response density matrix and of the liouvillian super - operator acting on it . the minimum dimension of such a representation is @xmath64 , where @xmath65 is the number of virtual ( conduction ) orbitals and @xmath66 the dimension of one - electron basis set.@xcite the inversion of the liouvillian appearing in eq . is a formidable task in typical large - scale plane - wave calculations , where the number of occupied states can be as large as several hundreds to a few thousands , and the number of virtual orbitals a hundred times as large . the recursion method by haydock , heine , and kelly @xcite offers an elegant solution to a similar problem , namely the calculation of a diagonal element of the resolvent of a hermitian matrix , in terms of a continued fraction , whose coefficients are frequency - independent . the _ lanczos bi - orthogonalization algorithm_,@xcite allows one to generalize this procedure to the calculation of _ off - diagonal _ elements of the resolvent of a _ non - hermitian _ matrix . the resulting numerical workload for calculating the full spectrum in a whole wide frequency range is comparable to that of a _ single _ ground - state ( or static response ) calculation . other flavours of the lanczos - type algorithm can be found in refs . . we want to calculate matrix elements such as : @xmath67 where @xmath68 is a @xmath69 non - hermitian matrix , and @xmath70 and @xmath26 are generic @xmath71-dimensional arrays . to this end we define two sets of _ lanczos vectors _ , @xmath72 and @xmath73 , through the recursive relations:@xcite @xmath74 where one defines @xmath75 , @xmath76 , and the @xmath77 , @xmath78 , and @xmath79 _ lanczos coefficients _ are determined by the _ bi - orthogonality _ conditions @xmath80 , and @xmath81 . the set of vectors and coefficients generated through the recursion relations ( [ eq : lanczos_chain_1]-[eq : lanczos_chain_2 ] ) is often referred to as a _ lanczos chain_. the details of this algorithm are reviewed _ e.g. _ in ref . [ ] , and its specialization to tddft is presented in refs . [ ] . for the purposes of the present paper , we limit ourselves to observe that the lanczos vectors thus generated have the property that they provide a tridiagonal representation of the @xmath68 matrix . more specifically , if we define the @xmath82 matrices @xmath83 and @xmath84 ( @xmath85 being the number of lanczos iterations ) , one has : @xmath86 where @xmath87 is the tridiagonal matrix @xmath88 in this lanczos representation , the matrix element of eq . can be expressed as:@xcite @xmath89 where @xmath90 and @xmath91 is the @xmath85-dimensional vector defined as:@xcite @xmath92 the right - hand side of eq . ( [ eq : resolvent_g ] ) can be conveniently computed by solving , for any given value of @xmath93 , the equation : @xmath94 and calculating the scalar product : @xmath95 the vector @xmath91 , eq . ( [ eq : zeta_coef_for_g ] ) , can be computed on the fly during the lanczos recursion , through the relation @xmath96 . in practice , the procedure outlined above is performed in two steps . in the first step , which is by far the most time consuming , one generates the tridiagonal matrix @xmath87 , eq . ( [ eq : tridiagonal_matrix ] ) , and the vector @xmath97 , eq . ( [ eq : zeta_coef_for_g ] ) . in the second step @xmath98 is calculated from eq . ( [ eq : resolvent_liouvillian_lanczos_2_for_g ] ) upon the solution of eq . ( [ eq : eta_post_processing_for_g ] ) , for different frequencies @xmath93 . in practice , a small imaginary part @xmath99 is added to the frequency argument , @xmath100 , so as to regularize the function @xmath98.@xcite setting @xmath99 to a non - zero value amounts to broadening each individual spectral line or , alternatively , to convoluting the function @xmath98 with a lorentzian . because of the tridiagonal form and the small dimension of the matrix @xmath87 ( a few hundreds to a few thousands ) , the second step is essentially gratis . different responses to a same perturbation can be computed simultaneously from a same lanczos recursion , by computing different @xmath101 vectors on the fly . equation shows that the response density matrix is uniquely determined by the two sets of functions @xmath102 and @xmath103 . it is convenient to consider a linear combination of these functions , defined as : @xmath104 the two sets @xmath105 and @xmath106 are called respectively the _ upper _ and _ lower _ component of the _ standard batch representation _ ( sbr)@xcite of the response density matrix _ super - vector _ : @xmath107.@xcite the sbr of a hermitian operator , @xmath108 , has vanishing lower component , @xmath109 , while that of its commutator with the unperturbed density matrix [ see eq . ( [ eq : liouville_eq_ft_1_general ] ) ] has vanishing upper component , @xmath110 \sbr \bigl\ { 0 , \ { \hat{p}_c \ , \hat a \ , \varphi_v^\circ(\mathbf{r } ) \ } \bigr\}$ ] . the sbr of the liouvillian super - operator has the block form:@xcite @xmath111 where the @xmath112 and @xmath113 super - operators are defined by their action on response batches , @xmath114 @xmath115 is the hxc kernel of eq . ( [ eq : hxc_kernel ] ) , and @xmath116 is the hxc potential ( see eq . ) generated by the response charge density distribution whose sbr is ( see eq . ): @xmath117 according to the above equations , operating with the liouvillian on a test super - vector essentially requires the calculation of the hxc potential response , its application to each valence ks orbital , as well as the operation of the unperturbed hamiltonian onto twice the number of valence ks states . the starting super - vector of the lanczos recursion is the right - hand side of eq . whose sbr is : @xmath118 because of the special block structure of the liouvillian , eq . ( [ eq : liouvillian_sbr ] ) , the sbr of odd lanczos iterates have vanishing upper components , whereas the even ones have vanishing lower components . as a consequence , the number of response wavefunctions onto which the unperturbed hamiltonian must operate per lanczos iteration is halved . also , the diagonal elements of the resulting tridiagonal matrix ( the @xmath119 coefficients ) are all vanishing . it was previously noted that the components of the vector @xmath97 , eq . ( [ eq : zeta_coef_for_g ] ) , decrease rather rapidly to zero , whereas the @xmath78 ( and @xmath79 ) coefficients oscillate around two distinct values for odd and even iterations , whose average is approximatively equal to one half of the kinetic - energy cutoff ( in a plane - wave implementation ) , and whose difference is approximately twice as large as the excitation gap in insulating or semiconducting materials.@xcite this finding can be used to speed up considerably the calculation by adopting a suitable extrapolation technique . in practice , the lanczos recursion is stoped after @xmath120 iterations , such that the components of the @xmath101 array are small enough . the dimension @xmath85 of the linear system , eq . , is then set to a very large ( and to a large extent arbitrary ) value . the @xmath101 components from @xmath121 to @xmath85 are set to zero , whereas the corresponding @xmath122 and @xmath123 coefficients are set to the average of the values that have been actually computed . the accuracy of the calculated spectrum is then checked _ a posteriori _ with respect to the value of @xmath120 . in many applications it turns out that @xmath120 may vary from a few hundreds up to a few thousands ( depending on the plane - wave kinetic energy cutoff ) , and @xmath85 is a ( to a large extent arbitrary ) number reaching up to several thousands . as the solution of tridiagonal systems can be performed very efficiently via standard factorization techniques , the numerical overhead of this procedure is negligible . more on lanczos extrapolation can be found in refs . . in a periodic solid the unperturbed ks orbitals are @xmath124 , where @xmath125 , @xmath126 is a band index and @xmath127 a point in the brillouin zone . these ks orbitals can be cast into the bloch form : @xmath128 where @xmath129 is the lattice - periodic function . similarly , the total perturbing potential can be conveniently decomposed into _ bloch components _ : @xmath130 where @xmath131 is also lattice - periodic , and the sum extends over the first brillouin zone . a similar decomposition can be applied to the external and hxc response potentials . the response of each ks orbital can be correspondingly expressed as a linear combination of the responses to each bloch component of the perturbing potential : @xmath132 where @xmath133 is a lattice - periodic response orbital that satisfies the equation : @xmath134 in eq . , as well as in the rest of this paper , quantum - mechanical operators bearing a wave - vector subscript ( such as @xmath135 ) or superscript ( such as @xmath136 ) are thought to operate on lattice - periodic functions , and are defined in terms of their coordinate representations as : @xmath137 the projector onto the conduction manifold in eq . can be expressed in terms of the periodic parts of the unperturbed bloch functions as : @xmath138 where the sum extends over all the occupied bands . a similar decomposition into bloch components holds for the response density matrix , which reads in this case : @xmath139 where @xmath140 the anti - resonant contribution to the density - matrix response in eq . satisfies the equation : @xmath141 which can be obtained from eq . by complex conjugation and simple manipulations deriving from time - reversal invariance of the unperturbed system ( @xmath142 ) and the reality of the perturbing potential ( @xmath143 ) . in analogy with eq . , eq . shows that the response density matrix of a periodic solid to a perturbation of wave - vector @xmath144 is uniquely determined by the two sets of response orbitals @xmath145 and @xmath146 . note that @xmath126 and @xmath127 are running indices , whereas @xmath144 is fixed . the sbr can in this case be defined as : @xmath147 the two sets of response orbitals , @xmath148 and @xmath149 satisfy the coupled set of equations : @xmath150 where @xmath151 , and @xmath152 and @xmath153 are the super - operators defined by the relations : @xmath154 and @xmath155 is the hxc potential generated by the response charge density : @xmath156 equations , , and are closely parallel to eqs . , , and of sec . [ sec : batch_repr_general ] . in practice , the sum over @xmath127 points is limited to the portion of the brillouin zone that is irreducible with respect to the small group of @xmath144 and the resulting function symmetrized accordingly , in close analogy with time - independent dfpt for lattice - dynamical calculations.@xcite more about the exploitation of crystal symmetry in the calculation of dynamical charge - density susceptibilities can be found in ref . . the @xmath157 component of the charge - density susceptibility is obtained from eq . as the response of the @xmath158 fourier component of the charge - density operator , whose coordinate representation reads @xmath159 , to a monochromatic perturbation , @xmath160 . the sbr of the periodic part of @xmath161 is @xmath162 . the final expression for the susceptibility is : @xmath163 where @xmath164 is the solution of eq . , obtained when the periodic part of the external perturbing potential is @xmath165 . in practice the susceptibility in eq . is computed following the procedure outlined in sec . [ sec : lanczos_method ] ( see eq . ): @xmath166 where @xmath167 is a tridiagonal matrix of dimension @xmath85 of the form ( [ eq : tridiagonal_matrix ] ) , and @xmath168 is an @xmath85-dimensional array whose coefficients @xmath169 are defined as : @xmath170 the liouville - lanczos approach for eel and ixs spectroscopies can be extended to metals by a suitable generalization of the smearing technique introduced by de gironcoli in the static case for lattice - dynamical calculations.@xcite in the smearing approach , each ks energy level is broadened by a smearing function @xmath171 , which is an approximation to the dirac @xmath172-function in the limit of vanishing smearing width @xmath173 . the monochromatic @xmath144 component of the charge - density response eq . ( [ eq : sbr_charge - density_2 ] ) can then be cast into the form : @xmath174 where the functions @xmath175 and @xmath176 satisfy the equations : @xmath177 ( cf . with eqs . and ) , where @xmath178 @xmath179 $ ] and @xmath180 $ ] being smooth approximations to the step - function , and @xmath181 is the fermi energy . it can be easily verified that the coefficients @xmath182 vanish when any of its indices refers to an unoccupied state . therefore , the operator @xmath183 involves only a small number of partially occupied bands , and the first - order variation of the wavefunctions and of the charge density can be computed avoiding any explicit reference to unoccupied states , much in the same way as for insulating materials . more details about the liouville - lanczos approach for metals can be found in ref . . the technique described above has been implemented in the quantum espresso suite of computer codes,@xcite and is scheduled to be distributed in one of its future releases . we now proceed to validate it by calculating the loss function in bulk silicon and aluminum , for which several tddft studies exist , and whose spectra are known to be accurately described within the adiabatic local density ( lda ) and generalized gradient ( gga ) approximations ( see , _ e.g. _ , refs . for si , and for al ) . all the calculations have been performed within the lda approximation , using the perdew - zunger parameterization of the electron - gas data,@xcite norm - conserving pseudopotentials from the quantum espresso database@xcite and plane - wave basis sets up to a kinetic - energy cutoff of 16 ry . the first brillouin zone has been sampled with a monkhorst - pack ( mp ) @xmath127 point mesh , supplemented , in the case of al , by the methfessel - paxton smearing technique@xcite with a broadening parameter @xmath184 ry . the frequency argument of the susceptibility has been assumed to have a small imaginary part , @xmath99 , thus resulting in a lorentzian smearing of the spectra ( see sec . [ sec : lanczos_method ] ) . for both si and al we have used the experimental lattice parameters ( 10.26 a.u.@xcite and 7.60 a.u.,@xcite respectively ) , which is very close to the theoretical one and resulting in no appreciable difference in the computed spectra . figure [ fig : si_conv_iter ] shows the convergence of the loss spectrum of si , as calculated for a transferred momentum @xmath185 a.u . along the [ 100 ] direction , as a function of the number of lanczos iterations . after 400 iterations the spectrum displays spurious wiggles , which disappear by increasing the number of iterations up to 1500 . also displayed are results obtained by the extrapolation procedure outlined at the end of sec . [ sec : batch_repr_general ] , performed with @xmath186 lanczos iterations and extrapolating the results up to a linear system of dimension @xmath187 . we see that the numerical workload can be considerably reduced without any appreciable loss of accuracy . in fig . [ fig : si_conv_k ] we show the convergence of the loss function with respect to the @xmath127 point sampling of the brillouin zone . the @xmath188 mp @xmath127 point mesh is not dense enough to obtain a well - converged spectrum , due to the presence of spurious wiggles , which disappear by increasing the size of the mp mesh up to @xmath189 . in fig . [ fig : si_theor_vs_exp ] we compare our present results with those obtained from the conventional approach based on the dyson - like equation for the susceptibility@xcite and with experiment.@xcite the agreement is excellent in both cases . all the salient features observed in the experiments at small transferred momentum ( panel ( a ) ) are correctly predicted : the main plasmon peak around 20 ev , a shoulder around 15 ev , and a weak peak around 6.5 ev . we attribute the slight differences between the two theoretical spectra to the slightly different technical details used in the two works . in particular , the authors of ref . mimicked electron- and hole - lifetime effects with an energy - dependent broadening , in contrast to the constant lorentzian broadening , @xmath190 ry , used in our calculations . at larger momentum transfer ( panel ( b ) ) the interaction of the plasmon with the electron - hole continuum broadens the spectrum.@xcite the agreement with experiment,@xcite remarkable also in this case , is enhanced by increasing the lorentzian broadening up to @xmath191 ry , which allowed us to reduce the size of the mp mesh down to @xmath192 without any appreciable loss of accuracy . figure [ fig : al_conv_iter ] shows the convergence of the loss function of al , calculated at a transferred momentum @xmath193 a.u . along the [ 100 ] direction , as a function of the number of lanczos iterations . although the qualitative behavior is similar to that observed in si ( wiggles showing up for a small number of iterations disappear by increasing this number ) , the convergence appears to be faster in the present case . as for the large - iterate behavior of the lanczos coefficient , we observe that , in contrast to si , in al the odd and even coefficients oscillate around a same value , which is also in this case of the order of one half the plane - wave kinetic - energy cutoff . this is due to the vanishing of the gap , as discussed in ref . . figure [ fig : al_conv_k ] shows the convergence with respect to the size of the @xmath127 point mesh : very satisfactory convergence is achieved with @xmath189 mp mesh and a broadening parameter @xmath194 ry . in fig . [ fig : al_theor_vs_exp ] we compare the loss function of al as calculated by the present method for two different values of the transferred momentum along the [ 100 ] direction , with ixs experiments and with previous theoretical work . at small transferred momentum ( panel ( a ) ) theoretical predictions agree remarkably well with each other ( the slight discrepancies being attributable to the usual small differences between the technical details of the calculations ) and with experiment . both theoretical spectra display a small blueshift ( @xmath195 ev ) of the plasmon peak with respect to experiments . at larger transferred momentum ( panel ( b ) ) the theoretical spectra display a feature at @xmath196 ev , which is not observed experimentally . we attribute the remaining discrepancies to the lifetime effects,@xcite which have been treated in our calculations by a constant lorentzian broadening parameter ( @xmath197 ry , requiring a @xmath198 mp mesh ) . we believe that the liouville - lanczos approach introduced in this paper will open new perspectives in the calculation of loss spectra in extended systems . its main features are the adoption of a representation for the charge - density response borrowed from density - functional perturbation theory , and of a lanczos recursion scheme for computing selected elements of the inverse of ( very ) large matrices . the combination of these two elements permits to compute the loss spectrum of a given system , for a given transferred momentum , and for an _ entire wide frequency range _ , with a numerical workload of the same order as needed for a standard ground - state calculation for a same system ( the pre - factor being only a few times larger ) . in principle , the convergence of the computed loss spectra with respect to the length of the lanczos chains depends on the spectral range : the lower the frequency , the faster the convergence , as it was already observed for optical spectra in finite systems.@xcite in practice , however , adoption of the extrapolation techniques explained in sec . [ sec : extrapolation ] substantially alleviates this dependence . also , the spectral range accessible to eel / ixs spectroscopies is limited by the so - called @xmath199-sum rule:@xcite @xmath200 \omega \ , d\omega = - \frac{\pi}{2 } \ , \omega_p^2 , \label{eq : f_sum_rule}\ ] ] where @xmath201 is the plasma frequency , @xmath202 being the average electron density , _ i.e. _ the number of electrons ( _ valence _ electrons , in a pseudopotential calculation ) per unit volume.@xcite of course , the spectral range that needs to be sampled by lanczos recursion is correspondingly limited . the liouville - lanczos approach introduced in this paper also lends itself to an easy generalization to those methods ( such as hybrid functionals or the static bethe - salpeter equation bse ) that require the full density - matrix ( rather than just charge - density ) response , which is in fact as easily accessible to the batch representation utilized here.@xcite further generalization to frequency - dependent xc kernels ( or to the bse with dynamical screening ) may simply require computing the loss function at shifted frequencies ( @xmath203 ) , as proposed _ e.g. _ in ref . , or further methodological developments . further work is required to clarify this issue . all in all we believe that the advances presented in this paper will allow for the simulation of complex , possibly nano - structured , surfaces , as well as of systems where valence and shallow - core loss spectra overlap . examples of the former include low miller index surfaces or plasmonic materials , while bulk bismuth is an example of the latter . work is in progress on both lines . we thank s. de gironcoli , a. dal corso , and l. reining for valuable discussions . support from the anr ( project pnano accattone ) and from dga are gratefully acknowledged . computer time was granted by genci ( project no . 2210 ) . the work of i.t . and n.v has been performed under the auspices of the _ laboratoire dexcellence en nanosciences et nanotechnologies labex nanosaclay_. n.v . thanks marco saitta for discussions about tddfpt at an early stage of the project . s.b . gratefully acknowledges hospitality at the _ laboratoire des solides irradis _ of the cole polytechnique , where this paper was written .	the liouville - lanczos approach to linear - response time - dependent density - functional theory is generalized so as to encompass electron energy - loss and inelastic x - ray scattering spectroscopies in periodic solids . the computation of virtual orbitals and the manipulation of large matrices are avoided by adopting a representation of response orbitals borrowed from ( time - independent ) density - functional perturbation theory and a suitable lanczos recursion scheme . the latter allows the bulk of the numerical work to be performed at any given transferred momentum only once , for a whole extended frequency range . the numerical complexity of the method is thus greatly reduced , making the computation of the loss function over a wide frequency range at any given transferred momentum only slightly more expensive than a single standard ground - state calculation , and opening the way to computations for systems of unprecedented size and complexity . our method is validated on the paradigmatic examples of bulk silicon and aluminum , for which both experimental and theoretical results already exist in the literature .	10,269	249
physical models often involve phenomenological parameters or auxiliary fields characterizing the background spacetime or the background media . in most cases , dynamics of the model depend smoothly ( continuously and differentiably ) on the values of the background parameter . a non - smooth functional dependence is a rather rare phenomenon , but if it exists , it usually represents a keystone issue of the model . the examples of such non - smooth behavior are well known in solid state physics as phase transitions at critical points . another similar issue is the scalar higgs model of spontaneous symmetry breaking . in this paper , we present a simple phenomenological model of an electromagnetic medium that allows wave propagation only for a sufficiently big value of the medium parameter . for zero values of the parameter , our medium is the ordinary sr ( or even gr ) vacuum with the standard dispersion relation @xmath0 . however even infinitesimally small variations of the parameter modify the dispersion relation in such a way that it does not have real solutions , i.e. , the medium becomes to be completely opaque . for higher values of the parameter , the dispersion relation is modified once more and once again it has real solutions . it is well known that the dispersion relation can be treated as an effective metric in the phase space . in our model , the vacuum lorentz metric is spontaneously transformed into the euclidean one and returns to be lorentzian for a sufficiently big value of the parameter . we consider the standard electromagnetic system of two antisymmetric fields @xmath1 and @xmath2 that obey the vacuum maxwell system @xmath3}=0\,,\qquad h^{ij}{}_{,j}=0\,.\ ] ] the fields are assumed to be related by the local linear constitutive relation , @xcite,@xcite , @xmath4 due to this definition , the constitutive tensor obeys the symmetries @xmath5 the electromagnetic model ( [ max ] ) with the local linear response ( [ cons ] ) is intensively studied recently , see @xcite , @xcite , @xcite , and especially in @xcite . by using the young diagram technique , a fourth rank tensor with the symmetries ( [ sym ] ) is uniquely irreducible decomposed into the sum of three independent pieces . @xmath6 the first term here is the principal part . in the simplest pure maxwell case it is expressed by the metric tensor of gr @xmath7 in the flat minkowski spacetime with the metric @xmath8 , it reads @xmath9 in quantum field description , this term is related to the photon . the third term in ( [ decomp ] ) is completely skew symmetric . consequently , it can be written as @xmath10 the pseudo - scalar @xmath11 represents the axion copartner of the photon . it influences the wave propagation such that birefringence occurs @xcite , @xcite . in fact , this effect is absent in the geometric optics description and corresponds to the higher order approximation , @xcite , @xcite , @xcite . we turn now to the second part of ( [ decomp ] ) , that is expressed as @xmath12 this tensor has 15 independent components , so it may be represented by a traceless matrix @xcite , @xcite . this matrix reads @xmath13 the traceless condition @xmath14 follows straightforwardly from ( [ skewon - matr ] ) . in order to describe the influence of the skewon on the wave propagation , it is convenient to introduce a covector @xmath15 consider a medium described by a vacuum principal part ( [ princ - part - m ] ) and a generic skewon . the dispersion relation for such a medium takes the form , @xcite , @xcite , @xmath16 here the scalar product @xmath17 and the squares of the covectors @xmath18 and @xmath19 are calculated by the use of the metric tensor . it can be easily checked that eq.([disp ] ) is invariant under the gauge transformation @xmath20 with an arbitrary real parameter @xmath21 . this parameter can even be an arbitrary function of @xmath22 and of the medium parameters @xmath23 . with this gauge freedom , we can apply the lorenz - type gauge condition @xmath24 and obtain the dispersion relation in an even more simple form @xmath25 this expression yields a characteristic fact @xcite : the solutions @xmath26 of the dispersion relation , if they exist , are non - timelike , that is , spacelike or null , @xmath27 we will proceed now with the form ( [ disp ] ) and with the skewon covector expressed as in ( [ skewon - cov ] ) . we can rewrite the dispersion relation as @xmath28 consequently , the real solutions exist only if @xmath29 our crucial observation that the first term here is quartic in the skewon parameters @xmath30 while the second term is only quadratic . under these circumstances , the first term can be small for for sufficiently small skewon parameters and the inequality ( [ ineq ] ) breaks down . for higher values , the first term becomes to be essential and the inequality is reinstated . we now present a model where this possibility is realized , indeed . consider a symmetric traceless matrix with two nonzero entries @xmath31 we denote the components of the wave covector as @xmath32 . the skewon covector has two nonzero components @xmath33 consequently , @xmath34 hence the inequality ( [ ineq ] ) takes the form @xmath35 observe that for every choice of the wave covector this expression is of the form @xmath36 with positive coefficients @xmath37 . quite surprisingly , this functional expression repeats the well known curve of the higgs potential . .,title="fig:",width=245 ] the dispersion relation as it is given in eq.([disp ] ) reads @xmath38 we rewrite it as @xmath39 consequently : * for @xmath40 , we return to the unmodified light cone @xmath41 . * for @xmath42 , except for the trivial solution @xmath43 , there are no real solutions of eq([disp2 ] ) at all . * for @xmath44 , there are two real solutions : @xmath45 for the numerical images of these algebraic cones , see fig . 3 and fig . 4 . ) with @xmath46 and @xmath47 signs respectively . the parameter @xmath48 . @xmath49 is directed as @xmath50-axis , @xmath51 are directed as @xmath52 and @xmath53 axes respectively . @xmath54 . , title="fig:",width=170 ] ) with @xmath46 and @xmath47 signs respectively . the parameter @xmath48 . @xmath49 is directed as @xmath50-axis , @xmath51 are directed as @xmath52 and @xmath53 axes respectively . @xmath54 . , title="fig:",width=170 ] ) with @xmath46 and @xmath47 signs respectively . the parameter @xmath48 . @xmath49 is directed as @xmath50-axis , @xmath55 are directed as @xmath52 and @xmath53 axes , respectively . @xmath56 . , title="fig:",width=170 ] ) with @xmath46 and @xmath47 signs respectively . the parameter @xmath48 . @xmath49 is directed as @xmath50-axis , @xmath55 are directed as @xmath52 and @xmath53 axes , respectively . , title="fig:",width=170 ] in both cones , the skewon interchanges the time axis with the spatial @xmath52-axis . these 3-dimensional cones are tangential to one another when the discriminant in eq . ( [ sym-15 ] ) is zero . it gives a 2-dimensional cone @xmath57 the expressions in eq.([sym-15 ] ) can be treated as finlsler metric elements . due to the fact that they have non - compact 2d - sections as in fig . 2 and compact 2d - sections as in fig . 3 , the corresponding finsler metric tensors are of the lorentz signature type . we construct a model of the electromagnetic vacuum with a skewon field that has the following features : * there is a gap for values of the parameters near zero , where the wave propagation is forbidden . * birefringence of the light propagation . * full interchange between time and spatial direction . * a continuous 2-dimensional variety of optic axes instead of distinct optic axes appearing in anisotropic optics . * the light cones are non - convex . electromagnetic media with an additional skewon field provide a rich class of models of wave propagation with rather unusual features , see @xcite , @xcite . recently the observational restrictions on such models were discussed in @xcite and @xcite . in this paper , we show that higgs - type potential can appear in a simple electromagnetic model by a minimal modification of the vacuum constitutive relation . my acknowledgments to f. hehl ( cologne / columbia , mo ) , yu . obukhov ( cologne / moscow ) , v. perlick ( zarm , bremen ) , c. lmmerzahl ( zarm , bremen ) , and y. friedman ( jct , jerusalem ) for valuable discussions . i acknowledge the gif grant no . 1078 - 107.14/2009 for financial support . y. n. obukhov , t. fukui and g. f. rubilar , phys . rev . d * 62 * , 044050 ( 2000 ) . y. n. obukhov and g. f. rubilar , phys . d * 66 * , 024042 ( 2002 ) . c. lmmerzahl and f. w. hehl , phys . d * 70 * ( 2004 ) 105022 . ni , phys . lett . a * 378 * , 1217 ( 2014 ) . s. di serego alighieri , w .- t . ni and w .- p . pan , `` new constraints on cosmic polarization rotation from b - mode polarization in cosmic microwave background , '' arxiv:1404.1701 .	in the framework of standard electrodynamics with linear local response , we construct a model that provides spontaneously broken transparency . the functional dependence of the medium parameter turns out to be of the higgs type .	2,832	47
.the `` half - split '' @xmath3 magic square of lie algebras . [ cols="^,^,^,^,^",options="header " , ] an analogous problem has been analyzed for the @xmath4 magic square , which is shown in table [ 2x2 ] ; the interpretation of the first two rows was discussed in @xcite ; see also @xcite . dray , huerta , and kincaid showed first @xcite ( see also @xcite ) how to relate @xmath5 to @xmath6 , and later @xcite extended their treatment to the full @xmath4 magic square of lie groups in table [ 2x2 ] . in the third row , their clifford algebra description of @xmath7 is equivalent to a symplectic description as @xmath8 , with @xmath9 . explicitly , they represent @xmath10 , where @xmath11 , in terms of actions on @xmath12 matrices of the form @xmath13 where @xmath14 is a @xmath4 hermitian matrix over @xmath15 , representing @xmath16 , @xmath17 , @xmath18 denotes the @xmath4 identity matrix , and tilde denotes trace - reversal , that is , @xmath19 . the matrix @xmath20 can be thought of as the upper right @xmath12 block of an @xmath21 clifford algebra representation , and the action of @xmath10 on @xmath20 is obtained as usual from ( the restriction of ) the quadratic elements of the clifford algebra . the generators @xmath22 can be chosen so that the action takes the form @xmath23 where the case - dependent signs are related to the restriction from @xmath21 matrices to @xmath12 matrices . following sudbery @xcite , we define the elements @xmath24 of the symplectic lie algebra @xmath25 by the condition @xmath26 where @xmath27 solutions of ( [ sp4def ] ) take the form , corresponding to @xmath28 . such elements can however also be generated as commutators of elements of the form ( [ ablock ] ) , so we do not consider them separately . ] @xmath29 where both @xmath30 and @xmath31 are hermitian , @xmath32 , and @xmath33 . but generators of take exactly the same form : @xmath34 represents an element of @xmath16 , @xmath30 and @xmath31 are ( null ) translations , and @xmath35 is the dilation . direct computation shows that the generators @xmath24 of do indeed satisfy ( [ sp4def ] ) ; the above construction therefore establishes the isomorphism @xmath36 as claimed . we can bring the representation ( [ pdef ] ) into a more explicitly symplectic form by treating @xmath14 as a vector - valued 1-form , and computing its hodge dual @xmath37 , defined by @xmath38 where @xmath39 is the levi - civita tensor in two dimensions . using the identity @xmath40 we see that @xmath41 takes the form @xmath42 which is antisymmetric , and whose block structure is shown in figure [ square ] . the diagonal blocks , labeled @xmath43 and @xmath44 , are antisymmetric , and correspond to @xmath45 and @xmath46 , respectively , whereas the off - diagonal blocks , labeled @xmath47 and @xmath48 , contain equivalent information , corresponding to @xmath37 . note that @xmath37 does not use up all of the degrees of freedom available in an off - diagonal block ; the set of _ all _ antisymmetric @xmath12 matrices is _ not _ an irreducible representation of @xmath25 . the action of @xmath25 on @xmath49 is given by @xmath50 for @xmath51 , that is , for @xmath24 satisfying ( [ sp4def ] ) . ) can be used if desired to determine the signs in ( [ soact ] ) . ] when working over @xmath52 or @xmath53 , the action ( [ sp4act ] ) is just the antisymmetric square @xmath54 of the natural representation @xmath55 , with @xmath56 . antisymmetric matrix in terms of @xmath4 blocks . a binary labeling of the blocks is shown on the left ; on the right , blocks with similar shading contain equivalent information.,title="fig:",height=96 ] antisymmetric matrix in terms of @xmath4 blocks . a binary labeling of the blocks is shown on the left ; on the right , blocks with similar shading contain equivalent information.,title="fig:",height=96 ] before generalizing the above construction to the @xmath3 magic square , we first consider the analog of @xmath37 . let @xmath57 be an element of the albert algebra , which we can regard as a vector - valued 1-form with components @xmath58 , with @xmath59 . the hodge dual @xmath60 of @xmath61 is a vector - valued 2-form with components @xmath62 where @xmath63 denotes the levi - civita tensor in three dimensions , that is , the completely antisymmetric tensor satisfying @xmath64 and where repeated indices are summed over . we refer to @xmath60 as a _ cubie_. we also introduce the dual of @xmath63 , the completely antisymmetric tensor @xmath65 satisfying @xmath66 and note the further identities @xmath67 in particular , we have @xmath68 operations on the albert algebra can be rewritten in terms of cubies . for instance , @xmath69 from which the components of @xmath70 can also be worked out . in the special case where the components of @xmath61 and @xmath71 commute , contracting both sides of ( [ epssq ] ) with @xmath72 yields @xmath73 or equivalently @xmath74 providing two remarkably simple expressions for the freudenthal product , albeit only in a very special case . we will return to this issue below . the action of @xmath75 on cubies is given by @xmath76 [ e6lemma ] consider the expression @xmath77 which is completely antisymmetric , and hence vanishes unless @xmath78 , @xmath79 , @xmath80 are distinct . but then @xmath81 which vanishes , since @xmath82 . thus , ( [ e6act ] ) becomes @xmath83 as claimed , where we have used both ( [ phiadj ] ) and ( [ trphi ] ) . a similar result holds for the action of @xmath84 . antisymmetric tensor in terms of @xmath85 `` cubies '' . a binary labeling of the cubies is shown on the pulled - apart cube on the left ; on the right , cubies with similar shading contain equivalent information.,title="fig:",height=192 ] antisymmetric tensor in terms of @xmath85 `` cubies '' . a binary labeling of the cubies is shown on the pulled - apart cube on the left ; on the right , cubies with similar shading contain equivalent information.,title="fig:",height=192 ] the representation ( [ thetadef ] ) can be written in block form , which we also call @xmath86 , namely require nested matrix transformations of the form ( [ e7mat ] ) . ] @xmath87 where @xmath88 denotes the @xmath3 identity matrix . by analogy with section [ sp4 ] , we would like @xmath86 to act on @xmath60 , which has 3 indices , and the correct symmetries to be an off - diagonal block of a rank 3 antisymmetric tensor @xmath89 , whose components make up a @xmath90 cube , which we divide into @xmath85 cubies , as shown in figure [ cube ] ; compare figure [ square ] . we identity the diagonal cubies , labeled @xmath91 and @xmath92 , with @xmath93 and @xmath94 , respectively , the cubie labeled @xmath95 with @xmath60 , the cubie labeled @xmath96 with @xmath97 , and then let antisymmetry do the rest . explicitly , we have @xmath98 where we have introduced the convention that @xmath99 , and where the remaining components are determined by antisymmetry . is a _ cube _ , and has components @xmath100 with @xmath101 , whereas @xmath63 , @xmath102 , and @xmath103 are the components of _ cubies _ , which are subblocks of @xmath89 , with @xmath104 . ] in the complex case , we could begin with the natural action of @xmath86 on 6-component complex vectors , and then take the antisymmetric cube , that is , we could consider the action @xmath105 with @xmath106 , or equivalently @xmath107 the action of the dilation @xmath108 on @xmath89 is given by ( [ pact ] ) . [ dilemma ] from ( [ e7mat ] ) , we have @xmath109 with the sign being negative for @xmath110 and positive for @xmath111 . thus , ( [ pact ] ) becomes @xmath112 where the signs depend on which of @xmath113 , @xmath79 , @xmath80 are `` small '' ( @xmath114 ) or `` large '' ( @xmath115 ) . examining ( [ pblocks ] ) , it is now easy to see that @xmath116 , @xmath117 , @xmath118 , and @xmath119 , exactly as required by ( [ freudx])([qfreud ] ) . if the elements of @xmath120 commute with those of @xmath89 , then the action of the translations @xmath121 and @xmath122 on @xmath89 is given by ( [ pact ] ) . [ trlemma ] set @xmath121 and consider the action of @xmath86 on @xmath45 , @xmath61 , @xmath71 , and @xmath46 , needing to verify ( [ freudx])([qfreud ] ) with @xmath123 , @xmath124 , and @xmath125 . from ( [ e7mat ] ) , we have @xmath126 since @xmath127 has one `` small '' index and one `` large '' index , it acts as a lowering operator , e.g. mapping cubie @xmath96 to @xmath91 , and thus maps @xmath128 . in particular , this confirms the lack of a term involving @xmath127 in ( [ qfreud ] ) . considering terms involving @xmath46 , we look at cubie @xmath95 , where the only nonzero term of ( [ pact ] ) is @xmath129 which verifies ( [ freudx ] ) in this case . we next look at cubie @xmath91 , where ( [ pact ] ) becomes @xmath130 which is clearly antisymmetric , so we can use ( [ eps6 ] ) and ( [ sinv ] ) to obtain @xmath131 which is ( [ pfreud ] ) , where we have used commutativity only in the last equality . finally , turning to cubie @xmath96 , ( [ pact ] ) becomes @xmath132 or equivalently , using ( [ sinv ] ) and ( [ cfreud ] ) , @xmath133 which is ( [ freudy ] ) . this entire argument can be repeated with only minor changes if @xmath122 . over @xmath134 or @xmath53 , we re done ; lemmas [ e6lemma ] , [ dilemma ] , and [ trlemma ] together suffice to show that the action ( [ pact ] ) is the same as the freudenthal action ( [ freudx])([qfreud ] ) . unfortunately , the action ( [ pact ] ) fails to satisfy the jacobi identity over @xmath135 or @xmath136 . however , we can still use lemmas [ e6lemma ] , [ dilemma ] , and [ trlemma ] to reproduce the freudenthal action in those cases , as follows . the action of @xmath137 on @xmath89 is determined by @xmath138 when acting on elements of the form ( [ pblocks ] ) , which extends to all of @xmath0 by antisymmetry . [ e6lemma2 ] from ( [ e7mat ] ) , we have @xmath139 inserting ( [ thetaphi ] ) into ( [ cubeact ] ) now yields precisely ( [ e6cact ] ) when acting on @xmath61 ; the argument for the action on @xmath71 is similar . furthermore , using a argument similar to that used to prove lemma [ e6lemma ] to begin with , ( [ pact ] ) acts on @xmath45 via @xmath140 which is completely antisymmetric in @xmath113 , @xmath79 , @xmath80 , and therefore proportional to @xmath141 . the argument for the action on @xmath46 is similar , with @xmath142 replaced by @xmath84 . although ( [ cubeact ] ) itself is only antisymmetric in its last two indices , that suffices to define an action on cubies @xmath91 , @xmath95 , @xmath96 , and @xmath92 ; the action on the remaining 4 cubies is uniquely determined by requiring that antisymmetry be preserved . we now have all the pieces , and can state our main result . the lie algebra @xmath0 acts symplectically on cubes , that is , @xmath143 acts on cubes via ( [ cubeact ] ) , as do real translations and the dilation , and all other @xmath0 transformations can then be constructed from these transformations using linear combinations and commutators . [ thm ] lemmas [ dilemma ] and [ trlemma ] are unchanged by the use of ( [ cubeact ] ) rather than ( [ pact ] ) , since the components of @xmath86 commute with those of @xmath89 in both cases , and lemma [ e6lemma2 ] verifies that @xmath144 acts via ( [ cubeact ] ) , as claimed . it only remains to show that the remaining generators of @xmath0 can be obtained from these elements via commutators . using ( [ freudx])([qfreud ] ) , it is straightforward to compute the commutator of two @xmath0 transformations of the form ( [ thetadef ] ) . letting @xmath145 be a boost , so that @xmath146 and @xmath147 , and using the identity @xmath148 for any @xmath149 , we obtain @xmath150 = ( 0,0,{{\cal a}}\circ{{\cal q}},0)\ ] ] we can therefore obtain the null translation @xmath151 for any _ tracefree _ albert algebra element @xmath152 as the commutator of @xmath153 and @xmath154 ; a similar argument can be used to construct the null translation @xmath155 . thus , _ all _ generators of @xmath0 can be implemented either as a symplectic transformation on cubes via ( [ cubeact ] ) , or as the commutator of two such transformations . we have showed that the algebraic description of the minimal representation of @xmath156 introduced by freudenthal naturally corresponds geometrically to a symplectic structure . along the way , we have emphasized both the similarities and differences between @xmath0 and @xmath157 . both of these algebras are _ conformal _ ; their elements divide naturally into generalized rotations ( @xmath144 or @xmath158 , respectively ) , translations , and a dilation . both act naturally on a representation built out of vectors ( @xmath3 or @xmath4 hermitian octonionic matrices , respectively ) , together with two additional real degrees of freedom ( @xmath45 and @xmath46 ) . in the @xmath4 case , the representation ( [ pdef ] ) contains just one vector ; in the @xmath3 case ( [ pcdef ] ) , there are two . this at first puzzling difference is fully explained by expressing both representations as antisymmetric tensors , as in ( [ squarie ] ) and ( [ pblocks ] ) , respectively , and as shown geometrically in figures [ square ] and [ cube ] . in the complex case , we have shown that the symplectic action ( [ pact ] ) exactly reproduces the freudenthal action ( [ freudx])([qfreud ] ) . the analogy goes even further . in @xmath159 dimensions , there is a natural map taking two @xmath78-forms to a @xmath160 matrix . when acting on @xmath89 , this map takes the form @xmath161 where @xmath162 now denotes the volume element in six dimensions , that is , the completely antisymmetric tensor with @xmath163 . it is not hard to verify that , in the complex case , ( [ pstarp ] ) is ( a multiple of ) @xmath164 , as given by ( [ pstarp0])([pstarp4 ] ) . similarly , the quartic invariant ( [ quartic ] ) can be expressed in the complex case as @xmath165 up to an overall factor . neither the form of the action ( [ pact ] ) , nor the expressions ( [ pstarp ] ) and ( [ quarticc ] ) , hold over @xmath135 or @xmath136 . this failure should not be a surprise , as trilinear tensor products are not well defined over @xmath135 , let alone @xmath136 . nonetheless , theorem [ thm ] does tell us how to extend ( [ pact ] ) to the octonions . although it is also possible to write down versions of ( [ pstarp ] ) and ( [ quarticc ] ) that hold over the octonions , by using case - dependent algorithms to determine the order of multiplication , it is not clear that such expressions have any advantage over the original expressions ( [ pstarp0])([pstarp4 ] ) and ( [ quartic ] ) given by freudenthal . despite these drawbacks , it is clear from our construction that @xmath0 should be regarded as a natural generalization of the traditional notion of a symplectic lie algebra , and fully deserves the name @xmath166 we thank john huerta for discussions , and for coining the term `` cubies '' . this work was supported in part by the john templeton foundation .	we explicitly construct a particular real form of the lie algebra @xmath0 in terms of symplectic matrices over the octonions , thus justifying the identifications @xmath1 and , at the group level , @xmath2 . along the way , we provide a geometric description of the minimal representation of @xmath0 in terms of rank 3 objects called _ cubies_.	5,237	103
the interaction of intense laser pulses with overdense plasmas has attracted much interest for the fast ignitor concept in inertial fusion energy @xcite . the interaction of ultrashort intense laser pulses with thin solid targets have also been of great interest for the application to high energy ion sources @xcite . ultraintense irradiation experiments using an infrared subpicosecond laser , e.g. , nd : glass ( @xmath3 1,053 nm ) or ti : sapphire ( @xmath3800 nm ) lasers , whose powers and focused intensities exceed 100 tw and @xmath4 w/@xmath5 , are possible using chirped pulse amplification techniques @xcite . in these experiments , the classical normalized momentum of electrons @xmath6 , where @xmath7 is the electron mass , @xmath8 is the speed of light , @xmath0 is the laser intensity in w/@xmath5 , and @xmath9 is the wavelength in @xmath10 m . on the other hand , a krf laser ( @xmath3 248 nm ) has an advantage as the fast ignitor in that the critical density is close to the core , and hot electron energies are suitable since the critical density of the krf laser is ten times greater than that of an infrared laser @xcite . the peak intensities of krf laser systems were only the order of @xmath11 w/@xmath5 , namely @xmath12 @xcite . therefore , the dependence of the laser plasma interactions on the laser wavelength was not investigated in @xmath13 . recently , the laser absorption and hot electron generation have been studied by the high intensity krf laser system of which focused intensity is greater than @xmath14 w/@xmath5 @xcite . however , the production of hot electrons by the high intensity krf laser has not been fully understood yet . namely , it has been not clear that the effects of laser wavelength on hot electrons produced by ultrashort intense laser pulse on solid - density targets . the absorption , electron energy spectrum , and hot electron temperature have usually been investigated and scaled using the parameters @xmath1 , @xmath15 , and @xmath16 @xcite , where @xmath17 , @xmath18 , and @xmath19 are the electron density , critical density , and density scale length , respectively . critical density absorption of the laser light converts laser energy into hot electrons having a suprathermal temperature @xmath20 approximately proportional to @xmath21 for @xmath22 , and @xmath23mc^2 $ ] at moderate densities @xcite , where @xmath24 kev , @xmath7 is an electron rest mass . the scaling of the hot electron temperature has been supported by experiments of nd : glass and ti : sapphire lasers @xcite . on the other hand , the results of one - dimensional simulation for normal incidence in the density region @xmath25 and the normalized intensity @xmath26 have shown that @xmath27 mc^2 $ ] , where @xmath28 is the electromagnetic fields at the surface of the overdense plasma , @xmath29 and @xmath30 , which depend weakly on @xmath1 and @xmath15 @xcite . @xmath31 is weakly depend on the angle of incidence , absorption rate , and @xmath32@xcite . the hot electron temperature is scaled by the amplitude of electromagnetic fields at the plasma surface rather than that in vacuum ; namely , the hot electron temperature is slightly dependent on the wavelength . in addition , at the interaction of intense laser pulses with solid density plasma which has a sharp density gradient the hot electron temperature is scaled by @xmath33 rather than @xmath1 @xcite . in the present paper , we study the absorption of ultrashort intense laser pulses on overdense plasmas for different laser wavelengths ( @xmath2 = 0.25 , 0.5 , and 1 @xmath10 m ) using a particle - in - cell ( pic ) simulation . in order to investigate hot electron generation for oblique incidence , we use the relativistic 1 and @xmath34 dimensional pic simulation with the boost frame moving with @xmath35 parallel to the target surface , where @xmath8 and @xmath36 are the speed of light and an angle of incidence@xcite . in the simulation , the target is the fully ionized plastic and the electron density @xmath37 . the density correspond to @xmath38 , @xmath39 , and @xmath40 for @xmath3 0.25 , 0.5 , and @xmath41 m , respectively . the density profile has a sharp density gradient , @xmath42 for @xmath43 and @xmath44 for @xmath45 . in order to clarify the boundary effect , ions are fixed , namely , the boundary does not move all the time . the laser pulse starts at @xmath46 and propagates towards @xmath47 . the laser intensity rises in 5 fs and remains constant after that . the irradiated intensity @xmath48 w/@xmath5 and the angle of incidence @xmath49 and @xmath50 ( p - polarized ) , respectively . @xmath51 2.3 , 9.2 , and 36 for @xmath3 0.25 , 0.5 , and 1.0 @xmath10 m , respectively . however , @xmath52 for all wavelength . normalized electron energy distributions after 50 fs are shown in fig.1(a ) and 1(b ) for @xmath49 and @xmath50 , respectively . the hot electron temperatures are 140 and 340 kev for @xmath49 and @xmath50 , respectively . the hot electron temperatures does not depend on the laser wavelength . the result is well agreement with that of a simple sharp boundary theory . on the other hand , the absorption depends on the laser wavelength , @xmath53 0.9 - 1.8% , 2.2 - 3.0% , and 3.6 - 4.3% and @xmath54 2.6 - 4.1% , 5.3 - 6.7% , and 7.8 - 9.0% for @xmath3 1.0 , 0.5 , and 0.25 @xmath10 m , respectively . the effects of laser wavelength on hot electrons produced by ultrashort intense laser pulse on solid - density targets are studied by the use of a pic simulation . as a result , the dependence to the wavelength of hot electron temperature strongly depend on the boundary condition , even in the one dimensional case , namely , all are not determined only by @xmath1 . the density profiles of both preformed plasma @xcite and multi - dimensional effects such as surface deformation @xcite are very important in the actual experiments . 99 m. tabak _ et al . _ , phys . plasmas * 1 * , 1626 ( 1994 ) ; r. kodama _ et al . _ , nature * 412 * , 798 ( 2001 ) . e.takahashi _ et al . _ , _ proceedings of the third international conference on inertial fusion sciences and applications ( ifsa2003 ) _ , editors : b. a. hammel d. d. meyerhofer j. meyer - ter - vehn h. azechi , p.406 ( american nuclear society , inc . , 2004 ) . _ , _ proc . of the seventh international symposium of the graduate university for advanced studies on science of super - strong field interactions , hayama , japan , 2002 _ , editors : k. nakajima and m. deguchi , p.290 ( american institute of physics , 2002 )	hot electron temperatures and electron energy spectra in the course of interaction between intense laser pulse and overdense plasmas are reexamined from a viewpoint of the difference in laser wavelength . the hot electron temperature measured by a particle - in - cell simulation is scaled by @xmath0 rather than @xmath1 at the interaction with overdense plasmas with fixed ions , where @xmath0 and @xmath2 are the laser intensity and wavelength , respectively .	2,017	113
pairing correlations play an essential role in the determination of the ground - state structure of the vast majority of atomic nuclei . among its treatments , an easy but still enough accurate one is to consider only single - particle states near the fermi level while the effects of the other states are assumed to be absorbed in the strength of an effective pairing interaction . in such a treatment , our experiences in hartree - fock(hf)+bcs calculations suggest that at least half of a major shell above the fermi level must be considered for meaningful estimations . as a consequence , one has to explicitly consider positive - energy states if the fermi level of neutrons is higher than the negative of half of the major shell spacing ( @xmath0 mev ) . this condition applies to about half of the @xmath1 nuclides which exist between proton and neutron drip lines in the nuclear chart , not only to nuclei near the neutron drip line or outside the s - process path . thus there are plenty of necessities for theoretical frameworks which can describe pairing correlations involving the continuum states . if the pairing correlation significantly involves the continuum states , the hf+bcs approximation becomes inadequate because the occupation of unbound hf orbitals leads to the unwanted dislocalization of the nucleon density . this is a serious problem in coordinate - space treatments , which is more favorable to describe loosely bound systems like drip - line nuclei than expansions in harmonic - oscillator eigenstates ( except the transformed oscillator basis of ref . the reason why a dislocalized solution becomes the variational minimum is that , in order to separate the variational equations into hf and bcs , one has to neglect the effects that the matrix elements of pair - scattering processes are affected by the changes in the wave functions of the orbitals involved @xcite . to fully take into account these effects leads to the hartree - fock - bogoliubov ( hfb ) equation , with which the density is localized whenever the fermi levels are negative @xcite . the hfb method in the coordinate space was first formulated using the quasiparticle states and solved for spherically symmetric states in ref . @xcite . although spherical solutions can be obtained easily with present computers ( for zero - range forces ) , deformed solutions are still difficult to obtain because there are quite a large number of quasiparticle states even for a moderate size of the normalization box ( i.e. , the cavity to confine the nucleons to discretize the positive - energy single - particle states ) . an orthodox approach to face this difficulty is the two - basis method @xcite , in which the quasiparticle states are expanded in bound and unbound hf orbitals . this method requires heavy numerical calculations because there are a pile of unbound hf orbitals below an energy cut - off of even only a few mev . an alternative approach is the canonical - basis hfb method . according to the bloch - messiah theorem @xcite , every hfb solution obtained as the vacuum of a set of bogoliubov quasiparticles has an equivalent expression in terms of a bcs - type wave function . the single - particle states appearing in this expression are called the hfb canonical basis . in the canonical - basis hfb method , one obtains the solution in the canonical form without using the quasiparticle states . this method appeared originally in ref . @xcite to obtain spherical solutions . however , there are no sever difficulties in obtaining hfb solutions for spherical nuclei using any other methods . applications to deformed nuclei have been done by us @xcite using a three - dimensional cartesian mesh representation@xcite . incidentally , a different line of application is also found in literature @xcite . let us explain the advantage of the canonical - basis method over the two - basis method concerning the treatment of the pairing in the continuum using fig . [ fig : two_basis_hfb ] . on both sides of the figure , the ordinate represents the expectation value of the mean - field ( hf ) energy , while the abscissa stands for the radius @xmath2 from the center of the nucleus . @xmath3 and @xmath4 mean a cut - off energy and the fermi level , respectively . on the left - hand side , the wavy horizontal lines stand for energy levels and their spatial extent for the hf potential denoted by a solid curve . the wave functions of positive energy states extend to the wall of the box and their level density is much larger than that of negative energy states . in the two - basis hfb method , one has to mix these positive - energy orbitals to construct localized canonical - basis orbitals , which is a numerically demanding task . on the right - hand side , the wavy lines represent the hfb canonical - basis orbitals . unlike hf orbitals , they are spatially localized for both negative and positive energies . because of this localization , the level density is much smaller than the unbound hf orbitals . therefore , one needs much less orbitals . more specifically , the number of necessary single - particle states to describe the hfb ground state of a nucleus is proportional to the volume of the nucleus in the canonical - basis method while it is related to the volume of the normalization box in the other methods . incidentally , with the dash curve , we suggest the existence of some potential which binds the high - lying canonical - basis orbitals . the identity of this potential is unveiled in sec . [ sec : canorb ] . in this paper we formulate the canonical - basis hfb method on a cubic mesh and develop an efficient gradient method to obtain its solutions . in order to decrease unphysical influences of high - momentum components due to zero - range interactions , we introduce a momentum dependent term to the pairing interaction and show how it suppresses a problematic behavior of wave functions peculiar to the canonical - basis hfb method . as a byproduct , we show a clear way to understand the nature of high - lying canonical - basis orbitals . the contents of this paper is as follows . in sections [ sec : cbhfb ] [ sec : gradientmethod ] , we give the formulation and general considerations . discussions using results of numerical calculations are collected in section [ sec : results ] . conclusions are given in section [ conclusions ] . in appendix [ sec : meshprec ] we examine the errors due to the mesh representation and the box boundary condition . appendix [ sec : pointcollapse ] gives a model analysis of the problem peculiar to the method , i.e. , a phenomenon that canonical - basis orbitals collapse to a spatial point once their occupation probabilities fall below some critical value when pure delta - function forces are used . some of the contents of this paper have already been discussed in our earlier reports @xcite . for example , ref . @xcite includes the first report on the canonical - basis hfb method on a cubic mesh . @xcite reported on the discovery of the point - collapse problem and the tests of the state - dependent cut - off factors and pairing - density - dependent forces as the remedies . in this paper we have rewritten those contents to include the momentum - dependent interaction and , by including the rewritten contents , organized the formalism for consistency and selfcontainedness . all the numerical calculations shown in this paper have been performed using the renewed form of the interaction . incidentally , some important parts of those reports are not repeated in this paper . the examples are an analysis of the spherical two - basis method concerning the precision of density tails @xcite and numerical demonstrations of the point collapses of high - lying orbitals @xcite . to begin with , let us formulate hf and hfb methods in the coordinate - space representation in order to elucidate the difficulty of the quasiparticle - based hfb method compared with the hf method and to propose its possible solution in terms of the canonical - basis hfb method . for the sake of simplicity , we consider only one kind of nucleons in this section . the number of that kind of nucleons are designated by @xmath5 . the @xmath6-component of the spin of a nucleon is represented by @xmath7 (= @xmath8 ) . in hf , one should minimize @xmath9 among all the @xmath5-body single slater - determinant states , @xmath10 by varying @xmath11 under orthonormality conditions @xmath12 = @xmath13 . the operator @xmath14 creates a nucleon with a wave function @xmath15 . the ket state @xmath16 stands for the state in which no nucleons exist . the distribution function of the number density of the nucleons is related to the single - particle wave functions as @xmath17 there is arbitrariness in the selection of @xmath18 as for unitary transformations among them . @xmath19 are called hf orbitals when they diagonalize the hf hamiltonian . in hfb , the solution takes the following form , @xmath20 where @xmath21 is the annihilation operator of a positive - energy bogoliubov quasiparticle with two types of amplitudes @xmath22 and @xmath23 for particle - like and hole - like parts of the excitation . @xmath24 is the number of the basis states of the representation used and is by far larger than @xmath5 . one should vary @xmath25 under an orthonormality condition @xmath26 and a constraint on the expectation value of the number of nucleons , @xmath27 where the nucleon density for state ( [ eq : vac_hfbq ] ) is expressed as latexmath:[\[\label{eq : dens_hfbq } \rho ( { \bm{r } } ) = \sum_{i=1}^{n_{\rm b } } \sum_{s } and hfb lies in the number of necessary single - particle wave functions . it is @xmath5 in the former and @xmath29 in the latter . obviously , the latter is much larger . owing to the bloch - messiah theorem @xcite , the state ( [ eq : vac_hfbq ] ) can be transformed ( as for the ordinary ground states of even - even nuclei ) into the following form , @xmath30 where @xmath31 and @xmath32 create a nucleon with wave functions @xmath33 and @xmath34 , respectively , which are called the canonical basis of the hfb vacuum @xmath35 . they may be called the natural orbitals@xcite instead , because they are the eigenstates of the one - body density matrix . in this paper , we often call them canonical - basis orbitals or canonical orbitals . it is required @xmath36 for the exact equivalence between eqs . ( [ eq : vac_hfbq ] ) and ( [ eq : vac_cbhfb ] ) to hold in general . in this paper we assume that @xmath35 is invariant under the time reversal operation . in this case , @xmath19 and @xmath37 are the time - reversal state to each other and only one of them has to be considered explicitly in the variational procedure . the nucleon density can be expressed as @xmath38 in the time - reversal invariant case . in order to obtain solutions expressed in the canonical form without using the information of the quasiparticle states , one should vary @xmath39 under three kinds of constraints , i.e , the orthonormality condition , @xmath40 a condition on the expectation value of the number of nucleons , @xmath41 and the normalization of the @xmath42-@xmath43 factors , @xmath44 . reinhard _ et al . _ wrote that the advantage of the representation ( [ eq : vac_cbhfb ] ) over ( [ eq : vac_hfbq ] ) is that one has to consider only a single set of wave functions @xmath18 , not a double set @xmath45 @xcite . in our opinion , however , one may expect much greater benefit from the canonical - basis representation . namely , @xmath46 may be truncated as @xmath47 = @xmath48 @xmath49 @xmath50 to a very good approximation . the reason is the localization of the density . the density distribution of hfb solutions can be shown to behave asymptotically for large @xmath2 as @xmath51 as far as the fermi level @xmath52 is negative @xcite , where @xmath53 is the nucleon mass . consequently , @xmath19 appearing on the right - hand side of eq . ( [ eq : dens_cbhfb ] ) must be a localized function as @xmath54 on the left - hand side . on the other hand , the orthogonality relation ( [ eq : ortho_cbhfb ] ) restricts the number of wave functions which can exist in the vicinity of the nucleus . therefore the number of canonical orbitals can not be very large . in mesh representations , @xmath24 is proportional to the volume of the box while @xmath55 is proportional to the volume of the nucleus . the latter is @xmath56 times as small as the former in typical calculations . situation is quite different in the quasiparticle formalism : the number of quasiparticle states is proportional to the volume of the box . this is because the localization of the density demands only that of @xmath57 through eq . ( [ eq : dens_hfbq ] ) while @xmath58 does not have to be localized in general . the orthogonality condition ( [ eq : norm_qp ] ) involves both @xmath57 and @xmath58 . this enables many quasiparticle states having similar @xmath57 to be orthogonal to each other by differing their @xmath58 . in this paper , we employ the skyrme interaction @xcite , which is a density- and momentum - dependent zero - range interaction : @xmath59 where @xmath60 is the spin exchange operator , @xmath61 with @xmath62 and @xmath63 the position vectors of the interacting two nucleons , @xmath64 is the nucleon density at @xmath62 ( @xmath65 ) , and @xmath66 @xmath67 is the relative momentum between the two nucleons . since it is hermitian under the box ( vanishing and periodic ) boundary conditions , we have not specified in which way ( to bra or ket states ) it operates in eq . ( [ eq : mf_int ] ) unlike in , e.g. , ref . zero - rangeness makes the mean - field potentials local , which is an essential advantage for coordinate - space solutions . among the terms of the skyrme force , those with strength parameters @xmath68 , @xmath69 , and @xmath70 act only on s - wave relative motions . the term with @xmath69 serves to take into account the effects of the finite - rangeness in terms of the lowest order momentum dependence , while the term with @xmath70 expresses a density dependence . the term with strength @xmath71 acts on relative p - wave states through a different form of momentum dependence . note that the isospin channel ( @xmath72=0 or 1 ) is uniquely determined through the requirement of the two - body antisymmetry . owing to the p - wave as well as the s - wave interaction terms , the skyrme force can act on all the four kinds of spin - isospin channels . the complete skyrme interaction includes a spin - orbit term @xmath73 , which we have excluded from eq . ( [ eq : mf_int ] ) . this elimination decreases the size of the numerical calculations by a factor of four ( two from spin up and down components , another two from real and imaginary parts of the spatial wave function ) . due to this size reduction , we can perform very severe tests of the canonical - basis hfb method by , e.g. , taking into account unusually many canonical orbitals or expressing wave functions on a very large mesh . as the parameters of the force , we choose the siii set @xcite except for the spin - orbit term , i.e. , @xmath68 = @xmath74 mev @xmath75 , @xmath70 = @xmath76 mev @xmath77 , @xmath78 , and @xmath79 . as for the parameters appearing in the momentum dependent terms , the following two combinations are sufficient to treat @xmath80 even - even nuclei , @xmath81 whose values are @xmath82 = 44.375 mev @xmath83 and @xmath84 = @xmath85 mev @xmath83 . as is usually done , we employ different parameters between the mean - field and the pairing interactions . we use the following interaction for the pairing channel , @xmath86 , \label{eq : pair_int } \ ] ] where @xmath87 is the strength and @xmath88 is the projector into spin - singlet states . terms in the braces represent dependences on particle and pairing densities . the interaction becomes repulsive where @xmath89 or @xmath90 . for the particle - density dependence , we use @xmath91 = 0.32 @xmath92 to vanish the volume - changing effect@xcite ( this choice , twice of the matter density , is also recommended for a different reason @xcite . ) the dependence on the pairing density @xmath93 ( to be defined by eq . ( [ eq : pairingdensity ] ) ) has been introduced in ref . @xcite to prevent unphysical behaviors of high - lying canonical orbitals discussed in appendix [ sec : pointcollapse ] . this term is squared because @xmath93 can become negative in principle . except in appendix [ sec : pointcollapse ] , we use @xmath94 . the momentum dependent term acts on s - wave relative motions and quenches interactions between nucleons in high - momentum states . the interaction vanishes at relative momentum @xmath95 at low - density points . if @xmath96 , this pairing force is reduced to that introduced in @xcite . the typical values of the parameters are @xmath97 mev @xmath75 and @xmath95=2 @xmath98 but different values are also used . the @xmath99=0 and @xmath72=1 part of the skyrme force of eq . ( [ eq : mf_int ] ) may be rewritten as eq . ( [ eq : pair_int ] ) without the pairing density dependent term . using different parameterizations between the two forces has an advantage of avoiding confusions . when one considers both protons and neutrons , the state of the nucleus is usually assumed to be a product of two bcs - type states of eq . ( [ eq : vac_cbhfb ] ) , one for the protons and the other for the neutrons : @xmath100 where the index q distinguishes between protons ( p ) and neutrons ( n ) . @xmath101 creates a proton having a wave function @xmath102 while @xmath103 creates a neutron with a wave function @xmath104 . this product form matches our choice of the pairing interaction of eq . ( [ eq : pair_int ] ) which acts only on @xmath105 pairs . in this paper we treat only @xmath5=@xmath106 nuclei without coulomb interaction for the sake of simplicity . in this case , the wave functions are the same between protons and neutrons . moreover , because the potentials are independent of the spin , the wave functions @xmath33 can be factorized into a product of a spin wave function and a real - number function of the position , which we write @xmath107 hereafter . more specifically , we assume @xmath102 = @xmath104 = @xmath108 and @xmath109 = @xmath110 = @xmath111 . the isoscalar pairing may take over the isovector pairing in @xmath112 nuclei @xcite . nevertheless , we assume eq . ( [ eq : vac_pn ] ) because the aim of this paper is to examine a method whose most important applications are to the nuclei near the neutron drip line . our hamiltonian @xmath113 consists of a kinetic energy term and two - body interaction terms . it should be understood that both are expressed in the second quantized form because hfb states do not have a fixed number of particles . the kinetic energy term is @xmath114 in the one - body expression , for which we assume the mass of a nucleon @xmath53 to be the average of the proton and the neutron bare masses ( @xmath115 and @xmath116 ) with an approximate correction factor for the center - of - mass motion of a nucleus of mass number @xmath117 : @xmath118 for the interaction terms , we use the interaction ( [ eq : mf_int ] ) for the mean - field type contractions and the interaction ( [ eq : pair_int ] ) for the pairing type contractions in evaluating the matrix elements of the two - body interactions using wick s theorem . the total energy for the state ( [ eq : vac_pn ] ) can be expressed in terms of a single space integral as , @xmath119 where the integrand is given by @xmath120 . \label{eq : ham_dens } \ ] ] on the right - hand side of the above equation , functions of position @xmath121 are @xmath122 which are called @xcite the ( particle ) density , the kinetic - energy density , the pairing density , and the pairing kinetic - energy density , respectively . there is a set of single - particle operators ( @xmath123 , @xmath124 , and @xmath125 ) which are useful to obtain mean - field solutions . they can be obtained by taking the functional derivative of @xmath126 with respect to a canonical wave function @xmath19 . this turns out to be equivalent to operating a state - dependent single - particle hamiltonian @xmath123 to the wave function : @xmath127 here , we have taken the familiar point of view that @xmath19 and @xmath128 can be formally regarded as independent variables , although @xmath19 is actually a real function . one can express @xmath123 , in turn , as a linear combination of state - independent single - particle hamiltonians as , @xmath129 where @xmath124 and @xmath125 are called the mean - field and pairing hamiltonians . they are given by @xmath130 with @xmath131 .\end{aligned}\ ] ] we call @xmath132 and @xmath133 the mean - field and the pairing potentials , respectively . when there are no pairing correlations , the mean - field hamiltonian @xmath124 is identical to the hf single - particle hamiltonian . therefore , its expectation value , @xmath134 may be called the single - particle energy of the @xmath46th canonical orbital . the negative of the expectation value of the pairing hamiltonian , @xmath135 has a meaning of the pairing gap of the orbital . the pairing gap should be positive or zero . indeed , @xmath136 could take negative values only if the wave function would consist of very high - momentum components @xcite . a related discussion is given in sec . [ sec : canorb ] . the quasiparticle states of eq . ( [ eq : ani_op_qp ] ) are the eigenstates of their own hamiltonian , i.e. , so - called hfb super matrix composed of @xmath124 and @xmath125 : @xmath137 where @xmath52 is the fermi level . in small subspaces like several major shells , the easiest way to obtain an hfb solution is to solve eq . ( [ eq : hfb_matrix ] ) as an eigenvalue problem of a hermitian matrix . in large spaces like mesh representations , however , it is easier to obtain the canonical orbitals directly , rather than via quasiparticle states . such a direct method is explained in the next section . incidentally , if one considers seriously that @xmath19 is a real function , one may notice that the functional derivative should be doubled since @xmath138 as @xmath139 and @xmath140 . ( [ eq : commentfuncderiv ] ) seems to be a more appropriate expression of what our numerical calculation program actually does . however , the factor @xmath141 at the beginning of the right - hand side as well as the factor @xmath142 which reflects the spin - isospin degeneracy do not matter because they can be conveniently absorbed into the step - size parameter of the gradient method ( see sec . [ sec : gradientmethod ] ) . in this section we discuss on the cut - off schemes for zero - range pairing interactions in relation to the canonical - basis hfb method . delta function forces without a cut - off lead to a divergence of the strength of the pairing correlation . in order to prevent the divergence , one usually takes only those quasiparticles whose excitation energy @xmath143 is lower than some cut - off energy @xmath144 in constructing the ground state @xcite . namely eq . ( [ eq : vac_hfbq ] ) is modified to @xmath145 where @xmath146 is the heaviside ( step ) function . in this paper we do not discuss how one should change the strength as a function of @xmath147 for renormalization . instead , we regard @xmath147 as one of the constants to define the force . in order to implement the cut - off , one should pay attention to the following two peculiarities of the canonical - basis hfb method . i ) the introduction of @xmath147 causes a serious inconvenience to this method , i.e. , the absence of the quasiparticle hamiltonian . for this reason , it is preferable to use the number of canonical orbitals @xmath55 instead of @xmath147 . ii ) for delta - function pairing interactions , substitution of @xmath147 with @xmath55 leads to a physically meaningless situation , in which canonical orbitals are collapsed to a spatial point ( a point collapse phenomenon , hereafter ) . to overcome this difficulty , we modify the pairing interaction by adding a repulsive momentum - dependent term . in the rest of this section we discuss these two problems . \i ) let us explain how the introduction of the cut - off causes the absence of the quasiparticle hamiltonian . an analogy with the bcs approximation indicates a natural way to introduce an energy cut - off to the canonical - basis method . namely , the smooth cut - off method@xcite modifies the seniority interaction as @xmath148 where @xmath149 is a function of single - particle energy @xmath150 and takes on @xmath151 well below a chosen cut - off energy and smoothly becomes zero above it . the smoothness is indispensable to the stability of the solution . in analogy , we may modify the pairing density as @xmath152 with @xmath153 in eq . ( [ eq : cut_func_k ] ) , we assume a dependence on the kinetic energy ( @xmath154 ) , not on the mean - field energy @xmath150 as in eq . ( [ eq : cut_sen_force ] ) , to avoid a highly complicated expression of the gradient for the latter case . ( in bcs , such complications are simply neglected . ) the gradient is given for @xmath155 by @xmath156 \psi_i .\ ] ] this method works well to prevent the point collapses in numerical tests using @xmath157 fm taken from the gogny force @xcite . this cut - off scheme has an disadvantage that the hfb super matrix in eq . ( [ eq : hfb_matrix ] ) can not be defined , because in eq . ( [ eq : grad_cutoff ] ) the pairing hamiltonian ( the part proportional to @xmath158 in square brackets ) is dependent on the canonical - orbital index @xmath46 . this means that there is no common pairing hamiltonian to all the quasiparticles . consequently one does not have well - defined bogoliubov quasiparticle states and can not utilize them to refine the hfb solution , which is a rather serious drawback as discussed in the next section . if one can use @xmath55 in place of @xmath147 , however , one can recover a unique quasiparticle hamiltonian . incidentally , a similar kind of ambiguity arises from the cut - off scheme in terms of the quasiparticle energies according to eq . ( [ eq : vac_hfbq_cut ] ) . in this case , the canonical orbitals do not have well - defined hamiltonians . it may not matter , however , because one does not need hamiltonians to define canonical orbitals when they can be obtained by transforming the quasiparticle states . \ii ) let us discuss the phenomenon of the point collapse . in quasiparticle ( or two - basis ) hfb method , specifying the number of quasiparticle ( hf single - particle ) states to be considered has the same consequences as imposing an energy cut - off because the states to be chosen are anyway the eigenstates of the lowest energies of the quasiparticle ( hf ) hamiltonian . on the other hand , the canonical - basis hfb method is very peculiar concerning this point . the canonical orbitals are not the eigenstates of a single operator but each of them has its own hamiltonian @xmath123 as a function of the occupation probability given by eq . ( [ eq : statedependenthamiltonian ] ) . @xmath123 becomes @xmath124 and @xmath125 in the two extreme situations @xmath159=1 and 0 , respectively . it is only when one considers a small number of orbitals that all of them are the lowest - energy approximate eigenstates of @xmath124 . otherwise , some of the orbitals may become the eigenstates of @xmath125 rather than @xmath124 : we show in appendix [ sec : pointcollapse ] that , if the occupation probability of an orbital becomes sufficiently small , it spontaneously decreases the probability further toward zero ( to decrease the total energy of the nucleus ) and change its hamiltonian to @xmath125 . this occurs when the pairing force is a delta function ( with or without particle - density dependence ) , for which @xmath125 is a local potential and thus its eigenstates are delta functions in the spatial coordinate . this means that their @xmath160 is infinitely high . on the other hand , the rest of the orbitals are left in low - energy subspace of @xmath124 . therefore the restriction on @xmath55 results in utterly different solutions from those obtained by using a cut - off energy explicitly . this peculiarity of the canonical - basis method is essential to the implementation of the method . in the quasiparticle and the two - basis methods , the unphysical divergence of the pairing correlation strength occur only in the limit of @xmath161 ( supposing @xmath55 is used instead of @xmath147 ) . in the canonical - basis method , in addition to this divergence , a different type of unphysical behavior ( the point collapse ) can also happen for @xmath55 fixed at finite numbers . one has to take special care of the latter problem if one tries to implement the canonical - basis method . now , if one can suppress the point collapse , one is allowed to use @xmath55 as the control parameter of the cut - off energy . appendix [ sec : pointcollapse ] shows that this can be achieved , e.g. , by introducing a sufficiently strong repulsive momentum dependence to the delta - function pairing interaction . with this term applied to finite ( small or large ) numbers of canonical orbitals , unphysical situations including the point collapse never happen . ( if it is applied to infinite number of orbitals , not the point collapse but the divergence occurs @xcite . ) in this paper we employ this momentum - dependent force . in addition to the role to enable a cut - off in terms of @xmath55 , equally important is the byproduct that it also clarifies the nature of high - lying canonical orbitals by providing a kinetic energy term to the pairing hamiltonian @xmath125 . further explanations using a numerical example are given in sec . [ sec : canorb ] . incidentally , appendix [ sec : pointcollapse ] also gives a discussion on an alternative way in terms of a pairing - density dependent force to enable cut - off in terms of @xmath55 . before ending this section , we mention the prospect about the renormalization of the pairing force strength . considering the recent quantitative success of the regularization method of delta - function forces based on the thomas - fermi approximation @xcite , we think it is worth trying in the canonical - basis method in future . however , it will require a special treatment . the thomas - fermi approximation makes the interaction strength a decreasing function of the particle density . one might expect this modification of the interaction would hinder the point collapse . actually , dependences on particle density can not prevent it because the collapse makes only the pairing density divergent but leaves the particle density unchanged ( because @xmath162 and @xmath163 as shown in appendix [ sec : pointcollapse ] ) . therefore one has to keep the momentum - dependent term . this will lead to a modification of the regularization procedure . we explain in this section the details of a procedure to obtain the canonical - basis solution of the hfb equation directly , not by way of quasiparticle states . what should be done is to minimize @xmath126 of eq . ( [ eq : tot_eng ] ) under constraints of eqs . ( [ eq : ortho_cbhfb ] ) and ( [ eq : num_cbhfb ] ) . equivalently , one may introduce a routhian , @xmath164 and minimize it without constraints . @xmath4 is probably the most familiar lagrange multiplier , whose physical meaning is the fermi level . in the definition ( [ eq : routhian ] ) , @xmath165 lagrange multipliers @xmath166 obeying hermiticity , @xmath167 are introduced instead of @xmath168 independent multipliers . the hermiticity ensures the equality between the number of constraints and the number of independent multipliers . moreover , it makes @xmath169 real so that two conditions , @xmath170 and @xmath171 , become equivalent and thus one has to consider only one of them . note that @xmath13 is subtracted from @xmath12 , in contrast to ref . this subtraction is necessary to treat @xmath166 not as constants like @xmath4 but as functionals of the wave functions . the stationary condition of @xmath169 results in two kinds of equations . one is @xmath172 , which concerns the occupation amplitudes @xmath173 and is fulfilled by @xcite @xmath174 among the double sign , the minus sign corresponds to the minimum . the relative sign between @xmath173 and @xmath175 = @xmath176 should be plus , i.e. , @xmath177 ( unless @xmath178 ) . the stationary condition for a wave function @xmath107 leads to the following equation : @xmath179 owing to the state dependence of @xmath123 , the orthogonality condition becomes nontrivial to fulfill . in hf , the orthogonality is automatically satisfied because @xmath19 are eigenstates of the same hermite operator @xmath124 , which are orthogonal to one another . the orthogonalization procedure is needed only because of the instability for decaying into pauli - forbidden configurations . on the other hand , in the canonical - basis hfb method , the orthogonalization is indispensable and the explicit functional form of @xmath166 is the most important secret of the method . reinhard _ et al . _ have proposed @xcite @xmath180 let us give our reasoning on how the above form can be deduced . this consideration plays the crucial role in order to modify the form for the sake of a faster convergence . the requirement that eq . ( [ eq : gradient ] ) must hold at the solution ( where @xmath181 ) means , @xmath182 by taking the overlap with @xmath183 , one obtains , @xmath184 eqs . ( [ eq : lambda1 ] ) and ( [ eq : lambda2 ] ) are equivalent at the solution because @xmath166 is defined to be hermite by eq . ( [ eq : hermiticity ] ) . however , one should employ eq . ( [ eq : lambda1 ] ) rather than eq . ( [ eq : lambda2 ] ) because the former , but not the latter , is hermite at points other than the solution . one can utilize the gradient method to obtain the hfb solutions in the canonical - basis formalism . let us describe this method briefly : small variations in @xmath19 and @xmath128 change the routhian as , @xmath185 which indicates the steepest descent direction to be , @xmath186 one takes a small distance movement toward this direction , i.e. , @xmath187 with @xmath188 by repeating the evaluation of @xmath189 and the movement , one eventually reaches the minimum of @xmath169 . incidentally , in the braces on the right - hand side , the term including @xmath190 in the second member of eqs . ( [ eq : gradient ] ) is dropped because we know empirically that the orthogonality is stable without this term . in eq . ( [ eq : gradstep ] ) , @xmath191 is a parameter introduced to control the size of a movement . one should regard that the prefactors on the right - hand side of eq . ( [ eq : commentfuncderiv ] ) are included in this parameter . one may call @xmath191 the imaginary - time - step size because the first order approximation in @xmath191 of the imaginary - time evolution method @xcite becomes essentially the same procedure as the gradient method : @xmath192 in order to justify the negligence of the higher - order terms in eq . ( [ eq : imag_evo ] ) , it must hold@xcite @xmath193 where @xmath194 is the maximum kinetic energy ( used as the upper bound of the single - particle energy ) . for the cartesian mesh representation with a mesh spacing @xmath195 , @xmath196 where @xmath197 is given by eq . ( [ eq : mf_b ] ) . by choosing @xmath64 between @xmath198 ( free space ) and @xmath64=0.16 @xmath92 ( nuclear matter density ) , one obtains @xmath194 = 1254 mev for the siii force with @xmath199 @xmath98 . for the calculations shown in this paper , we have used @xmath200 , i.e. , @xmath201 sec . kinetic energy also arises from the pairing hamiltonian @xmath125 . however , an estimation by replacing @xmath197 of eq . ( [ eq : ekinmax ] ) with @xmath202 of eq . ( [ eq : pair_b ] ) results in an order of magnitude smaller value . what is presented above may be called a nave version of the gradient method . the convergence to the hfb solution turns out to be very slow with this nave method . a numerical example will be given in sec . [ sec : conv ] . the origin of this slowness can be traced back to the factor @xmath158 in @xmath123 , which is just a linear combination of @xmath124 and @xmath125 with coefficients @xmath159 and @xmath203 . the effects of @xmath123 can be very weak for canonical orbitals having small @xmath173 , which leads to the smallness of their changes in a gradient - method step . now , steepest - descent paths depend on the choice of the variables . for example , eq . ( [ eq : gradstep ] ) is obtained when one uses @xmath19 as the variables . if one uses norm - resized wave functions @xmath204 , where @xmath205 is a scaling factor , a gradient - method step becomes @xmath206 , which is equivalent to @xmath207 thus the path is changed . by choosing @xmath205 to cancel the small factors in @xmath123 , one can accelerate the otherwise slow convergence , which we call the accelerated gradient method . in this paper , we take @xmath208 where the factor @xmath209 may be chosen empirically to maximize the speed of convergence . we choose @xmath210 . this factor seems to take care of the fact that @xmath125 is smaller than @xmath124 by an order of magnitude . with this choice for the scaling factor , @xmath211 for deeply bound orbitals and @xmath212 for high - lying orbitals . thus , all the orbitals evolve at roughly the same pace . incidentally , by using the approximate equality symbol in eq . ( [ eq : alphai ] ) , we mean the utilization of some common recipes for numerical calculations like taking a moving average and restricting on the maximum values . it should be noticed here that one should also modify the form of multipliers @xmath166 when one introduces the acceleration factors @xmath213 . in principle one may reach the solution by using eq . ( [ eq : lambda1 ] ) because transformations of variables do not change the location of the minimum of @xmath169 . nevertheless , one should modify eq . ( [ eq : lambda1 ] ) for a practical reason . in calculating the gradient vector whose exact form is given by the second member of eqs . ( [ eq : gradient ] ) , the last term takes much more computing time than the first two terms due to @xmath214 . one can drop the last term if the orthogonality relation ( [ eq : ortho_cbhfb ] ) is fulfilled along the path of the evolution . let s suppose that the relation is satisfied before a gradient - method step is taken and require that it is conserved without the dropped term to the first order in @xmath191 after the step , i.e. , @xmath215 with @xmath216 and @xmath217 given by eq . ( [ eq : grad_psi ] ) . substituting @xmath19 and @xmath218 in eq . ( [ eq : orthog_path ] ) and requiring the hermiticity ( [ eq : hermiticity ] ) result in @xmath219 ) , fulfills the requirement that it should agree with the expression ( [ eq : lambda2 ] ) at the solution . the original form differs from the new form , however , before reaching the solution if @xmath220 . consequently , only the new form conserves the orthogonality during the course of the evolution . we have indeed suffered from large errors of orthogonality by using the original form . on the other hand , by using the new form , we have observed that the error does not grow but decreases without performing explicit orthogonalizations . the reason for this stability will probably be found in the second order terms in @xmath191 neglected in eq . ( [ eq : orthog_path ] ) . there is an auxiliary method to speedup the convergence . it is to diagonalize the hfb super matrix given by eq . ( [ eq : hfb_matrix ] ) in the subspace spanned by @xmath55 canonical orbitals . then one obtains @xmath55 quasiparticle states with positive energy , whose two - component wave functions defined in eq . ( [ eq : ani_op_qp ] ) are expressed as , @xmath221 where @xmath222 is the @xmath46th normalized eigenvector of eq . ( [ eq : hfb_matrix ] ) . the hfb ground state is constructed as the vacuum of these quasiparticles as in eq . ( [ eq : vac_hfbq ] ) . by diagonalizing the one - body density matrix of this vacuum in this subspace , @xmath223 , one obtains a renewed set of canonical orbitals as the eigenvectors and @xmath159 as the eigenvalues . if the initial set of canonical orbitals are taken from the exact solution , the renewed and the initial sets are identical . however , if the initial set is taken before the state converges to the solution , the renewed set corresponds to a better solution than the initial set . it is because the above procedure gives the exact variational minimum @xcite in the subspace spanned by the initial canonical orbitals . the gradient - step method takes care of both the variation inside this subspace and the optimization of the subspace itself . the diagonalization performs only the former part but perfectly in a single step . this diagonalization method is not indispensable but useful to obtain the solutions . it makes the convergence more robust and somewhat quicker if it is performed after every @xmath224100 gradient - method steps . its effect seems to saturate with this interval since using shorter periods does not lead to noticeable improvements . a numerical example is given in sec . [ sec : conv ] . incidentally , the idea of this diagonalization method originates in the two - basis formalism of hfb @xcite , in which the quasiparticle hamiltonian is diagonalized in the subspace spanned by low - lying hf orbitals ( eigenstates of @xmath124 ) , not the canonical orbitals as in this paper . the relation between these quasiparticles and the true quasiparticles defined in the full space is discussed in sec . [ sec : cbqp ] . in this section , we show the results of numerical calculations using a newly developed computer program @xcite of the canonical - basis hfb method in a three - dimensional cartesian mesh representation . the parameters of the model are as follows : for the mean - field interaction , skyrme siii force is employed but its spin - orbit term and the coulomb force are not included . for the pairing interaction , @xmath225 @xmath92 and @xmath226 are used . the value of @xmath87 is changed depending on @xmath95 (= 1.7 5 @xmath98 ) and @xmath55 (= 14 210 ) to make the average pairing gap to be 2 2.5 mev . unless the values are specified , @xmath95=2 @xmath98 , @xmath227 mev @xmath75 , and @xmath228 are used . with @xmath228 , the number of orbitals amounts to three times as large as the number of the nucleons , since each orbital can be occupied by four nucleons due to the spin - isospin degeneracy . the calculated state is the axially symmetric prolate solution of @xmath229si@xmath230 . the single - particle wave functions are expressed on a cartesian mesh having ( in the default set up ) @xmath231 mesh points with mesh spacing @xmath195=0.8 fm . as the boundary condition , we assume a cubic box whose infinitely high walls are located at the 0th and the 30th mesh points . the center of mass of the nucleus is placed at the center of the box . the 17-point formulae @xcite are employed to calculate the first and second derivatives . in applying these formulae , wave functions are assumed to be antisymmetrically extended beyond the walls . spatial integrals are done using the mid - point ( = trapezoidal ) rule . in fig . [ fig : conv ] , examples of the convergence curves are shown for several versions of the gradient method . the calculations are done using the default parameters . the initial wave functions of the canonical orbitals are taken from the eigenstates of the standard harmonic oscillator potential with a quadruple deformation of @xmath232 and @xmath233 . the @xmath234 symmetry is conserved to a very good accuracy during the course of the gradient - method evolutions . for the size of an imaginary time step , we use @xmath200 , defined by eq . ( [ eq : delta_tau ] ) , for all the curves . this value of @xmath235 roughly optimizes the convergence speed for the fastest method . for slower methods , @xmath235 may be slightly larger but such a small change does not affect our discussions below . in connection with the step size , we also adopt a recipe from the hf+bcs method of ref . @xcite , according to which the changes of @xmath124 and @xmath125 after each gradient - method step should be further damped compared with the changes of @xmath19 ( by a factor of 0.4 in our case ) . it is a precaution against occasional instabilities like oscillating behaviors . this additional damping does not affect our conclusion on the comparison of the methods , either . si . see text for explanations . , width=8 ] the top portion of fig . [ fig : conv ] shows the total energy @xmath126 of eq . ( [ eq : tot_eng ] ) . the middle portion shows the maximum over @xmath236 of the error of the second equality of eqs . ( [ eq : gradient ] ) ( neglecting the contribution from the error of orthogonality ) , @xmath237 which is an indicator of the accuracy of hfb solutions . as for hf , this quantity is reduced to the maximum over @xmath238 of @xmath239 which is just the energy width of @xmath240 for @xmath124 . the bottom portion shows the mass quadrupole deformation parameter @xmath241 defined as the general definition @xcite times @xmath242 disregarding nucleon form factors . hf result for @xmath241 , converging to 0.44 , is omitted . the abscissae are common to all the portions and designate the number of gradient - method steps . thin solid lines in all the three portions are the result of the nave ( i.e. , with @xmath243 ) gradient method . these curves demonstrate that one can indeed obtain an hfb solution with the canonical - basis hfb method on a mesh , because after a million steps , @xmath126 and @xmath241 appear to reach a plateau and @xmath244 becomes as small as 0.1 kev . we have also obtained similar convergence curves for other quantities like the axial asymmetry @xmath245 and the r.m.s . radius . there is a serious problem , however , of the obvious slowness of the convergence . dot curves show the convergence to an hf solution with pairing interactions turned off . ( with `` w / d '' in the legend in the top portion of the figure we mean that diagonalization of @xmath124 is done in the subspace of occupied @xmath246 hf orbitals to renew the orbitals after every 100 gradient - method steps . without this diagonalization , @xmath19 are not hf orbitals and do not satisfy @xmath247 . this diagonalization also makes the convergence somewhat quicker . ) one can see that the hf energy reaches a plateau by three orders of magnitude faster than the nave hfb method . in the middle portion , while hf can achieve precision of 0.1 kev with only 1400 steps , hfb requires 55000 steps for 10 kev , @xmath248 steps for 1 kev , and @xmath249 steps for 0.1 kev precisions . a method to improve the convergence speed is the diagonalization of the hfb super matrix in the subspace spanned by @xmath55 canonical orbitals . dot - dash curves are obtained by performing this diagonalization after every 100 steps . one can see that the convergence speed is improved by an order of magnitude . however , this progress is not satisfactory yet compared with the hf curve . dash curves uses the accelerated gradient method ( i.e. , @xmath213 , expressed as `` w / a '' in the legend ) , which brings about a speedup by two orders of magnitude . a comparison with dot - dash curves suggests that the variation of the canonical - basis subspace itself is a more important factor than a minimization inside the subspace . using both the diagonalization and the acceleration methods leads to thick solid curves ( expressed as `` w / a , d '' in the legend ) . steep changes in every 100 steps are due to the diagonalizations . one can see that these curves converge almost as quickly as the dot curves of the hf method . these results demonstrate that canonical - basis hfb can be solved without very heavy numerical computations by adopting the two improvements described above . in this subsection , we use fig . [ fig : canorb ] to discuss the properties of the canonical orbitals in connection with the mean - field and the pairing hamiltonians . an analysis using these two hamiltonians leads to a comprehensive understanding of the nature of high- as well as low - lying canonical orbitals . the calculation has been done for @xmath250si with @xmath251 canonical orbitals , which is 30 times as many as the number of the orbitals below the fermi level . for the combination of this @xmath55 and @xmath252 @xmath98 , we have adjusted the strength of the pairing force to be @xmath253 mev so that the average pairing gap have a reasonable size , 2.0 mev . the result shown in fig . [ fig : canorb ] has been obtained after @xmath254 gradient - method steps : because of the slow convergence of very high - lying orbitals ( even using the accelerated method ) , we continued the gradient - method evolution until no changes are going on in any of @xmath150 . the last change took place during @xmath255th to @xmath256th steps , in which two orbitals gradually increased their energies from @xmath257 mev to @xmath258 mev . during this change @xmath244 increased slightly up to @xmath259 mev . in the last @xmath260 steps there were no changes and @xmath244 was decreased to @xmath261 mev . the top - left panel shows the logarithm of the occupation probability @xmath159 . the abscissa ( common to all the four panels in the left column ) is the label @xmath46 of canonical orbitals , which are sorted in the descending order of @xmath159 . first seven orbitals have occupation probabilities close to unity . above them , the decrease of the probability looks rather steady . it is in fact a result of a mixing of several sequences : the top - right panel plots the same @xmath159 data versus @xmath150 ( only @xmath262 part is shown ) , in which one can see three sequences . let us denote the number of the oscillator quanta by @xmath263 . we have confirmed that the top sequence corresponds to @xmath264 subshells ( @xmath265 are completely seen in @xmath266 region ) . the lower bunch of dots are for @xmath267 . for @xmath268 mev , @xmath269 sequence is bifurcated from @xmath270 sequences . these observations indicate that the canonical orbitals have the same shell structure as that of the harmonic oscillator at least up to several tens of mev . then , a question arises what generates this shell structure . the second and third top portions give clues to answer this question . the panels in the second top row show the mean - field hamiltonian @xmath124 and its expectation value and width for each canonical orbital . in the right panel , the mass term @xmath271 and the potential term @xmath132 of @xmath124 are plotted as functions of the distance from the center of the nucleus . their profiles in the symmetry ( @xmath6 ) axis as well as those in the equatorial ( @xmath272 ) plane are shown . in the left panel , mean - field energy of each canonical orbital is designated using a dot and a vertical line . the dot corresponds to the expectation value @xmath150 while the vertical line connects between @xmath273 with @xmath274 one can see that first seven orbitals , which are below the fermi level , have very narrow widths ( 30 760 kev ) : they are approximate eigenstates of @xmath124 , i.e. , the ( occupied ) hf orbitals . above the fermi level , the width becomes larger with increasing expectation value and thus the wave functions of canonical orbitals begin to diverge from those of ( unoccupied ) hf orbitals . at the crossing point from negative to positive @xmath150 , there seems to be only a smooth continuation of this trend . this is an essential difference between the hfb canonical and the hf orbitals . concerning the hf orbitals , @xmath150 may be estimated in the thomas - fermi approximation , where the number of levels of @xmath124 below @xmath275 is given as , @xmath276 the widths of hf orbitals are always zero . the solid curve designates the function @xmath277 . it is owing to the finite volume of the normalization box that @xmath278 is not infinite for positive @xmath275 . one can see a distinct difference between this curve for the hf orbitals and the dots for the hfb canonical orbitals : the level density is by far sparser in the latter . ( it is indeed the reason to use canonical orbitals as discussed in sec . [ sec : cbhfb ] . ) it seems difficult to relate the level structure of canonical orbitals to @xmath124 . incidentally , one can see a bifurcation of the sequence of the dots ( @xmath46 , @xmath150 ) at @xmath279 . its origin is the subshell structure : upper sequence corresponds to high - angular momentum ( @xmath280 ) subshells while the lower one to @xmath267 subshells . the third panels from the top show similar graphs to the second top panels but for the pairing hamiltonian @xmath125 . in the right panel , attention should be paid to the scale of the ordinate , which is smaller by an oder of magnitude than that for @xmath124 . it is because the expectation value of @xmath125 is nothing but the negative pairing gap ( i.e. , @xmath281 ) and thus its size is only a few mev . in the left panel , dots are plotted at @xmath282 and vertical lines connect between @xmath283 , where @xmath284 one can see an opposite trend of the width between @xmath124 and @xmath125 : in the former ( latter ) , the width is smaller for lower ( higher ) expectation values . it suggests a view that high - lying canonical orbitals are closely related to @xmath125 . this can be understood in terms of the state dependent hamiltonian , @xmath285 . because of this form , it is close to @xmath124 ( @xmath125 ) for deeply bound ( highly excited ) canonical orbitals . the solid curve designates the thomas - fermi estimation of @xmath286 , the number of levels of @xmath125 below @xmath287 . @xmath288 can be obtained by replacing @xmath289 with @xmath290 , respectively , in eq . ( [ eq : thomas_fermi ] ) . one can see that this curve agrees quite well with the dots . ( sorting the orbitals more appropriately in the ascending order of @xmath282 makes the agreement better . ) it supports the view that @xmath125 roughly determines the level structure of canonical orbitals . it also guarantees that the application of the thomas - fermi approximation to @xmath125 provides a rather precise method to count the number of canonical orbitals . one can see another difference between @xmath124 and @xmath125 that @xmath150 becomes positive for @xmath291 while @xmath282 is still negative at @xmath292 . it means that @xmath125 has by far more number of bound states than @xmath124 . its origins in the thomas - fermi approximation are the larger ratio @xmath293 than @xmath294 inside the nucleus and the vanishing behavior of @xmath295 as @xmath296 . it is an interesting question whether there are finite or infinite number of canonical orbitals with @xmath297 . non - zero occupation probability means @xmath298 and thus @xmath299 . therefore one has to count the number of orbitals with negative @xmath287 . to give a thomas - fermi estimation of this number , let us suppose the form of density tails to be , @xmath300 for the kinetic - energy density inside the nucleus , a reasonable estimation is @xmath301 , where @xmath302 is the fermi momentum . using a relation @xmath303 and assuming @xmath304 appropriate in the peripheral region , one can show that @xmath305 is independent of @xmath2 . consequently , the number of orbitals becomes proportional to the volume of the integral region @xmath306 . the explicit form of the result is @xmath307 therefore , the number of non - zero - occupation canonical orbitals is infinite . the bottom panels show the radial distributions of the nucleon density ( right panel ) and that of each canonical orbital ( left panel ) . the former is shown in two directions while the latter is integrated over the angles as , @xmath308 and thus normalized as @xmath309 . vertical lines are drawn in the interval of @xmath2 where @xmath310 @xmath98 . the width of the lines are proportional to @xmath311 , except that the width is saturated at 0.8 ( in the scale of the abscissa ) for @xmath312 @xmath98 . one can see that the canonical orbitals are localized in the neighborhood of the nucleus and only gradually shifted outward with increasing @xmath46 . it is not due to the finite box size because even the 210th orbital is located far from the boundary , which is at 12 to 12@xmath313 fm . the location of the maximum of @xmath311 agrees fairly well with the classical turning point of @xmath133 for energy @xmath282 ( as seen in the third top panel in the right column ) . this vindicates the view that @xmath125 determines the spatial extent of high - lying canonical orbitals . let us give another fact to confirm this view . evaluating eq.([eq : gammazero ] ) using @xmath252 @xmath98 , @xmath314 @xmath98 ( see next subsection ) , and @xmath315 @xmath98 , one obtains @xmath316 = 0.14 @xmath92 . for @xmath317 , the value of @xmath306 becomes equal to the volume of a sphere of radius 7.1 fm . this roughly agrees with the fact that the 210th orbital has the maximum density at @xmath318 fm . the discussions of this subsection can be summarized into three points : i ) canonical orbitals well below the fermi level are localized by the mean - field potential . ii ) highly excited canonical orbitals are localized by the pairing potential . iii ) the pairing hamiltonian can have infinite number of localized orbitals due to the vanishing mass term at large radii . it should be noted here that we do not insist that @xmath125 alone determines high - lying orbitals to the details . one has to keep in mind an asymmetry between low- and high - lying orbitals . because lower - lying orbitals have much stronger influences on the total energy of the nucleus , the determinations of their wave functions are hardly affected by the orthogonality conditions with the higher - lying orbitals . however , the opposite is not true : high - lying orbitals must be orthogonal to the orbitals below the fermi level , which are determined mainly by @xmath124 . consequently , although the most essential points can be explained by the idea of the change of the hamiltonian from @xmath124 to @xmath125 , there may still remain something delicate in the treatments of high - lying canonical orbitals compared with those of low - lying ones . wave functions of bound states have an asymptotic form for large radii of @xmath319 in hf , negative energy orbitals have the form ( [ eq : psi_ir ] ) with @xmath320 determined by @xmath150 as @xmath321 . in hfb , the hole component of quasiparticle wave functions ( @xmath322 ) has this form with @xmath323 determined by the excitation energy @xmath324 as @xmath325 @xcite . as for the canonical orbitals of hfb , however , one can not relate @xmath320 to any sort of energies in a simple manner due to the complex influences of the requirement of orthogonality , which is rooted in the lack of a common hamiltonian to all the orbitals . the only possible statement is that it is not larger than the smallest value of @xmath323 for quasiparticles , whose lower bound is given by the fermi level as in eq . ( [ eq : density_tail ] ) . in this subsection , we compute @xmath320 of canonical orbitals numerically using angle - averaged density : @xmath326 . in order to see the behaviors at large radii , we have used a large box containing @xmath327 mesh points . the edge of the box is @xmath328=46.4 fm . the nearest walls from the center of the nucleus is at a distance of 23.2 fm and the farthest ( i.e. , the corners ) at 40.2 fm . for the sake of using the large box , we restrict the number of orbitals to @xmath228 , for which an adequate strength of the pairing interaction is @xmath227 mev . figure [ fig : wf_tail ] shows @xmath329 of canonical orbitals , all of which have exponentially damping tails . this result demonstrates that the canonical - basis hfb method can convert gaussian tails of initial wave functions , for which we use eigenstates of a harmonic oscillator , into the correct exponential form . it is evident that the first orbital changes its slope at around @xmath330 fm . by looking at the other curves minutely , one can also find such changes in several other orbitals . therefore , we give two discussions , first on the behaviors near the nucleus and second on those far away . figure [ fig : wf_kappa ] shows @xmath320 as a function of @xmath150 . open circles denote @xmath320 determined by the logarithmic derivative of @xmath331 near the nucleus ( @xmath332 fm ) . one can see less than 21 circles because several pairs of circles overlap exactly due to the degeneracy for the sign of @xmath333 ( @xmath6-component of the orbital angular momentum ) . the dash and dot curves represent functions @xmath334 , and @xmath335 , respectively . the open circles seem to agree fairly well with a function @xmath336 which coincides with the dash curve for @xmath337 and the dot curve for @xmath338 . by noting that the dash curve expresses @xmath323 for bound hf orbitals , one may understand the @xmath337 part of eq . ( [ eq : kappa_i ] ) as indicating that canonical orbitals below the fermi level have approximately the same tails as hf orbitals . an alternative interpretation is also possible which refers to the full domain of eq . ( [ eq : kappa_i ] ) : canonical orbitals with @xmath150 = @xmath339 have the same @xmath323 as the hole component @xmath322 of a quasiparticle state with @xmath340 . it may suggest that canonical orbitals below ( above ) the fermi level are related to the hole components of hole - like ( particle - like ) quasiparticle excitations . however , in order to confirm it , we need quasiparticle states in the full space , which are outside the scope of this paper . cross symbols plotted in fig . [ fig : wf_kappa ] are @xmath320 calculated at a farther radius ( @xmath341 fm ) . in this case , concerning the deepest and highest orbitals , @xmath323 drops toward the minimum value at @xmath342 . this may be ascribed to the fact that the canonical orbitals are constructed by mixing up the hole part of all the quasiparticle states and thus the component having smallest @xmath323 dominates in all the canonical orbitals at large radii . probably , all the canonical orbitals have the same @xmath323 ( @xmath343 where @xmath344 is the smallest quasiparticle energy ) at sufficiently large radii . let us discuss on the quasiparticle states defined by eq . ( [ eq : hfb_matrix ] ) in the subspace spanned by @xmath55 canonical orbitals . these quasiparticles and the true quasiparticles defined in the full space are quite different . the former are only a set of tools for the minimization in a small subspace , while the latter have physical information on the excitation modes of the nucleus . the width @xmath345 ( in the full space ) of the excitation energy @xmath143 of the @xmath46th quasiparticle ( in the subspace ) is given by @xmath346 @xmath347 where the operation of @xmath124 and @xmath125 should be evaluated in the full space ( i.e. , in the mesh representation ) , not in the @xmath55-dimensional subspace . quasiparticles can be characterized by the norms of the particle and hole components as well as by the excitation energy . these norms are defined as @xmath348 for the particle and hole components , respectively . they are normalized as @xmath349 according to eq . ( [ eq : norm_qp ] ) . quasiparticle states are called to be particle - like ( hole - like ) if @xmath350 ( @xmath351 ) @xmath352 . figure [ fig : cbqpe ] shows @xmath324 and @xmath353 for @xmath251 ( with @xmath253 mev @xmath75 ) . one can see that hole - like quasiparticles have very small widths . even the deepest hole state ( 1s ) has a relatively small width ( 1.7 mev ) . on the other hand , the widths of particle - like quasiparticles amount to a few tens of mev : they are very different from the true quasiparticles defined in the full space . for the sake of comparison , quasiparticle levels obtained in a smaller subspace ( @xmath228 , @xmath227 mev @xmath75 ) are also shown in the top - right window . by comparing the results of two calculations , one can see that increasing the number of canonical orbitals leads to the addition of high - excitation particle - like quasiparticles , while it does not change very much the hole - like states and low - excitation particle - like states . thus , enlargement of the canonical - basis subspace is a very inefficient way to decrease the energy widths of particle - like quasiparticle states . the differences from true quasiparticles become more evident by looking at the wave functions . for @xmath354 , the particle component @xmath355 should be an oscillating function of @xmath2 @xcite . however , @xmath355 of quasiparticles defined in the canonical - orbital subspace has an exponentially damping tail . it is only natural because they are expressed as linear combinations of such functions . it should be stressed again that the quasiparticles in the canonical - orbital subspace are nevertheless a useful auxiliary tool to obtain the hfb ground state . however , if one needs true quasiparticle excitation modes , the quasiparticles in the subspace are useless . one has to calculate with some other method the true eigenstates of the quasiparticle hamiltonian . a good news is that the hamiltonian ( being composed of @xmath124 and @xmath125 ) is already obtained and thus iterations for self - consistency are not necessary . this will make the numerical calculations rather inexpensive . our canonical - basis hfb method has several parameters to specify the pairing force , i.e. , the over - all strength @xmath87 , the parameter for the momentum dependence @xmath95 , the number of canonical orbitals having non - zero occupation probabilities @xmath55 , and parameters for density dependences . however , available experimental data do not provide sufficient information on the pairing correlation for precise determinations of so many parameters . therefore we tune only @xmath87 precisely so that the resulting pairing gap has a desired value , while assuming some reasonable values for the other parameters . however , this does not guarantee that their values may be chosen completely arbitrarily . in this and next subsections we discuss on the choices of the values of @xmath95 and @xmath55 , respectively . pairing - channel interactions were taken into account when skyrme forces sgii @xcite and skp @xcite were determined . their values of @xmath95 (= @xmath356 ) are 2.5 @xmath98 and 4.3 @xmath98 , respectively . from this result , values from 2 to 5 @xmath98 seem equally reasonable . from a pragmatic point of view to use the force in the canonical - basis method , @xmath95 must be smaller than 6 @xmath98 in order to avoid the point collapse ( see appendix [ sec : pointcollapse ] ) . values smaller than 1.5 @xmath98 are not preferable , either , because such small @xmath95 makes the solutions very difficult to obtain . if @xmath87 is adjusted for each value of @xmath95 to reproduce the same value of the average pairing gap , which we define @xcite as , @xmath357 resulting properties of the nucleus are almost independent of @xmath95 except the pairing density , which becomes more diffuse when smaller @xmath95 is used . unfortunately , pairing density is rather difficult to determine experimentally . incidentally , the properties of each canonical orbital are rather sensitive to @xmath95 : @xmath159 and @xmath358 increase while @xmath150 decreases with decreasing @xmath95 . only @xmath136 stays roughly constant . nevertheless , properties of the nucleus ( except the pairing density profile ) are hardly changed , as far as the average pairing gap is kept unchanged . from the allowed interval 1.5 @xmath98 @xmath359 6 @xmath98 , we have chosen 2 @xmath98 in this paper . there is no physical reason for this choice . however , smaller values of @xmath95 lead to a favorable situation that the dependence on @xmath55 becomes weaker and the determination of @xmath55 can be less precise . [ fig : nw_gap ] shows the dependence of @xmath360 of eq . ( [ eq : avegap ] ) on @xmath55 . solid circles , open circles , and triangles are obtained with @xmath95= 5 , 3 , and @xmath361 @xmath98 , respectively . the values of @xmath87 are given in the legends of the figure in units of mev @xmath75 . here , @xmath87 is not changed as a function of @xmath55 but is determined for each @xmath95 such that the resulting @xmath360 is roughly the same at @xmath362 . one can see in the figure that , for @xmath363 @xmath98 , @xmath364 is almost zero around @xmath365 . the emergence of this plateau is explained by the vanishing of the pairing interaction at relative momentum @xmath95 . for much larger @xmath55 , @xmath360 will increase again infinitely @xcite . we next study the dependence on the number of canonical orbitals @xmath55 . here we adjust the strength @xmath87 for each value of @xmath55 to keep @xmath360 constant . the used values of @xmath87 are shown in fig . [ fig : nw_pfs ] . in fig . [ fig : nw_etot ] we show the total energy of @xmath250si as a function of @xmath55 . it is obtained as follows : starting from harmonic oscillator eigenstates , we obtain an hfb solution for @xmath366 . then , we remove the canonical orbital with the smallest @xmath159 and again obtain the hfb solution for @xmath367 by the gradient method . by repeating this procedure , we obtain solutions for @xmath368 . this figure shows that the magnitude of the influence of @xmath55 on the total energy is 300 kev or less when @xmath87 is determined for the average pairing gap . other bulk properties of the nucleus are also roughly independent of @xmath55 : deformation parameters @xmath241 and @xmath245 are within @xmath369 and @xmath370 , respectively . r.m.s mass radius is @xmath371 fm . of the canonical orbitals on the number of the orbitals @xmath55 for @xmath250si . see text for explanations . , width=6 ] such insensitivity can also be seen in the properties of individual canonical orbitals . in fig . [ fig : nw_spl ] , mean - field energies @xmath150 of canonical orbitals are plotted versus @xmath55 . one can see that @xmath150 below 5 mev stays almost constant . higher levels are not so constant but only slightly decreasing . we have also confirmed that other quantities also remain roughly constant . therefore , unlike @xmath95 , the effects of changing @xmath55 is almost canceled by the adjustment of @xmath87 . finally we demonstrate the applicability of the canonical - basis hfb method to nuclei heavier than @xmath250si . we have encountered no special difficulties at least up to @xmath372 except the increase in the computation time per gradient - method step . for example , fig . [ fig : mass_gap ] shows the dependence of the average pairing gap @xmath360 on the mass number @xmath117 . in the calculations , the normalization box is gradually expanded with increasing @xmath117 : the edge of the box is at least 2.5 times as large as the liquid - drop - model diameter . we fix @xmath87 at @xmath373 mev @xmath374 . instead , we change @xmath55 versus @xmath117 . when we assume a relation @xmath375 , we can not reproduce the empirical trend that @xmath360 is a decreasing function of @xmath117 . we can improve the result by using different relations , e.g. , @xmath376 . however , the goal , a universal pairing force , should be more than an empirical formula expressing @xmath55 as a function of @xmath117 . it will be one of our future challenges . in this paper we have presented a method to obtain solutions of the hfb equation expressed in the canonical form , i.e. in terms of bcs - type wave functions , without using quasiparticle states . the method is fit to three - dimensional coordinate space representations and is advantageous to describe simultaneously arbitrary deformations , long and short density tails , and pairing correlations involving states in the continuum of the hf hamiltonian . we have improved the speed of the convergence to hfb solutions by a few orders of magnitude . for this purpose , we have modified the gradient method and derived the appropriate form of the lagrange multiplier functionals for the orthogonality between canonical orbitals . the two - basis method has also been adapted by replacing the hf basis with the canonical basis to further speedup the convergence . for delta - function pairing interactions , the wave functions of canonical orbitals have been shown to shrink infinitely into a spatial point once their occupation probabilities become less than some critical value . this problem is peculiar to the canonical - basis hfb method . to avoid it , a repulsive momentum dependent term has been added to the pairing interaction . as a byproduct of the introduction of this term , the nature of high - lying canonical orbitals has been greatly clarified as approximate eigenstates of the pairing hamiltonian . we have demonstrated that the level density and the spatial extent of high - lying canonical orbitals can be computed by applying the thomas - fermi approximation to the pairing hamiltonian . the obtained canonical orbitals have been examined in many aspects . one of the results is that canonical orbitals closer to the fermi level have wave functions with longer tails . at very large radius , however , all the orbitals seem to change the slope of their tails to that of the orbital closest to the fermi level . another result is that quasiparticle states can not be efficiently expanded in the canonical basis . therefore the canonical orbitals perfectly describing the ground state are not sufficient to treat excitations . the pairing force strength has been adjusted to reproduce the average pairing gap as a function of the number of canonical orbitals , which is used instead of an energy cut - off . properties of the nucleus have been found to be almost independent of the number of canonical orbitals and rather insensitive to the relative strength of the momentum dependent to independent terms of the pairing force . although most of our discussions refer to calculations for @xmath250si , we have also checked the applicability of the method to heavy nuclei up to @xmath117=252 . checks have also been done concerning the precision of the mesh representation in connection with this method . we believe that the canonical - basis hfb method has the potential to become the standard approach to treat neutron - rich nuclei in hfb . as a next step , we are going to extend the formalism and the computer program concerning the treatment of @xmath377 nuclei and the inclusion of the spin - orbit and the coulomb interactions . in this appendix , we examine the precision of the cartesian mesh representation when it is applied to the canonical - basis hfb method . discussions are given concerning the dependencies on i ) the mesh spacing , ii ) the approximation formulae for the derivatives , and iii ) the size of the normalization box . the pairing force strength used is @xmath378 mev @xmath75 with @xmath228 . the state is the prolate ground state of @xmath250si . \i ) we discuss the dependence on the mesh spacing @xmath195 . for this purpose , we calculate the hf and hfb solutions for various value of @xmath195 while keeping the size of the box constant . for the discrete approximation of the first and second derivatives , we use the 17-point formula @xcite . according to the calculation using the finest mesh ( @xmath379 fm ) , @xmath380 mev , @xmath381 mev , and @xmath382 . the mesh is changed in the following manner : we use @xmath383 mesh points in @xmath384 , @xmath385 , and @xmath6-directions . then the length of the edge of the normalization box is @xmath386 = @xmath387 . we fix @xmath386 at 24 fm so that increasing @xmath383 from 15 to 79 corresponds to decreasing @xmath195 from 1.5 fm to 0.3 fm . the error is defined as the deviation from the result with @xmath195=0.3 fm . the center of mass of the nucleus is placed at the center of the box . the top portion of fig . [ fig : mesh_dep ] shows the error of the total energy @xmath126 as a function of the mesh spacing @xmath195 . the error of the corresponding hf calculation is also shown for comparison . before obtaining this result , we suspected that hfb solutions suffer from larger errors than hf solutions because the former involve states above the fermi level while the latter depend on only levels below it . higher levels contain more amount of high momentum components which are less accurately treated with a finite mesh spacing . however , this figure shows that the errors are of the same order . the principal reason may be that errors from high - lying levels are reduced by factors @xmath159 or @xmath158 . the bottom portion of fig . [ fig : mesh_dep ] shows the error of the quadrupole deformation parameter @xmath241 . unlike @xmath126 , the error of @xmath241 takes both positive and negative values . for this quantity , the hfb errors are several times as large as the hf errors on the average . this may be related to the strong influence of the pairing correlation on deformations . \ii ) we compare some approximation formulae given in ref . @xcite for the evaluation of the derivatives . note that the fourier transformation method ( or the lagrange - mesh method @xcite applied to uniform - spacing meshes ) assumes a different boundary condition from the other formulae . it assumes the periodic boundary condition and the edge of the box is also slightly shortened to @xmath388 . the error is defined as the deviation from the most precise result , which is obtained with the 17-point formula and @xmath195=0.3 fm . we have not chosen the fourier method for this purpose because of a large error due to numerical cancellations for more than several tens of mesh points . in fig . [ fig : fml_dep ] , the error of the total energy is shown for each approximation formula and for three values of the mesh spacing @xmath195 . this figure tells , e.g. , if one uses the 17-point formula with @xmath199 fm for @xmath250si , the error of the total energy is 70 kev . one can observe two trends : first , for all the three mesh spacings , the error decreases as the formula involves more number of mesh points ( fourier transformation method uses all the mesh points in the box ) . second , for every formula , the error decreases when the mesh spacing is decreased . therefore , one can decrease the error of the total energy either by employing a more precise formula or by diminishing the mesh spacing . on the other hand , the other quantities can not always be improved significantly by using more precise formulae . for example , with @xmath195=1.0 fm , the error of the pairing gap @xmath360 does not continue to decrease but seems to almost saturate around 10 kev for more than 9-point formulae ( including the fourier transformation ) . we have also found a similar saturation phenomenon in the error of the deformation parameter @xmath241 , which is affected strongly by the pairing gap . the reason of this saturation is probably that the pairing gap is more sensitive than the total energy to high - lying states having large momentum and thus the lack of momentum larger than @xmath389 in the mesh space becomes the main source of the error , while the change of the formulae can only improve the treatment of momentum components smaller that @xmath389 . \iii ) we examine the errors due to the finite size of the normalization box . we consider @xmath250si , which has @xmath390 mev . naturally the errors depend on the fermi level . in this paper , however , we can treat only @xmath80 nuclei , all of which have roughly the same fermi level . we control the size of the box @xmath386 in terms of @xmath383 while fixing @xmath195 at 0.8 fm . we define the errors as the deviation from the result obtained with large @xmath386 ( @xmath391 fm ) . here , the comparison should be made within the same parity of @xmath383 @xcite . on the left- and right - hand sides of fig . [ fig : box_dep ] , we show the errors of the total energy @xmath126 and the r.m.s . radius , respectively . we have used @xmath392 fm in most of the calculations shown in this paper . these figures confirm that this value of @xmath386 leads to sufficiently precise results for both quantities . by comparing the errors of hf ( crosses ) and hfb ( circles ) solutions , one can see that the effects of the finite box size are of the same order between the two methods . this is because the density tail is determined by the fermi level , which is roughly the same whether there exist pairing correlations or not . in this appendix we study the mechanism of `` point collapse '' , a phenomenon that the wave function of a canonical orbital shrinks infinitely into a point in the coordinate space once its occupation probability becomes less than some critical value . it is a problem peculiar to the canonical - basis hfb method . for this purpose , we consider only one canonical orbital explicitly while representing all the others in terms of uniform densities , @xmath393 , @xmath394 , @xmath395 , and @xmath396 . the explicitly considered orbital is assumed to have a wave function whose spatial part is a gaussian wave packet parameterized with a size parameter @xmath397 as , @xmath398 the r.m.s . values of the coordinates and the wave numbers of this wave packet are , respectively , @xmath399 = @xmath400 = @xmath401 = @xmath402 and @xmath403 = @xmath404 = @xmath405 = @xmath406 . one can express the densities of the total system as , @xmath407 where @xmath408 is the occupation probability of this orbital and @xmath409 . the change in the total energy due to the presence of the explicitly considered orbital is given by @xmath410 r^2 dr,\ ] ] where @xmath411 has been defined by eq . ( [ eq : ham_dens ] ) , while @xmath412 by replacing @xmath64 , @xmath413 , @xmath93 , and @xmath414 in the equation with @xmath393 , @xmath395 , @xmath394 , and @xmath396 , respectively . for @xmath95 = @xmath415 = @xmath416 , one can obtain the following expression for @xmath417 ( @xmath418 is assumed for the density dependence of the skyrme force ) : @xmath419 where @xmath420 among the nine coefficients , @xmath421 and @xmath422 are negative ( at low density places ) while the others are positive . fig . [ fig : shrink ] shows @xmath417 as a function of @xmath397 for several values of @xmath408 . in the calculations , we employ a pairing force having @xmath423 mev @xmath424 , @xmath225 @xmath92 , and @xmath425 . this strength @xmath87 would give @xmath426 mev for @xmath427 system with @xmath228 if point collapses would not occur . the effects of the momentum and the pairing density dependent forces are discussed later . the mean - field force is siii without the spin - orbit term . @xmath428 is used in eq . ( [ eq : nucleonmass ] ) for the nucleon reduced mass . for the background densities , we employ @xmath429 @xmath92 and @xmath430 @xmath431 , which are typical values at peripheral regions where point collapses occur in actual hfb calculations for finite nuclei . ( a fermi gas relation @xmath432 is assumed to determine @xmath413 from @xmath64 . ) one can see from fig . [ fig : shrink ] that the total energy becomes minimum at the bottom of a dip for small values of @xmath408 . the dip emerges at @xmath433 fm when @xmath408= @xmath434 and , as @xmath408 is decreased , its depth increases while the location of the bottom moves to smaller values of @xmath397 . this minimum in the dip corresponds to hfb `` solutions '' in finite nuclei reported in refs . @xcite , in which the wave function of a high - lying orbital has shrunken into a mesh point . hfb states having physically reasonable wave functions turn themselves into such strange states in the following abrupt way : in the course of gradient - method evolutions , suddenly after an almost stationary situation has continued , @xmath275 of a high - lying orbital soars up to a few hundred mev and simultaneously the total energy decreases by a few mev . the dip is caused by the zero - rangeness of the pairing force and thus does not correspond to physically meaningful states . physical solutions should be found at @xmath435 fm , where @xmath417 is slightly negative . therefore one needs to eliminate the effects of such dips in order to obtain well - behaving hfb solutions in the canonical - basis representation . it is worth pointing out here that a wave packet with @xmath397=0.3 fm can be expressed fairly precisely on a mesh with spacing @xmath195=0.8 fm . it is because the maximum wave number expressible with the mesh is @xmath436 and thus the minimum size of the wave packet of eq . ( [ eq : wavepacket ] ) can be estimated roughly by a relation @xmath403 = @xmath406 @xmath224 @xmath436 , which leads to @xmath437 fm for @xmath195=0.8 fm . therefore collapses to wave packets as small as @xmath438 fm are a practical problem even when @xmath195=0.8 fm . before discussing how to remove the dip , let us clarify in an analytical way the mechanism which gives rise to the dip . we consider the limit of @xmath439 , in which the nature of the dip becomes most manifest . therefore , the following considerations are for the untruncated hilbert space , not for mesh representations . in order to study this limit , we assume a scaling behavior @xmath440 where @xmath397 is the location of the bottom of the dip . @xmath441 and @xmath442 are positive constants to be determined . one can easily confirm that , for @xmath443 , all the nine terms in the right - hand side of eq . ( [ eq : detot ] ) have positive powers of @xmath397 and thus converge to zero as @xmath444 . consequently , it must be satisfied that @xmath445 for a point - collapse solution to exist . for @xmath446 , @xmath447 + ( lower order terms in @xmath406 ) and thus the energy diverges to @xmath448 as @xmath444 . the only possibility for a point - collapse solution to exist is in the case of @xmath449 , for which @xmath450 + ( terms with positive powers in @xmath397 ) . from a condition @xmath451 , one immediately finds that @xmath417 becomes minimum at @xmath452 . the minimum value in the limit @xmath444 becomes @xmath453 its numerical value is @xmath454 mev for the parameters used . the essential points clarified here are i ) the minimum of the total energy is realized in the limit @xmath455 with @xmath162 and ii ) the limiting value of the minimum energy is finite . one can also derive analytically the location @xmath397 of the barrier top which divides the physical and unphysical regions . from the condition , @xmath456 one can simply obtain the location of the local maximum as , @xmath457 by neglecting terms of @xmath458 . note that it is independent of @xmath43 ( if @xmath43 is sufficiently small ) . with the parameters used , @xmath459 fm . indeed in fig . [ fig : shrink ] , the location of the top of the barrier ( designated with open circles ) seems to converge to @xmath151 fm as @xmath408 is decreased . the height of the maximum is @xmath460 , which is @xmath461 and converges to zero as @xmath462 . therefore , for sufficiently small values of @xmath43 , hfb solutions easily fall into the unphysical dip since the mesh spacing is much smaller than @xmath463 fm . the first type of the interaction is the momentum dependent term of eq . ( [ eq : pair_int ] ) . with @xmath464 , but with @xmath94 , @xmath417 becomes to have additional terms as , @xmath465 where @xmath417 is given by eq . ( [ eq : detot ] ) and , @xmath466 since @xmath467 , @xmath468 , @xmath469 , and @xmath470 are positive . the dominant term in the limit @xmath444 is the one having coefficient @xmath469 ( @xmath470 ) for @xmath471 , which behaves as @xmath472 ( @xmath473 ) . these two terms overwhelm the negative terms in @xmath417 . consequently wave functions can not shrink to a point . however , a dip can still emerge with finite @xmath43 . figure [ fig : shrinkkc ] shows how the local minimum ( the bottom of the dip ) and the local maximum ( the top of the barrier ) of @xmath417 change versus @xmath408 . the left - hand portion shows the depth and the height ( @xmath417 ) , while the right - hand portion shows the locations ( @xmath397 ) . the solid curves are obtained with @xmath155 . the upper solid curve stands for the height of the local maximum , which converges to zero as @xmath474 , as already discussed analytically . ( this curve is almost unchanged when @xmath95 is decreased to finite values . ) the lower solid curve corresponds to the depth of the local minimum , which converges to @xmath454 mev , in accord with eq . ( [ eq : localmin ] ) . with finite values of @xmath95 , the minimum becomes shallower as @xmath408 is decreased and it disappears at some small value of @xmath408 . this disappearance is clearly seen in the dot curve ( corresponding to @xmath95=15 @xmath98 ) in the right - hand portion of the figure . a numerical search has shown that the dip exists only for @xmath475 @xmath98 . the result with this simplified model roughly agrees with realistic hfb calculations : we have observed that point collapses can occur for @xmath476 @xmath98 in calculations like those shown in fig . [ fig : nw_gap ] . the difference between 6 @xmath98 and 10 @xmath98 can be partially attributed to the finite - point approximation to the derivatives , which underestimates the kinetic energy by 30% at @xmath477 , as shown in fig . 20 of ref . @xcite . incidentally , the requirement of orthogonality among the canonical orbitals does not seem to change the conclusion drawn with this model . the first reason is that point collapses in light nuclei are not affected by orthogonality if the number of canonical orbitals @xmath55 is relatively small . this is because of the symmetry ( e.g. , @xmath234 ) of the solution : if a high - lying orbital is the only one in an irreducible representation of the symmetry ( e.g. , in @xmath478=@xmath479 sector when @xmath384-axis is the shortest axis ) , it is automatically orthogonal to the other orbitals . in this case , it shrinks not to a point but to many points ( e.g. , two points at @xmath480 , @xmath481 , @xmath169 @xmath224 nuclear radius , with opposite - sign amplitudes ) . indeed , we have experienced that point collapses occur only when @xmath55 is small for relatively small values of @xmath95 . the second reason is an empirical fact that , with large values of @xmath95 , many orbitals collapse one after another or simultaneously even for large @xmath55 : the process of point collapses under orthogonality condition may not be so complicated as it sounds in such cases . the second type of the interaction we examine is the pairing density dependent term of eq . ( [ eq : pair_int ] ) . with @xmath482 , but with @xmath483 , @xmath417 has additional terms to eq . ( [ eq : detot ] ) as , @xmath484 where @xmath485 all the added terms are positive . assuming @xmath440 , terms with coefficients @xmath486 and @xmath487 have negative powers of @xmath397 at @xmath449 . consequently point collapses can be avoided by using finite values of @xmath415 . figure [ fig : shrinkrp ] shows extrema of @xmath417 versus @xmath408 . the solid curve corresponds to @xmath94 and is the same as that in fig . [ fig : shrinkkc ] . the other types of curves are for finite values of @xmath415 . one can see that @xmath417 at the local minimum is already very small with @xmath488 @xmath92 ( dot curve ) , which is much larger than the typical values of @xmath93 and thus leads to only physically acceptable small amount of modification of the pairing interaction . this figure is consistent with the results of actual hfb calculations : from fig . 4 of ref . @xcite , one can see that point collapses can be avoided with @xmath489 @xmath92 . mathematically speaking , however , the local minimum continues to exist for arbitrary @xmath43 , which is a different behavior from that with finite @xmath95 . in the limit of @xmath462 , this minimum behaves as @xmath444 and @xmath490 . because the depth converges to zero from below , the point - collapse limit is unstable and is not very meaningful . so , let us give only briefly the result of an analytical consideration . assuming @xmath440 , one can show that the local minimum continues to exist in the limit @xmath444 with @xmath491 and @xmath441 determined by a cubic equation @xmath492 if @xmath493 ( i.e. , @xmath494 for @xmath495 ) . at the local minimum , @xmath496 and @xmath497 . one can see this scaling behavior with @xmath498 on the right - hand side of fig . [ fig : shrinkrp ] . in the present paper we have used the momentum dependent interaction because it has a physical origin , i.e. , the finite - rangeness . however , the pairing density dependent interaction may also be useful under different circumstances because the momentum dependent interaction leads to a divergence of the pairing energy in unrestricted hilbert space @xcite : the assumption of @xmath499 made in this paper is not adequate for orbitals with @xmath500 . to summarize , assuming a wave function of gaussian form , we have clarified the mechanism of the point - collapse phenomenon occurring in the canonical - basis hfb method due to delta interactions . we have also demonstrated the successes of momentum or pairing density dependent interactions in avoiding the collapse . stoitsov , w. nazarewicz , and s. pittel : phys . rev . * c58 * , 2092 ( 1998 ) . d. vautherin and d.m . brink , phys . * c5 * , 626 ( 1972 ) . j. dobaczewski , h. flocard and j. treiner : nucl . * a422 * , 103 ( 1984 ) . j. dobaczewski , w. nazarewicz , t.r . werner , j.f . berger , c.r . chinn , and j.decharg , phys . rev . * c53 * , 2809 ( 1996 ) b. gall , p. bonche , j. dobaczewski , h. flocard , and p .- h . heenen , z. phys . * a348 * , 183 ( 1994 ) . j. terasaki , p .- h . heenen , h. flocard and p. bonche : nucl . * a600 * , 371 ( 1996 ) . j. terasaki , h. flocard , p .- h . heenen and p. bonche : nucl . a621 * , 706 ( 1997 ) . n. tajima , in _ proceedings of the xvii rcnp international symposium on innovative computational methods in nuclear many - body problems _ , osaka , 1997 , edited by h. horiuchi _ et al . _ , ( world scientific , 1998 ) , p. 343 ; nucl - th/0002061 . m. yamagami , k. matsuyanagi , m. matsuo , nucl . * a693 * , 579 ( 2001 ) c. bloch and a. messiah , nucl . phys . * 39 * , 95 ( 1962 ) . reinhard , m. bender , k. rutz , and j.a . maruhn : z. phys . * a358 * , 277 ( 1997 ) . n. tajima , in _ high performance computing in riken 1997 _ , riken review * 19 * , 29 ( 1998 ) ; http://www.riken.go.jp/lab-www/library/publication/ review / conts / conts19.html n. tajima , in _ high performance computing in riken 1998 _ , riken review * 25 * , 75 ( 1999 ) ; http://www.riken.go.jp/lab-www/library/publication/ review / conts / conts25.html n. tajima , in _ proceedings of a riken international symposium on models and theories of the nuclear mass _ , 1999 , wako ; riken review * 26 * , 87 ( 2000 ) ; http://www.riken.go.jp/lab-www/library/publication/ review / conts / conts26.html p. bonche , h. flocard , p .- h . heenen , s.j . krieger , and m.s . weiss : nucl . a443 * , 39 ( 1985 ) . skyrme , phil . mag . * 1 * , 1043 ( 1956 ) . m. beiner , h. flocard , nguyen van giai , and p. quentin , nucl.phys . * a238 * , 29 ( 1975 ) . s. sakakihara and y. tanaka , nucl .phys . * a691 * , 649 ( 2001 ) . n. tajima , p. bonche , h. flocard , p .- h . heenen , and m.s . weiss : nucl . * a551 * , 434 ( 1993 ) . j. dobaczewski , w. nazarewicz , and m.v . stoitsov , presented at _ nato advanced research workshop on the nuclear many - body problem _ , brijuni , 2001 ; nucl - th/0109073 . goodman , adv . phys . * 11 * , 263 ( 1979 ) . s. takahara , n. onishi , and n. tajima , phys . * b331 * , 261 ( 1994 ) . j. decharg and d. gogny , phys . rev . * c21 * , 1568 ( 1980 ) . a. bulgac and y. yu , phys . lett . * 88 * , 042504 ( 2002 ) . a. bulgac , phys . * c65 * , 051305(r ) ( 2002 ) . y. yu and a. bulgac , phys . * 90 * , 222501 ( 2003 ) . m. grasso and m. urban , cond - mat/0305588 . n. tajima , s. takahara , and n. onishi : nucl . phys . * a603 * , 23 ( 1996 ) . k.t.r . davies , h. flocard , s. krieger , and m.s . weiss , nucl . * a342 * , 111 ( 1980 ) . p. ring and p. schuck : _ the nuclear many - body problem _ ( springer , new york , 1980 ) . n. tajima , progr . suppl . * 142 * , 265 ( 2001 ) . nguyen van giai and h. sagawa , phys . * b106 * , 379 ( 1981 ) . d. baye and p .- h . heenen : j. phys . * a19 * , 2041 ( 1986 ) . h. imagawa and y. hashimoto , phys . rev . * c67 * , 037302 ( 2003 )	a method is presented to obtain the canonical - form solutions of the hfb equation for atomic nuclei with zero - range interactions like the skyrme force . it is appropriate to describe pairing correlations in the continuum in coordinate - space representations . an improved gradient method is used for faster convergences under constraint of orthogonality between orbitals . to prevent high - lying orbitals to shrink into a spatial point , a repulsive momentum dependent force is introduced , which turns out to unveil the nature of high - lying canonical - basis orbitals . the asymptotic properties at large radius and the relation with quasiparticle states are discussed for the obtained canonical basis .	29,498	167
the detection of x - ray , optical and radio afterglows from some well - localized gamma - ray bursts ( grbs ) definitely shows that at least most long grbs are of cosmological origin ( e.g. , costa et al . 1997 ; frail et al . 1997 ; galama et al . 1998 ; akerlof et al . 1999 ; zhu et al . 1999 ) . the so called fireball model is thus strongly favoured . however , we are still far from resolving the puzzle of grbs ( piran 1999 ; van paradijs , kouveliotou & wijers 2000 ) . a major problem is that we do not know whether grbs are due to highly collimated jets or isotropic fireballs , so that the energetics involved can not be determined definitely ( e.g. , pugliese , falcke & biermann 1999 ; kumar & piran 2000 ; dar & rjula 2000 ; wang & loeb 2001 ; rossi , lazzati & rees 2002 ) . this issue has been extensively discussed in the literature . generally speaking , three methods may help to determine the degree of beaming in grbs . first , based on analytic solutions , it has been proposed that optical afterglows from a jetted grb should be characterized by a break in the light curve during the relativistic phase , i.e. , at the time when the lorentz factor of the blastwave is @xmath0 , where @xmath1 is the half opening angle ( rhoads 1997 ; kulkarni et al . 1999 ; mszros & rees 1999 ) . some grbs such as 990123 , 990510 are regarded as good examples ( castro - tirado et al . 1999 ; wijers et al . 1999 ; harrison et al . 1999 ; halpern et al . 2000 ; pian et al . 2001 ; castro ceron et al . 2001 ; sagar et al . 2001 ) . however , detailed numerical studies show that the break is usually quite smooth ( panaitescu & mszros 1998 ; moderski , sikora & bulik 2000 ) , and huang et al . went further to suggest that the light curve break should in fact occur at the trans - relativistic phase ( huang et al . 2000a , b , c , d ) . additionally , many other factors can also result in light curve breaks , for example , the cooling of electrons ( sari , piran & narayan 1998 ) , a dense interstellar medium ( dai & lu 1999 ) , or a wind environment ( dai & lu 1998a ; chevalier & li 2000 ; panaitescu & kumar 2000 ) . all these facts combine together to make the first method not so conclusive . second , gruzinov ( 1999 ) argued that optical afterglows from a jet can be strongly polarized , in principle up to tens of percents . some positive observations have already been reported ( wijers et al . 1999 ; rol et al . 2000 ) . but polarization can be observed only under some particular conditions , i.e. , the co - moving magnetic fields parallel and perpendicular to the jet should have different strengths and we should observe at the right time from the right viewing angle ( gruzinov 1999 ; hjorth et al . 1999 ; mitra 2000 ) . the third method was first proposed by rhoads ( 1997 ) , who pointed out that due to relativistic beaming effects , @xmath2-ray radiation from the vast majority of jetted grbs can not be observed , but the corresponding late time afterglow emission is less beamed and can safely reach us . these afterglows are called orphan afterglows , which means they are not associated with any detectable grbs . the ratio of the orphan afterglow rate to the grb rate might allow measurement of the grb collimation angle . great expectations have been put on this method ( rhoads 1997 ; mszros , rees & wijers 1999 ; lamb 2000 ; paczyski 2000 ; djorgovski et al . 2001 ) . in fact , the absence of large numbers of orphan afterglows in many surveys has been regarded as evidence that the collimation can not be extreme ( rhoads 1997 ; perna & loeb 1998 ; greiner et al . 1999 ; grindlay 1999 ; rees 1999 ) . recently dalal , griest & pruet ( 2002 ) argued that measurement of the grb beaming angle using optical orphan searches is extremely difficult . the main reason is that when the afterglow emission from a jet begins to go into a much larger solid angle than the initial burst does , the optical flux density usually becomes very low . generally speaking , this problem can be overcome by improving the detection limit . in fact , an interesting result was recently reported by vanden berk et al . ( 2002 ) , who discovered a possible optical orphan at @xmath3 . they suggested that grbs should be collimated . in this article , we will point out another difficulty associated with the third method : there should be many `` failed gamma - ray bursts ( fgrbs ) '' , i.e. , baryon - contaminated fireballs with initial lorentz factor @xmath4 . fgrbs can not be observed in gamma - rays , but their long - lasting afterglows are detectable , thus they will also manifest themselves as orphan afterglows . our paper is organized as follows . in section 2 we explain the concept of fgrbs . section 3 describes the difficulty in distinguishing an fgrb orphan and a jetted grb orphan . some possible solutions that may help to overcome the difficulty are suggested . section 4 is a brief discussion . occurring in the deep universe , grbs are the most relativistic phenomena ever known . the standard fireball model ( mszros & rees 1992 ; dermer , bttcher & chiang 1999 ) requires that to successfully produce a grb , the initial lorentz factor of the blastwave should typically be @xmath5 1000 during the main burst phase ( piran 1999 ; lithwick & sari 2000 ) . generally speaking the requirement of ultra - relativistic motion is to avoid the so called `` compactness problem '' . a modest variation in the lorentz factor will result in a difference of the opacity of the high - energy @xmath2-ray photons by a factor of @xmath6 ( totani 1999 ) . additionally , assuming synchrotron radiation , the observed peak frequency is strongly dependent on @xmath2 , @xmath7 ( mszros , rees & wijers 1998 ) . thus a lorentz factor of @xmath8 makes the blastwave very inefficient in emitting @xmath2-ray photons . so , to successfully produce a grb , we need @xmath5 1000 . however , theoretically it is not easy to construct a model to generate such ultra - relativistic motions . currently there are mainly two kinds of progenitor models , the collapse of massive stars ( with mass @xmath9 ) , or the collision of two compact stars ( such as two neutron stars or a neutron star and a black hole ) . since a baryon - rich environment is involved in all these models , some researchers are afraid that the baryon - contamination problem may exist . but this problem maybe is not as serious as we previously expected . let us first take the collapsar model ( macfadyen & woosley 1999 ) as an example . we can imagine that the baryon mass and energy released in different collapsar events should vary greatly , then @xmath10 of the resultant fireballs may also vary in a relatively wide range . in most cases , @xmath10 should be very low ( i.e. , @xmath4 ) , but there still could be a few cases ( e.g. , one percent or even one in a thousand ) in which the fireball is relatively clean so that the blastwave can be successfully accelerated to @xmath5 1000 and produces a grb . since the collapsar rate is high enough in a typical galaxy , there should be no problem that such collapsars can meet the requirement of grb rate ( i.e. , @xmath11 @xmath12 event per typical galaxy per year , under isotropic assumption ) . cases are similar in the collisions of two compact stars . in short , we can not omit an important fact : if grbs are really due to isotropic fireballs , then there should be much more failed grbs ( i.e. , fireballs with lorentz factor much less than one hundred , but still much greater than unity ) . these fgrb fireballs can contain similar initial energy as normal grb fireballs , i.e. , @xmath13 @xmath14 ergs , but they are polluted by baryons with mass @xmath15 @xmath16 . radiation from these fgrbs should mainly be in x - ray bands in the initial bursting phase , not in @xmath2-ray bands . in fact , bepposax team has reported the discovery of several anomalous events named as fast x - ray transients , x - ray rich grbs , or even x - ray - grbs . they resemble usual grbs except that they are extremely x - ray rich ( frontera et al . 2000 ; kippen et al . 2001 ; gandolfi & piro 2001 ) . observational data on this kind of events are being accumulated rapidly . recent good examples include grbs 011030 , 011130 and 011211 ( gandolfi et al . 2001a , b ; ricker et al . 2001 ) . we propose that these events are probably just fgrbs . huang et al . ( 1998 ) and dai , huang & lu ( 1999 ) have pointed out that for afterglow behaviour , the parameter @xmath17 is decisive , while @xmath18 is only of minor importance , especially at late stages . so , fgrbs should also be associated with prominent afterglows . in figure 1 we compare the theoretical optical afterglows from fgrbs with those from isotropic grbs and jetted grbs . we can see that the light curve of an fgrb afterglow differs from that of a successful isotropic burst only slightly , i.e. , only notable at early stages . in our calculations , we have used the methods developed by huang et al.(1999a , b , 2000b , d ) , i.e. , for the dynamical evolution of isotropic fireballs we use @xmath19 where @xmath20 is the swept - up mass and @xmath21 is the radiation efficience . ( 1 ) has been proved to be proper in both ultra - relativistic phase and non - relativistic phase ( huang , dai & lu 1999a , b ) . for jetted ejecta , the following equation is added ( huang et al . 2000b , d ) , @xmath22 where @xmath23 is the blastwave radius , and the co - moving sound speed @xmath24 is given realistically by @xmath25 with @xmath26 the adiabatic index . in fact , in beamed grb models , there should also be many fgrbs , i.e. , beamed ejecta with @xmath27 . we call them beamed fgrbs . afterglow from beamed fgrbs has also been illustrated in figure 1 . in this article emphasises will be put on isotropic fgrbs , so by using `` fgrbs '' we will only mean isotropic fgrbs unless stated explicitly . both fgrbs and jetted but off - axis grbs can produce isolated fading objects , i.e. , orphan afterglows . theoretically , when orphan afterglows are really discovered observationally , it is still risky to conclude that grbs are beamed . we should study these orphans carefully to determine whether they come from fgrbs or jetted grbs . however , we will show below that it is not an easy task . usually the light curve of grb afterglows is plotted as @xmath28 vs. @xmath29 , where @xmath30 is the flux density at observing frequency @xmath31 and @xmath32 is observer s time measured from the burst trigger . in such plots , the behaviour of afterglows from isotropic grbs and jetted ones are possibly quite different . the former is generally characterized by a simple flat straight line with slope @xmath33 @xmath34 and the latter can be characterized by a break in the light curve or by a steep straight line with a slope sharper than @xmath35 ( figure 1 , also see huang et al . 2000a , b , c , d ) . but for orphan afterglow observations , the derivation of such a @xmath28 @xmath29 light curve is not direct : we do not know the trigger time so that the exact value of @xmath32 for each observed data point can not be determined . as the first step , the best that we can do is to produce a light curve with a linear time axis , which , however , is of little help for unveiling the nature of the orphan . figure 2 illustrates the matter . in this figure we plot @xmath28 vs. @xmath32 for the two kinds of orphans theoretically . the uncerntainty in trigger time means the observed light curve can be shifted along x axis , while the unknown distance results in a shift along y axis . we see that after some simple manipulations , the segment ab on the dashed curve ( i.e. , from @xmath36 d to @xmath37 d ) can be shifted to a place ( a@xmath38b@xmath38 ) that differs from the solid line only slightly . it hints that a linear light curve as long as @xmath39 days is still not enough . note that the solid and the dashed curves in figure 2 are only two examples . the variation of some intrinsic parameters , such as @xmath17 , @xmath40 , @xmath10 , @xmath41 , @xmath42 , @xmath43 , @xmath44 and @xmath45 as defined in the caption of figure 1 , can change the shape of the two curves notably , thus brings in much more difficulties . in figure 3 , we compare the theoretical @xmath28 @xmath29 light curves of optical afterglows from fgrbs and jetted but off - axis grbs directly . to investigate the influence of the uncertainty in trigger time , we also shift the light curve of fgrbs by @xmath46 d , @xmath47 d and @xmath48 d intentionally . from the dashed curves , we can see that the shape of the fgrb afterglow light curve is seriously affected by the uncertainty of the trigger time . but fortunately , these dashed curves still differ from the theoretical light curve of the jetted grb orphan markedly , i.e. , they are much flatter at very late stages . this means it is still possible for us to discriminate them . in figure 4 , similar results to figure 3 are given , but this time the light curve of the jetted grb orphan is shifted . again we see that the two kinds of orphans can be discriminated by their late time behaviour . figures 3 and 4 explain what we should do when an orphan afterglow is discovered . first , we have to assume a trigger time for it arbitrarily , so that the logarithmic light curve can be plotted . we then need to change the trigger time to many other values to see how the light curve is affected . in all our plots , we should pay special attention to the late time behaviour , which will be less affected by the uncertainty in the trigger time . if the slope tends to be @xmath49 @xmath34 , then the orphan afterglow may come from an fgrb event . but if the slope tends to be steeper than @xmath35 , then it is very likely from a jetted but off - axis grb . in fact , from figure 1 , we know that for all kinds of grbs , either successful or failed , the optical afterglow approximately follows a simple power - law decay at late stages ( i.e. , @xmath50 ) so that the light curve is a straight line . in such a relation , if we shift the time by @xmath51 , @xmath52 then the line would become curved . the slope at each point on the curve is @xmath53 for positive @xmath51 values the lines bend up - ward , while for negative values the lines bend down - ward . it hints that in plotting the orphan afterglow light curve , we could select the trigger time properly to get a straight line at late stages , then we can determine not only the late time slope , but also the true trigger time . in other words , we can use @xmath54 as the condition to determine the trigger time and to get the straight line at late stages . however , we must bear in mind that it is in fact not an easy task . first , to take the process we need to follow the orphan as long as possible , and the simple discovery of an orphan is obviously insufficient . note that currently optical afterglows from most well - localized grbs can be observed for only less than 100 days . it is quite unlikely that we can follow an orphan for a period longer than that . second , since the orphan is usually very faint , errors in the measured magnitudes will seriously prevent us from deriving the straight line . due to all these difficulties , a satisfactory light curve is usually hard to get for most orphans . we see that measurement of the grb beaming angle using orphan searches is not as simple as we originally expected . in fact , it is impractical to some extent . recently it was suggested by rhoads ( 2001 ) that grb afterglows can be effectly identified by snapshot observations made with three or more optical filters . the method has been successfully applied to grb 001011 by gorosabel et al . it is believed that this method is also helpful for orphan afterglow searches . however , please note that a jetted grb orphan and an fgrb one still can not be discriminated directly . we have shown that the derivation of a satisfactory light curve for an orphan afterglow is difficult . the major problem is that we do not know the trigger time . anyway , there are still some possible solutions that may help to determine the onset of an orphan afterglow . firstly , of course we should improve our detection limit so that the orphan afterglow could be followed as long as possible . the longer we observe , the more likely that we can get the true late - time light curve slope . secondly , we know that fgrbs usually manifest themselves as fast x - ray transients or x - ray - grbs . if an orphan can be identified to associate with such a transient , then it is most likely an fgrb one . in this case , the trigger time can be well determined . thirdly , maybe in some rare cases we are so lucky that the rising phase of the orphan could be observed . for a jetted grb orphan the maximum optical flux is usually reached within one or two days and for an fgrb orphan it is even within hours . then the uncertainty in trigger time is greatly reduced . additionally , a jetted grb orphan differs markedly from an fgrb one during the rising phase . the former can be brightened by more than one magnitude in several hours ( see figures 1 4 ) , while the brightening of the latter can hardly be observed . so , if an orphan afterglow with a short period of brightening is observed , then it is most likely a jetted grb orphan . of course , we should first be certain that it is not a supernova . fourthly , valuable clues may come from radio observations . in radio bands , the light curve should be highly variable at early stages due to interstellar medium scintillation , and it will become much smoother at late times . so the variability in radio light curves provides useful information on the trigger time . and fifthly , in the future maybe gravitational wave radiation or neutrino radiation associated with grbs could be detected due to progresses in technology , then the trigger time of an orphan could be determined directly and accurately . in fact , with the successful detection of gravitational waves or neutrino emission , our understanding on grb progenitors will surely be promoted greatly ( paolis et al . 2001 ) . sixthly , the redshift of the orphan afterglow can help us greatly in determining the isotropic energy involved , which itself is helpful for inferring the trigger time . seventhly , the microlensing effect may be of some help . since the size of the radiation zone of a jetted grb orphan is much smaller than that of an fgrb one , they should behave differently when microlensed . finally , although a successful detection of some orphan afterglows does not directly mean that grbs be collimated , the negative detection of any orphans can always place both a stringent lower limit on the beaming angle for grbs and a reasonable upper limit for the rate of fgrbs . to successfully produce a grb , the blastwave should be ultra - relativistic , with lorentz factor typically larger than 100 1000 . however , in almost all popular progenitor models , the environment is unavoidably baryon - rich . we believe that only in very rare cases can an ultra - relativistic blastwave successfully break out to give birth to a grb , and there should be much more failed grbs , i.e. , fireballs with lorentz factor much less than 100 but still much larger than unity . in fact , this possibility has also been mentioned by a number of authors , such as mszros & waxman ( 2001 ) . owing to the existence of fgrbs , there should be many orphan afterglows even if grbs are due to isotropic fireballs . then the simple discovery of orphan afterglows does not necessarily mean that grbs be highly collimated . to make use of information from orphan afterglow surveys correctly , we should first know how to discriminate a jetted grb orphan and an fgrb one . this can be done only by checking the detailed afterglow light curve . however , we have shown that the derivation of a satisfactory light curve for an orphan afterglow is difficult . the major problem is that we do not know the trigger time . in section 3.2 , some possible solutions to the problem are suggested . unfortunately many of these solutions are still quite impractical in the foreseeable future , which means measure of grb beaming angle using orphan afterglow searches is extremely difficult currently . however , special attention should be paid to the second solution . usually , fgrbs manifested themselves as fast x - ray transients during the main burst phase , while jetted but off - axis grbs went unattended completely . if the fast x - ray transients ( or x - ray - grbs ) observed by bepposax are really due to fgrbs , then afterglows should be detectable . we propose that this kind of events should be followed rapidly and extensively in all bands , just like what we are doing for grbs . if observed , afterglows from these anomalous events can be used to check our concept of fgrbs , and even to test the fireball model under quite different conditions ( i.e. , when @xmath4 ) . also , these fgrbs can provide valuable information for our understanding of grbs , especially on the progenitor models . note that beamed fgrbs can also give birth to fast x - ray transients if they are directed toward us , but afterglows from such a beamed fgrb and an isotropic fgrb can be discriminated easily from the light curves ( see figure 1 ) . it is very interesting to note that optical afterglows from two x - ray - grbs , 011130 and 011211 , have been observed ( garnavich , jha & kirshner 2001 ; grav et al . their redshifts were measured to be @xmath55 0.5 and 2.14 respectively ( jha et al . 2001 ; fruchter et al . 2001 ) , eliminates the possibility that they were ordinary classic grbs residing at extremely high redshifts ( @xmath56 ) . we propose that they should be fgrbs ( either isotropic or beamed ) or just jetted grb `` orphan '' . however , the observational data currently available are still quite poor so that we could not determine their nature definitely . as for other x - ray - grbs without a measured redshift , the possibility that they were at redshifts of @xmath57 can not be excluded . finally , the concept of fgrbs is based on the fact that most popular progenitor models for grbs are baryon - rich . but cases are quite different for another kind of progenitor models where strange stars are involved . strange stars , composed mainly of u , d , and s quarks , are compact objects which are quite similar to neutron stars observationally ( alcock , farhi & olinto 1986 ) . a typical strange star ( with mass @xmath58 ) can have a normal matter crust of less than @xmath59 ( alcock , farhi & olinto 1986 ) , or even as small as @xmath60 ( huang & lu 1997a , b ) . then baryon contamination can be directly avoided if grbs are due to the phase transition of neutron stars to strange stars ( cheng & dai 1996 ; dai & lu 1998b ) or collisions of binary strange stars . in these models , there should be very few fgrbs . we thank an anonymous referee for valuable comments and suggestions . yfh thanks l. j. gou and x. y. wang for helpful discussion . this research was supported by the special funds for major state basic research projects , the national natural science foundation of china ( grants 10003001 , 19825109 , and 19973003 ) , and the national 973 project ( nkbrsf g19990754 ) . note added after acceptance * ( this paragraph might not appear in the published version ) * : the optical orphan at z=0.385 reported by vanden berk et al . 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ph/0112083 ) sagar r. , pandey s. b. , mohan v. , bhattacharya d. , castro - tirado a. j. , 2001 , bull . india , 29 , 1 sari r. , piran t. , narayan r. , 1998 , apj , 497 , l17 totani t. , 1999 , mnras , 307 , l41 vanden berk d. e. et al . , 2002 , apj submitted ( astro - ph/0111054 ) van paradijs j. , kouveliotou c. , wijers r. a. m. j. , 2000 , ara&a , 38 , 379 wang x. , loeb a. , 2001 , apj , 552 , 49 wijers r. a. m. j. et al . , 1999 , apj , 523 , l33 zhu j. et al . , 1999 ,	it is believed that orphan afterglow searches can help to measure the beaming angle in gamma - ray bursts ( grbs ) . great expectations have been put on this method . we point out that the method is in fact not as simple as we originally expected . due to the baryon - rich environment that is common to almost all popular progenitor models , there should be many failed gamma - ray bursts , i.e. , fireballs with lorentz factor much less than 100 1000 , but still much larger than unity . in fact , the number of failed gamma - ray bursts may even be much larger than that of successful bursts . owing to the existence of these failed gamma - ray bursts , there should be many orphan afterglows even if grbs are due to isotropic fireballs , then the simple discovery of orphan afterglows never means that grbs be collimated . unfortunately , to distinguish a failed - grb orphan and a jetted but off - axis grb orphan is not an easy task . the major problem is that the trigger time is unknown . some possible solutions to the problem are suggested . = -0.5 in stars : neutron ism : jets and outflows gamma - rays : bursts	10,260	343
the interaction between a movable mirror and the radiation field of an optical cavity has recently been the subject of extensive theoretical and experimental investigations . these optomechanical systems couple the mechanical motion to an optical field directly via radiation pressure buildup in a cavity . the coupling of mechanical and optical degrees of freedom via radiation pressure has been a subject of early research in the context of laser cooling @xcite and gravitational - wave detectors @xcite . recently there has been a great surge of interest in the application of radiation forces to manipulate the center - of - mass motion of mechanical oscillators covering a huge range of scales from macroscopic mirrors in the laser interferometer gravitational wave observatory ( ligo ) project @xcite to nano - mechanical cantilevers@xcite , vibrating microtoroids@xcite membranes@xcite and bose - einstein condensates @xcite . the quantum optical properties of a mirror coupled via radiation pressure to a cavity field show interesting similarities to an intracavity kerr - like interaction @xcite . recently , in the context of classical investigations of nonlinear regimes , the dynamical instability of a driven cavity having a movable mirror has been investigated @xcite . theoretical work has proposed to use the radiation - pressure coupling for quantum non - demolition measurements of the light field @xcite . it has been shown that ground state cooling of micro - mechanical mirror is possible only in the resolved side band regime ( rsb ) where the mechanical resonance frequency exceeds the bandwidth of the driving resonator @xcite . the cooling of mechanical oscillators in the rsb regime at high driving power can entail the appearance of normal mode splitting ( nms ) @xcite . recently , it was shown that an optical parametric amplifier inside a cavity considerably improves the cooling of a micro - mechanical mirror by radiation pressure @xcite . in this paper , we consider the dynamics of a movable mirror interacting with a nonlinear optical cavity mode and predict novel properties of the dynamics of the system . giant optical kerr nonlinearities are obtained by placing a @xmath0 medium inside a cavity@xcite . this gives rise to a strong nonlinear interactions between photons . a single photon in a cavity can block the injection of a second photon due to a photon blockade effect . we show that due to the photon blockade mechanism , as the kerr nonlinearity is increased , the nms progressively decreases . we consider an optical kerr medium with @xmath0 nonlinearity inside a fabry - perot cavity with one fixed partially transmitting mirror and one movable totally reflecting mirror in contact with a thermal bath in equilibrium at temperature @xmath1 , as shown in fig.1 . the movable mirror is treated as a quantum mechanical harmonic oscillator with effective mass @xmath2 , frequency @xmath3 and energy decay rate @xmath4 . the system is also coherently driven by a laser field with frequency @xmath5 through the cavity mirror with amplitude @xmath6 . it is well known that high - q optical cavities can significantly isolate the system from its environment , thus strongly reducing decoherence and ensuring that the light field remains quantum - mechanical for the duration of the experiment . we also assume that the induced resonance frequency shift of the cavity and the nonlinear interaction coefficient @xmath7 are much smaller than the longitudinal mode spacing , so that we restrict the model to a single longitudinal mode @xmath8 . we also assume that @xmath9 ( adiabatic limit ) ; @xmath10 is the speed of light in vacuum and @xmath11 the cavity length in the absence of the cavity field . the total hamiltonian of the system in a frame rotating at the laser frequency @xmath5 can be written as @xmath12 here @xmath13 and @xmath14 are the annihilation and creation operators for the cavity field respectively . also @xmath15 and @xmath16 are the position and momentum operators for the movable mirror . the parameter @xmath17 is the coupling parameter between the cavity field and the movable mirror and @xmath7 is the anharmonicity parameter and is proportional to the third - order nonlinear susceptibility @xmath0 of the kerr medium : @xmath18/2\epsilon_{0}v_{c}$ ] , @xmath19 is the dielectric constant of the medium and @xmath20 is the volume of the cavity . the input laser field populates the intracavity mode which couples to the movable mirror through the radiation pressure . the field in turn is modified by the back - action of the cantilever . it is important to notice the nonlinearity in eqn . ( [ eff ] ) arising from the coupling between the intracavity intensity and the position operator of the mirror . the system we are considering is intrinsically open as the cavity field is damped by the photon - leakage through the massive coupling mirror and the mirror is connected to a bath at finite temperature . in the absence of the radiation - pressure coupling , the cantilever would undergo a pure brownian motion driven by its contact with the thermal environment . the motion of the system can be described by the following quantum langevin equations : @xmath21 @xmath22 @xmath23 here @xmath24 is the input vacuum noise operator and it obeys the following correlation functions : @xmath25 @xmath26 the force @xmath27 is the brownian noise operator resulting from the coupling of the movable mirror to the thermal bath , whose mean value is zero , and has the following correlation function at temperature @xmath1 : @xmath28 d\omega,\ ] ] where @xmath29 is the boltzmann constant and @xmath1 is the thermal bath temperature . the steady state values of @xmath16 , @xmath15 and @xmath13 are obtained as : @xmath30 @xmath31 @xmath32 where @xmath33 is the effective cavity detuning which includes the radiation pressure effects . here @xmath34 denotes the new equilibrium position of the mirror while @xmath35 denotes the steady state amplitude of the cavity field . both @xmath34 and @xmath35 displays multistable behaviour due to the nonlinear interaction between the mirror and the cavity field . from the above equations we clearly see how the mirror dynamics affects the steady state of the intracavity field . the coupling to the mirror shifts the cavity resonance frequency and changes the field inside the cavity in a way to induce a new stationary intensity . the change occurs after a transient time depending on the response of the cavity and strength of the coupling to the mirror . here we show that the coupling of the mechanical oscillator and the cavity field fluctuations in the presence of the kerr medium leads to the inhibition of the normal mode splitting ( nms ) . the optomechanical nms however involves driving two parametrically coupled nondegenerate modes out of equilibrium . the nms does not appear in the steady state spectra but rather manifests itself in the fluctuation spectra of the mirror displacement . to this end , we write each canonical operator of the system as a sum of its steady - state mean value and a small fluctuation with zero mean value , @xmath36 , @xmath37 , @xmath38 and linearize to obtain the following heisenberg - langevin equations for the fluctuation operators @xmath39 @xmath40 @xmath41 here we will always assume @xmath42 . ( [ fluc1 ] , [ fluc2 ] , [ fluc3 ] ) and their hermitian conjugates constitute a system of four first order coupled operator equations , for which the routh - hurwitz criterion implies that the system is stable for the following conditions : @xmath43 @xmath44 @xmath45^{2}+[\delta+4 \eta \|a_{s}\|^{2}]^{2 } \right\rbrace + \dfrac{\hbar g_{m}^{2 } \eta}{m}(a_{s}^{2}+a_{s}^{2})^{2}-\omega_{m}^{2 } \eta^{2}(a_{s}^{2}+a_{s}^{2})^{2}-\dfrac{2 \hbar g_{m}^{2 } \|a_{s}\|^{2}}{m}(\delta+4 \eta \|a_{s}\|^{2})>0.\ ] ] the study of these conditions reveals the point at which the system enters an unstable regime . here , we will restrict ourselves to the stable regime . we now transform to the quadratures : @xmath46 , @xmath47 , @xmath48 and @xmath49 . the position fluctuations of the movable mirror in fourier space is given by @xmath50 \xi ( \omega)-i \hbar g_{m } \sqrt{2 \kappa}[(\omega+i \kappa+\delta)a_{s}^ { * } \delta a_{in}+(\omega+i \kappa-\delta)a_{s } \delta a_{in}^{\dagger}]\right\rbrace,\ ] ] where @xmath51[(\kappa - i \omega)^{2}+\delta'^{2}]-2 \hbar g'^{2}_{m } \delta''$ ] , @xmath52 , @xmath53 , @xmath54 , @xmath55 , @xmath56 and @xmath57 . in the above equation for @xmath58 , the term proportional to @xmath59 arises from thermal noise , while the term proportional to @xmath60 originates from radiation pressure . the displacement spectrum is obtained from @xmath61 together with the correlation functions : @xmath62 @xmath63 \delta(\omega+\omega).\ ] ] the displacement spectrum in fourier space is finally obtained as : @xmath64 where , @xmath65-i \omega \gamma_{eff},\ ] ] @xmath66 @xmath67 for two values of the nonlinear coefficient : @xmath68(thin line ) and @xmath69 ( thick line ) . parameters used are : @xmath70 , @xmath71 , @xmath72 and @xmath73 . clearly for a finite value of @xmath7 the nms slowly becomes less prominent . ] any information about the mirror s modified motion can be obtained from the study of @xmath74 . an immediate observation reveals that @xmath74 is peaked at a frequency @xmath75 . in the expression for @xmath74 , the first term represents the contribution due to the thermal contact with the bath while the second term is attributed to the contribution due to radiation pressure . in the absence of radiation pressure any dependence from the effective detuning @xmath76 vanishes and the resulting spectrum is simply that of a harmonic oscillator undergoing brownian motion at temperature @xmath1 . in fig.2 , we show the plot of @xmath74 as a function of @xmath77 for @xmath68(thin line ) and @xmath69(thick line ) . in the absence of kerr nonlinearity , a clear nms is observed in the displacement spectrum . the nms is associated to a mixing between the mechanical mode and the fluctuation around the steady state of the cavity field . in the presence of finite kerr nonlinearity , we notice the absence of nms . the absence of nms in the presence of kerr nonlinearity is understood as follows : if @xmath78 , the applied field will couple the vacuum state to the fock state with single photon resonantly . the higher lying photon - number states may be neglected since they are out of resonance . now if initially a photon from the driving field is injected in the cavity with a probability determined by the drive strength . however , injection of a second photon will be blocked , since the presence of two photons in the cavity will require an aditional @xmath79 energy , which can not be provided by the pump laser . only after the first photon leaves the cavity can a second photon be injected . the strong interactions between the photons therefore causes a photon ( kerr ) blockade of cavity transmission and this drastically reduces the photon number fluctuation . in any case if @xmath80 , the cavity would contain more than one photon and the above argument is still valid due to the photon - photon repulsion . we now return to the linearized heisenberg - langevin equations ( [ fluc1 ] , [ fluc2 ] , [ fluc3 ] ) and calculate the corresponding eigenfrequencies that determine the dynamics of nms . in particular , we focus on the following : ( i ) @xmath81 , ( ii ) @xmath82 , ( iii ) @xmath42 and ( iv ) @xmath83 . the two eigenfrequencies are found to be : @xmath84 where , @xmath85 and @xmath86 . there is another pair of eigenfrequencies @xmath87 . for @xmath68 , @xmath88 and @xmath89 , the square root term of @xmath90 is real for @xmath91 and shows nms . on the other hand , for @xmath92 , nms is exhibited for @xmath93 . next we analyze the influence of the kerr nonlinearity on the backaction cooling of the movable mirror . the effective temperature is defined by the total energy of the movable mirror , @xmath94 @xcite , where @xmath95 , @xmath96 and @xmath97 . this basically means that the effective temperature is proportional to the displacement spectrum . from fig.2 , we observe that the displacement spectrum for @xmath98 is always more than that for @xmath99 . from this we conclude that @xmath100 @xmath101 @xmath102 . we can come to this conclusion also from the fact that there is a reduction in the number of photons ( hence radiation pressure ) in the cavity due to photon - photon repulsion due to the presence of the kerr medium and hence an increase in the temperature of the movable mirror . an important point to note is that in order to observe the nms , the energy exchange between the two modes(mechanical and photon number fluctuation ) should take place on a time scale faster than the decoherence of each mode . also the parameter regime in which nms may appear implies cooling . on the negative detuning side , the observation of nms is prevented by the onset of parametric instability . to demonstrate that the dynamics investigated here are within experimental reach , we discuss the experimental parameters from @xcite : from @xcite , the mechanical frequency @xmath103 and @xmath104 . the coupling rate @xmath105 . from @xcite , the kerr nonlinearity is numerically estimated to be about @xmath106 for extremely strong photon - photon repulsion . here we take @xmath107 . the energy of the cavity mode decreases due to the photon loss through the cavity mirrors , which leads to a reduced atom - 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the searches for the higgs boson and new phenomena beyond the standard model of fundumental interactions are important tasks for the new particle accelerator the large hadron collider . the most popular direction beyond the sm is low energy supersymmetry . however , it is not clear how supersymmetry is realized . the simplest case the minimal supersymmetric standard model ( mssm)@xcite is studied in detailes , however possible deviations from it are of great interest as well . there exists a wide class of models which contain so called @xmath0-parity breaking interactions leading to the violation of lepton and baryon numbers@xcite . these models have a number of new coupling constants , some of them are badly constrained , for instance , by rare processes , other ones are less restricted . in this letter we consider the model with the neutrino yukawa interactions and related @xmath0-parity violating term in the superpotential . the interesting consequence of including the @xmath0-parity violating @xmath1 term is the modification of the tree - level expressions for the higgs boson masses and upper bound on the lightest higgs boson mass in the mssm . in this chapter we consider general features of supersymmetric theories related to the scalar potential , its minima , and masses of the physical higgs boson states . it is well known that in these theories the scalar potential is not an arbitrary function of the fields but is fixed by supersymmetry . the sources of the higgs scalar potential are @xmath3-terms coming from gauge - higgs interactions , @xmath2-terms originating from the superpotential , and soft supersymmetry breaking terms : @xmath4 in the mssm based on the @xmath5 symmetry group the @xmath3-term contribution reads @xmath6 where the sum is taken over the gauge groups , their generators , and scalar components of the chiral higgs superfields . from the @xmath0-parity conserving mssm superpotential @xmath7 only the higgs mixing @xmath8-term @xmath9 gives the contribution to the @xmath2-part of the @xmath10 : @xmath11 the higgs doublets are defined as @xmath12 and the `` dot '' operation ( @xmath13 ) denotes the @xmath14 convolution of the ( super)fields doublets with the help of the totally antisymmetric @xmath15-tensor . soft supersymmetry breaking is parametrized by the mass terms for the corresponding scalar components of the higgs superfields and an analogue of the higgs mixing term @xmath16 @xmath17 the final form of the higgs scalar potential then reads @xmath18 & + & b\mu \left(h^+_{\rm u}h^-_{\rm d}-h^0_{\rm u}h^0_{\rm d}\right ) + { \rm h.c . } \nonumber \\ & + & \frac{g^2+g^{\prime 2}}{8}\bigl(\|h^+_{\rm u}\|^2 + \|h^0_{\rm u}\|^2-\|h^0_{\rm d}\|^2-\|h^-_{\rm d}\|^2\bigr)^2 \nonumber \\ & + & \frac{g^2}{2}\left\|h^+_{\rm u}h^{0\dagger}_{\rm d}+h^0_{\rm u}h^{-\dagger}_{\rm d}\right\|^2 . \label{eq : vscal_mssm}\end{aligned}\ ] ] to calculate the masses of physical higgs boson states one has to take the second derivatives of the scalar potential ( [ eq : vscal_mssm ] ) with respect to the corresponding real and imaginary components of the higgs fields taken at the minima and then diagonalize the mass - squared matrices . then one gets the following expressions@xcite for the mass of the @xmath19-odd higgs boson @xmath20 @xmath21 for the heavy and the lightest neutral @xmath19-even higgs bosons @xmath22 and @xmath23 @xmath24 and for the pair of charged higgs bosons @xmath25 @xmath26 three mass eigenstates remain zero corresponding to the goldstone bosons eaten by @xmath27 gauge bosons in the higgs mechanism . here we have introduced the standard notation @xmath28 the ratio of the vacuum expectation values of the neutral components of the two higgs boson doublets . the assumption @xmath29 in eqn.([eq : masshh ] ) leads to the well - known tree - level expression for the lightest mssm higgs boson mass @xmath30 and the famous inequality @xmath31 however , radiative corrections coming mainly from the top - stop loops badly violate this inequality@xcite . for the loop corrected lightest higgs mass one has @xmath32 which shift the mass bound upwards@xcite . renormgroup resummation of all - order leading log contributions using effective potential approach slightly changes the predictions@xcite . besides , the mssm scenario with @xmath33 is excluded experimentally by non - observation of the higgs boson lighter than 114 gev . the superpotential of the standard model ( [ eq : wmssm ] ) is constructed under assumption that neutrinos are massless ( there are no yukawa interactions for the neutrinos which can generate dirac neutrino mass terms after the electroweak symmetry breaking ) and the @xmath0-parity is conserved . in this case it repeats ( up to notations ) the yukawa part of the standard model lagrangian . however , it is believed nowadays that neutrinos have masses , even tiny , then the neutrino yukawa term @xmath34 in the superpotential is possible and should be included ( @xmath35 here are @xmath14 singlet right - handed neutrino superfields , and @xmath36 are neutrino yukawa couplings ) . the latter implies that one also has to include a term @xmath37 to the @xmath0-violating part ( recall that @xmath38 and @xmath39 superfields have the same quantum numbers and no symmetries but lepton number are violated ) . therefore we consider the superpotential of the model in the form : @xmath40 the soft supersymmetry breaking lagrangian also includes the following terms : @xmath41 ( @xmath42 here are scalar components of the corresponding superfields ) . the model with this kind of superpotential has been previously considered and studied@xcite , however the authors were mainly interested in the solution of the @xmath8-problem rather than higgs mass predictions in the model . note also , that for simplicity we do not consider sneutrino v.e.v . the @xmath43 term gives the @xmath2-type contribution to the higgs self - coupling , and the higgs scalar potential now reads @xmath44 & = & \left(\|\mu\|^2 + m^2_{{\rm h}_{\rm u}}\right)\left(\|h^+_{\rm u}\|^2+\|h^0_{\rm u}\|^2\right ) + \left(\|\mu\|^2 + m^2_{{\rm h}_{\rm d}}\right)\left(\|h^0_{\rm d}\|^2+\|h^-_{\rm d}\|^2\right ) \nonumber \\[1 mm ] & + & b\mu \left(h^+_{\rm u}h^-_{\rm d}-h^0_{\rm u}h^0_{\rm d}\right ) + { \rm h.c . } \nonumber \\ & + & \frac{g^2+g^{\prime 2}}{8}\bigl(\|h^+_{\rm u}\|^2 + \|h^0_{\rm u}\|^2-\|h^0_{\rm d}\|^2-\|h^-_{\rm d}\|^2\bigr)^2 + \frac{g^2}{2}\left\|h^+_{\rm u}h^{0\dagger}_{\rm d}+h^0_{\rm u}h^{-\dagger}_{\rm d}\right\|^2 \nonumber \\ & + & \left\|\lambda_\nu^i \lambda_\nu^i \right\| \left\| h^+_{\rm u } h^-_{\rm d } - h^0_{\rm u } h^0_{\rm d } \right\|^2 . \label{eq : vscalhr}\end{aligned}\ ] ] the minimization conditions for the potential ( [ eq : vscalhr ] ) have an additional term due to new neutrino - higgs interaction ( the notation @xmath45 is introduced ) : @xmath46 now it is easy to calculate the masses of the higgs bosons in our model . taking the second derivatives of the potential ( [ eq : vscalhr ] ) with respect to @xmath47 and @xmath47 in the minimum one gets for the elements of the @xmath19-odd higgs boson mass - squared matrix : @xmath48 the mass eigenstates are then given by @xmath49 the zero eigenstate of the matrix ( [ eq : a0matrix11 ] ) @xmath50.\ ] ] is the the goldstone boson eaten by the @xmath51 boson in the higgs mechanism , while the combination @xmath52\ ] ] is the @xmath19-odd neutral higgs boson @xmath20 with the mass expression @xmath53 sometimes it is more convenient to use @xmath54 instead of the @xmath55 parameter , they are related via ( [ eq : massa0 ] ) . let us now look at the masses of @xmath19-even higgs boson @xmath56 and @xmath57 , the latter corresponds to the lightest higgs boson in the mssm . the mass - squared matrix is obtained by differentiating twice the potential ( [ eq : vscalhr ] ) with respect to the @xmath58 and @xmath59 and substituting the expression for the vacuum expectation values ( [ eq : mincon1 ] ) and ( [ eq : mincon2 ] ) . one has , for instance @xmath60 the neutral @xmath19-even higgs boson mass - squared matrix then reads @xmath61 the masses of physical eigenstates are then given by @xmath62 where the quantity @xmath63 related to the new @xmath0-violating neutrino - higgs interactions considered above reads @xmath64 in the limiting case @xmath65 one gets @xmath66 which reproduces the mssm tree - level upper bound for the lightest mssm higgs boson at @xmath67 @xmath68 thus we conclude that in our model the higgs mass constraint is less stringent . fig.[fig : hmass_vs_tan ] illustrates the dependence of the upper bound for the lightest supersymmetric higgs mass as a function of @xmath69 for various values of the @xmath70 parameter . it is easily observed that the bigger the value of @xmath70 , the higher the mass bound . in particular , this takes place for small values of @xmath69 , thus making the parameter space less constrained , and slightly open the scenario with low @xmath69 excluded in the mssm by non - observation of the higgs boson lighter than 114 gev ( fig.[fig : param_space ] ) . for the moderate and large @xmath71 the contribution proportional to @xmath72 is practically negligible , and the higgs mass bound as well as the excluded region in the parameter space practically coincides with those of the mssm . note , that the modified upper bound on the lightest higgs mass ( [ eq : massh_nu ] ) has the same form as in the next - to - minimal supersymmetric standard model ( nmssm ) @xmath73 where @xmath74 is the coupling of the three @xmath14 singlet higgs superfields . for completeness we also calculate the charged higgs boson masses which get negative contribution from the @xmath75 term . the charged higgs mass - squared matrix reads @xmath76 and the physical state mass is @xmath77 the zero mass eigenstate corresponds to the goldstone boson eaten by @xmath78-boson . we have not considered in this letter the possible bilinear @xmath0-parity violating term @xmath79 in the superpotential , since for our purposes it is practically irrelevant . its presence leads only to the redefinition of the higgs scalar potential parameters , while the higgs mass relations ( [ eq : massa0 ] ) , ( [ eq : masshh ] ) , and ( [ eq : masshch ] ) remain unchanged . our goal is to demonstrate the possibility of relaxing the upper bound on the lightest supersymmetric higgs . in conclusion we summarize the main results of this letter . we have shown that introducing the yukawa interactions of neutrinos @xmath80 leading to their dirac mass terms and consequently the possible @xmath0-parity violating terms @xmath81 modifies the expressions of the higgs boson masses since the latter gives @xmath2-type contribution to the higgs scalar potential . the final expressions for the higgs masses in our toy model are ( [ eq : massa0 ] ) , ( [ eq : masshh ] ) , ( [ eq : masshch ] ) and the tree - level bound on the lightest supersymmetric higgs boson @xmath82 gets the noticable shift upwards for small values of @xmath69 . the shift is compatible with the loop corrections to the tree - level mssm expression . this opens the part of the parameter space excluded by the non - observation of the light higgs . the situation looks like as in the nmssm ( [ eq : masshnmssm ] ) . including the bilinear @xmath0-parity violating term @xmath79 will not crucially affect our results . however , the more detailed study is needed , and the analysis of the full @xmath0-parity violating model with the right - handed neutrinos is under study and will be published elsewhere . authors are grateful to d.i . kazakov and a.v . bednyakov for fruitful discussions . financial support from the russian foundation for basic research ( grant # 08 - 02 - 00856 ) and the ministry of education and science of the russian federation ( grant # 3810.2010.2 ) is kindly acknowledged . nilles , _ phys . rept . _ * 110 * , 1 ( 1984 ) . haber , g.l . kane , _ phys . rept . _ * 117 * , 75 ( 1985 ) . gladyshev , d.i . kazakov , _ phys . atom . nucl . _ * 70 * , 1553 ( 2007 ) . aitchison , _ supersymmetry and the mssm : an elementary introduction _ , e - print : hep - ph/0505105 . r. barbier _ et al . _ , _ phys . rept . _ * 420 * , 1 ( 2005 ) , and references therein . j.r . ellis , g. ridolfi , f. zwirner , _ phys . * b262 * , 477 ( 1991 ) . a. brignole , j.r . ellis , g. ridolfi , f. zwirner , _ phys . * b271 * , 123 ( 1991 ) . gladyshev , d.i . kazakov , w. de boer , g. burkart , r. ehret , _ nucl . * b498 * , 3 ( 1997 ) . pierce , j.a . bagger , k.t . matchev , r. zhang , _ nucl . phys . _ * b491 * , 3 ( 1997 ) . m. carena , m. quiros , c.e.m . wagner , _ nucl . * b461 * , 407 ( 1996 ) . s. heinemeyer , w. hollik , g. weiglein , _ phys . lett . _ * b455 * , 179 ( 1999 ) . s. heinemeyer , w. hollik , g. weiglein , _ eur . phys . j. _ * c9 * , 343 ( 1999 ) . gladyshev , d.i . kazakov , _ mod . lett . _ * a10 * , 3129 ( 1995 ) . lopez - fogliani , c. munoz , _ phys . lett . _ * 97 * , 041801 ( 2006 ) . p. ghosh , s. roy , _ neutrino masses and mixing , lightest neutralino decays and a solution to the @xmath8 problem in supersymmetry _ , e - print : arxiv:0812.0084 [ hep - ph ] .	we consider the supersymmetric extension of the standard model with neutrino yukawa interactions and @xmath0-parity violation . we found that @xmath0-parity breaking term @xmath1 leads to an additional @xmath2-type contribution to the higgs scalar potential , and thus to the masses of supersymmetric higgs bosons . the most interesting consequence is the modification of the tree - level expression for the lightest neutral supersymmetric higgs boson mass . it appears that due to this contribution the bound on the lightest higgs mass may be shifted upwards , thus slightly opening the part of the model parameter space excluded by non - observation of the light higgs boson at lep in the framework of the minimal supersymmetric standard model .	4,841	179
the random - field ising model ( rfim ) @xcite has been extensively studied both because of its interest as a simple frustrated system and because of its relevance to experiments @xcite . the hamiltonian describing the model is @xmath4 where @xmath5 are ising spins , @xmath6 is the nearest - neighbors ferromagnetic interaction , @xmath7 is the disorder strength , also called randomness of the system , and @xmath8 are independent quenched random - fields ( rf s ) obtained here from a bimodal distribution of the form @xmath9.\ ] ] various rf probability distributions , such as the gaussian , the wide bimodal distribution ( with a gaussian width ) , and the above bimodal distribution have been considered @xcite . as it is well known , the existence of an ordered ferromagnetic phase for the rfim , at low temperature and weak disorder , follows from the seminal discussion of imry and ma @xcite , when @xmath10 . this has provided us with a general qualitative agreement on the sketch of the phase boundary separating the ordered ferromagnetic ( * f * ) phase from the high - temperature paramagnetic ( * p * ) phase . a sketch of the phase boundary of the 3d bimodal rfim , where @xmath11 is the critical disorder strength and @xmath12 the critical temperature of the pure 3d ising model . the question - mark refers to the mean - field prediction of a tricritical point ( tcp ) , where the transition supposedly changes from second - order at low - fields ( solid line ) to first - order at high - fields ( dotted line).,width=12 ] the phase boundary ( see figure [ fig:1 ] ) separates the two phases of the model and intersects the randomness axis at the critical value of the disorder strength @xmath11 . this value of @xmath11 is known with good accuracy for both the gaussian and the bimodal rfim to be @xmath13 @xcite and 2.21(1 ) @xcite , respectively . a most recent detailed numerical investigation of the phase boundary of the 3d bimodal rfim appears in reference @xcite . however , the general behavior of phases and phase transitions in systems with quenched randomness is still controversial @xcite , and one such lively example is the 3d rfim , which , despite @xmath14 years of theoretical and experimental study , is not yet well understood . in particular , the nature of its phase transition remains unsettled , although it is generally believed that the transition from the ordered to the disordered phase is continuous governed by the zero - temperature random fixed - point @xcite . for the bimodal rfim , the mean - field prediction @xcite of a first - order region separated from a second - order region by a tcp , remains today an open controversy . this main issue has regained interest after the recent observations @xcite of first - order - like features at the strong disorder regime . nowadays , this is the main conflict regarding the nature of the phase transition of the 3d bimodal rfim , although other controversies and scenarios exist in the literature , concerning mainly the intermediate regime of the phase diagram and a possible third spin - glass phase @xcite . thus , the possibility of a first - order transition at the strong disorder regime has been discussed in several papers and has been supported over the years by numerical and theoretical findings . the extreme sharpness of the transition reflected in some studies in the estimated very small values of the order - parameter exponent @xmath15 @xcite has also been reinforcing such first - order scenarios . in particular first - order - like features , such as the appearance of the characteristic double - peak ( dp ) structure of the canonical energy probability density function ( pdf ) , have been recently reported for both the gaussian and the bimodal distributions of the 3d rfim . particularly , wu and machta @xcite , using the wang - landau ( wl ) approach @xcite , reported such properties for the gaussian rfim at a strong disorder strength value @xmath2 below their critical randomness ( @xmath16 ) . moreover , hernndez and diep @xcite have emphasized that they have found evidence for the existence of a tcp in the phase diagram of the bimodal rfim , in agreement with the early predictions of mean - field theory @xcite . these authors have also observed , at the disorder strength value @xmath17 , using standard and histogram monte carlo methods @xcite and more recently the wl algorithm @xcite , the same first - order - like characteristic dp structure and concluded that there is a tcp at some intermediate value of the disorder strength . the existence of a dp structure in the canonical pdf is related to a convex dip in the microcanonical entropy and it is known that for some systems a mere observation of this structure is not sufficient for the identification of a first - order transition . the baxter - wu @xcite and four - state potts models in 2d @xcite are well - known examples of such systems undergoing , in the thermodynamic limit , second - order phase transitions . recently , behringer and pleimling @xcite have demonstrated for these two models that , the appearance of a convex dip in the microcanonical entropy can be traced back to a finite - size effect different from what is expected in a genuine first - order transition . in other words , the pseudosignatures of a first - order transition are finite - size effects , which can be understood within a scaling theory of continuous phase transitions and such first - order - like features cease to exist in the thermodynamic limit . similar first - order - like properties have been observed in many other finite systems , such as the well - known examples of the fixed - magnetization versions of the ising model , where it has been also shown that these finite - size effects disappear in the thermodynamic limit @xcite . the present paper , is the first extensive numerical investigation of this fundamental issue for the 3d bimodal rfim . we proceed , having in mind that a mere observation of a first - order structure is not sufficient for the identification of the transition . this is especially true for the present model , since its critical behavior is obscured by strong and complex finite - size effects , involving also the important issue of the lack of self - averaging @xcite . thus , for a clear identification of the order of the transition , we implement an appropriate version of the lee - kosterlitz ( lk ) free - energy barrier method @xcite . initially , we used a straightforward one - range ( one - r ) wl sampling on a set of a small number of rf realizations , at two values of the disorder strength , @xmath2 and @xmath3 . this attempt enabled us to observe the behavior of the free - energy barrier and the latent heat and indicated that the transition remains continuous at the strong disorder regime . by a second substantial attempt , using a combined more efficient numerical scheme , we simulated large numbers of rf realizations and verified that , indeed , the first - order - like transition signatures are finite - size effects that disappear in the thermodynamic limit . the remainder of the paper is as follows : subsection [ sec:2a ] gives a summary of the wl and lee methods . in particular , we explain , discuss , and give details of two different numerical strategies , called hereafter as one - r approach and high - level one - r approach . we continue , in subsection [ sec:2b ] , to present the application of the lk free - energy barrier method @xcite on the numerical data , obtained via a straightforward application of the one - r wl implementation on a small ensemble of rf realizations for two values of the disorder strength , @xmath2 and @xmath3 . the same method is applied in subsection [ sec:2c ] on the numerical data obtained via a new proposal capable to simulate large numbers of rf realizations , at the disorder strength value @xmath2 . as will be explained in subsection [ sec:2a ] , this latter strategy is an efficient and accurate wl approach , which combines in three stages , the multi - range ( multi - r ) wl algorithm , the high - level one - r wl approach , and a final quite long lee run , to obtain an alternative , and presumably most accurate , density of states ( dos ) . subtle points behind the necessity of implementing such an elaborate scheme will be discussed appropriately in the sequel . finally , we summarize our conclusions in section [ sec:3 ] . several sophisticated simulation techniques , such as cluster algorithms and flat - histogram approaches , have been used to study the rfim @xcite , while graph theoretical algorithms have been used to study properties of the ground - states of this model @xcite . entropic sampling methods such as the lee @xcite and wl @xcite methods are efficient alternatives for complex systems and systems that undergo first - order transitions . accordingly , we will implement a combination of such numerical approaches , based mainly on the wl method , to study the nature of phase transition of the 3d bimodal rfim at the strong disorder regime . the wl algorithm is one of the most refreshing improvements in monte carlo simulation and has been applied to a broad spectrum of interesting problems in statistical mechanics and biophysics @xcite . to apply the wl algorithm , an appropriate energy range of interest has to be identified . a wl random walk ( single spin flip ) is performed in this energy subspace . trials from a spin state with energy @xmath18 to a spin state with energy @xmath19 are accepted according to the transition probability @xmath20.\ ] ] during the wl process the dos @xmath21 is modified ( @xmath22 ) after each spin flip trial by a modification factor @xmath23 . the wl iterative process ( @xmath24 ) is defined as a process in which successive refinements of the dos are achieved by monotonically decreasing the modification factor @xmath25 . most implementations use an initial modification factor @xmath26 and a rule @xmath27 , while a @xmath28 flatness criterion ( on the energy histogram ) is applied in order to move to the next refinement level ( @xmath29 ) @xcite . the wl process is terminated in a sufficiently high - level , at which @xmath30 ( typically @xmath31 ) . note that the detailed balanced condition is satisfied in the limit @xmath32 . there have been several papers in recent years dealing with improvements and sophisticated implementations of the wl iterative process @xcite . some of these suggestions appeared in studies of efficiency and convergence of the wl iterative process @xcite , while others were proposed in applications of the wl scheme in simulating several models of statistical mechanics @xcite . in our recent study , of the first - order transition of the triangular saf model @xcite , we also used a final stage of an unmodified ( @xmath33 ) lee entropic simulation @xcite by applying after , a relatively long run , a lee correction to an already good approximation obtained by the wl process . this final lee entropic stage will be also followed here and it is hoped that through this practice we improve accuracy , but also obtain an idea of the level of approximation , since starting with a very accurate dos and using a sufficiently long run , the lee correction should produce an almost identical dos . in our implementation of this lee entropic stage , we start with a very good approximation of the dos [ @xmath34 , obtained by the wl process after a large number @xmath35 of wl iterations ( we choose @xmath36 ) in which we follow the above described reduction of the modification factor @xmath37 ( @xmath26 , @xmath27 , @xmath38 ) . this good estimate of the dos is used to determine the transition probabilities [ equation ( [ eq:3 ] ) ] for an unmodified ( @xmath33 ) random walk in energy space , as described by lee @xcite , in a process which obeys now the detailed balance condition ( equation ( 5 ) of reference @xcite ) and produces an almost flat energy histogram @xmath39 in the long run . note that , a completely flat histogram ( besides statistical fluctuations ) will be produced in case one is using the exact dos , as pointed out by lee @xcite . however , provided that the monte carlo time is long compared to the ergodicity time , we obtain a better estimate for the dos , i.e. @xmath40 , by the following prescription : @xmath41 the implementation described above is in fact very similar to the suggestion of reference @xcite for a repeated application of the above lee correction scheme , after a first stage of the wl process consisting of @xmath35 wl iterations . then , the successive lee corrections obtained by repeated applications of our equation equation ( [ eq:4 ] ) ( equation ( 15 ) of reference @xcite ) will improve the original dos , as shown in reference @xcite . this repeated application was started in @xcite at an early wl iteration level ( @xmath42 ) and was tested favorably compared to the simple wl process . the multi - r wl approach is the implementation of the method in which one splits the energy range in many subintervals @xcite . this is almost a necessity for very large lattices and the subintervals used are slightly overlapping . the dos s of the separate pieces are joined at the end of the process . this multi - r approach is , of course , a much faster process compared with a straightforward one - r implementation and in many cases has provided very accurate results for very large systems @xcite . although , several papers have pointed out problems with the accuracy , efficiency and convergence of the wl method @xcite , several important related questions are still unanswered , or at least , not well understood . possible distortions ( systematic errors ) induced on the dos by using a multi - r wl approach have not been adequately discussed in the literature . of course , boundary effects of wl sampling in restricted energy subspaces ( multi - r processes ) were observed , analyzed and successfully resolved in reference @xcite . however , subtle effects , coming from the inevitable breaking of the ergodicity of phase space , may be inherent in any restricted energy subspace or multi - r method . below , we will present a novel case coming from our recent studies of the rfim . an escape from this novel , rather discouraging case , will be proposed by compromising between the multi - r and the one - r approach . a typical sp energy pdf for a rf realization at the temperature corresponding to the specific heat peak . three almost coinciding energy pdf s are shown , corresponding to the three approaches discussed in the text , i.e. one - r , multi - r , and high - level one - r . the pdf s are expressed as a function of the energy per site @xmath43.,width=12 ] the wl method has been already applied to the rfim in several previous studies . two such recent investigations , directly related to this work , have been presented for the gaussian @xcite and the bimodal rfim @xcite . as pointed out in the introduction , both of these studies have observed and discussed first - order - like properties of the rfim at the strong disorder regime . the wl method was also implemented , in restricted energy subspaces , for the study of the bimodal rfim in our earlier studies @xcite . in these papers , a systematic restriction of the energy space , with increasing the lattice size , was used and explained in detail in order to further improve the efficiency of the wl method . this approach followed the general spirit of our earlier proposal of estimating the critical behavior of classical statistical systems via entropic simulation in dominant energy subspaces . this restrictive version , utilizes the so called critical minimum energy subspace ( crmes ) technique @xcite to locate and study finite - size anomalies of systems by carrying out the random walk only in the dominant energy subspaces . generally , our finite - size scaling studies have shown that this restrictive practice can be followed in systems undergoing second - order @xcite and also first - order transitions @xcite . furthermore , in our recent study of the phase diagram of the 3d bimodal rfim @xcite we have used a one - r and looser version of this restrictive scheme . in this case we have used the high - levels of the one - parametric wl method as a convenient entropic vehicle , by which the accumulation of the two - parametric , exchange - energy , field - energy , histograms would provide , via extrapolation , a good approximation for the two - parameter dos necessary to find several points of the phase diagram . different energy pdf s for two rf realizations estimated by the numerical approaches discussed in the text : one - r , multi - r , high - level one - r , and final lee . the pdf s are determined at the temperature where the two peaks are of equal height . note that , only the rf realization of panel ( b ) shows a large distortion of the dp structure in the case of the multi - r approach , when compared to the other ( one - r ) schemes.,width=16 ] since substantial histogram accumulation is necessary to overcome statistical errors in such an application , the faster multi - r approach was not used and for having a reliable extrapolation scheme the energy spectrum for the simulation was restricted only from the high - energy side , while the entire low - energy part of the spectrum down to the ground - state was included . for the restriction of the high - energy side we used our data from our previous study of the model at the value @xmath2 . however , for the larger lattice sizes , one can conveniently avoid the ground - state neighborhood . in the present study , we initially used the data of this last straightforward one - r approach @xcite to observe the behavior of the dp structure of some typical rf realizations at the strong disorder regime ( @xmath2 and @xmath3 ) . as already pointed out , these results appear in subsection [ sec:2b ] and are obtained by using a final of @xmath44 wl iteration levels for the smaller lattices up to @xmath45 and a final of @xmath46 wl iteration levels for the larger sizes ( @xmath47 ) . subsequently , and in order to simulate larger numbers of dp rf realizations , we decided to test carefully and then use a multi - r approach . thus , we compared the ( energy pdf ) dp s of some typical rf realizations obtained by the one - r approach with the dp s obtained by a usual multi - r approach ( corresponding to the same or an even higher level of the wl process ) . for some rf realizations the energy pdf graphs almost coincided and this was especially true for the single - peak ( sp ) rf realizations . but also for several dp rf realizations the corresponding graphs were close enough and within statistical errors . figures [ fig:2 ] and [ fig:3](a ) show two such examples , one corresponding to a sp rf realization and one to a dp rf realization . however , for some peculiar dp rf realizations the dp graphs resulting form the multi - r approach were dislocated and with a rather large deviation in their depth , when compared with the dp energy pdf graphs obtained for the same rf realizations by the one - r approach . figure [ fig:3](b ) shows a characteristic case , corresponding to a serious dislocation and underestimation of the dp structure . furthermore , it was observed that the dp details were very sensitive to the division of the energy range to subintervals , indicating that the distortion errors were due to the application of the multi - r approach . after several tests , we concluded that this peculiar problem is related to the division of the dp range in subintervals . it appears that for some rf realizations , the structure of the convex dip in the microcanonical entropy is not well estimated by using the multi - r approach within the dp range . we concluded that the details of the convex dip are sensitive to possible subtle violations of ergodicity , induced by the multi - r approach . thus , we tried to find an alternative that will not suffer from this problem and still be efficient enough so that we could simulate large numbers of rf realizations . the developed method will be called high - level one - r wl approach and is a further sophistication in the same spirit of our earlier practice in optimizing the wl entropic sampling . it combines the multi - r and the one - r approaches in an almost optimum way and seems to meet the needs of a careful estimation of the dp structure of the present model . the details of this approach , applied here only for @xmath2 , are as follows . for each lattice size , a wide energy subspace restricted mainly from the high - energy side is divided in relatively small subintervals , of the order of @xmath48 energy levels and a multi - r approach is applied up to the @xmath49 wl iteration level . this completes the first stage of the approach and the dos obtained is used to estimate for each particular rf realization the dp range . this identification is easily achieved by using our earlier practice for first - order transitions @xcite , by finding the appropriate temperature of equal height for the two peaks of the energy pdf . the energy pdf at this temperature is normalized so that the height of the two peaks is unity and corresponds to energies @xmath50 and @xmath51 . the dp range is now identified as follows : the left - end of the dp subspace is the energy @xmath52 , for which the density becomes greater than @xmath53 starting from @xmath54 and respectively the right - end of the dp subspace is the energy @xmath55 , for which also the density becomes greater than @xmath53 starting from @xmath56 . having this first approximate identification of the dp subspace , a one - r wl walk is again performed at the level @xmath49 in a subspace which is wider than the dp subspace by a factor of @xmath57 at each end . in other words , the left - end @xmath52 is shifted to the left by @xmath57 of the dp range and correspondingly the right - end @xmath55 is shifted to the right by the same amount . after the @xmath49 one - r approach the ends of the dp subspace are re - estimated and fixed . the one - r wl approach is then carried on only in this dp subspace for the higher levels @xmath58 , and @xmath44 . this completes the second stage of our approach . finally , an unmodified lee random walk is performed in this dp subspace , using the last approximation of the wl dos for the transition rates . the lee correction is applied at the end to produce an alternative estimate for the dos . the time duration of this last lee run is taken to be equal to the duration of the four last one - r wl iterations . for some rf realizations , this one - r process was pushed up to @xmath46 to observe differences and estimate statistical errors . in all cases , these statistical errors were very small , much smaller than the observed sample - to - sample fluctuations ( see also discussion in subsection [ sec:2c ] below ) . we give here some details for the sizes of the dp subspaces involved in the above scheme . for @xmath59 , the initial restricted energy subspace used for the multi - r process was of the order of @xmath60 energy levels , i.e. counting energy levels from the all minus spin state this was the subspace defined by the levels @xmath61 to @xmath62 . typically the size of the resulting dp subspace was of the order of @xmath63 energy levels , which is about @xmath64 of the initial restricted energy subspace . the left - end @xmath52 roughly fluctuated , for a sample of @xmath48 rf realizations , between the levels @xmath65 , while the right - end @xmath55 between the levels @xmath66 . respectively , for @xmath67 the initial restricted energy subspace used for the multi - r process was of the order of @xmath68 energy levels , defined by the levels @xmath69 to @xmath70 . typically , the size of the resulting dp subspace was again of the order of @xmath63 energy levels , which is about @xmath71 of the initial restricted energy subspace . in this case , @xmath52 fluctuated , again for a sample of @xmath48 rf realizations , between the levels @xmath72 , while @xmath55 between the levels @xmath73 . to conclude the above remarks , let us point out that , for a typical rf realization , when @xmath59 , a safe dp location is established after the @xmath49 one - r level and consists of about @xmath74 energy levels ( @xmath75 ) . it is quite astonishing that the same energy space requirements are needed also for @xmath67 and this is related to the final conclusion of this paper , that the dp peak width , in units of energy per site , tends to zero in the limit @xmath76 . the above remarks clarify also the reasons behind the efficiency of the present proposal ( high - level one - r approach ) . typically , for one rf realization of a lattice size @xmath45 at the disorder strength value @xmath2 ( figure [ fig:3 ] ) , the simulation time @xmath77 for the one - r wl process , in all the energy subspace ( @xmath78 ) , was of the order of @xmath79 hours performed in a pentium iv 3ghz . the simulation times corresponding to the other cases presented in panel ( b ) of figure [ fig:3 ] are as follows : multi - r approach @xmath80 and high - level one - r wl approach together with the final lee run @xmath81 . we may also point out , that very recently , fernndez @xcite , have found that the phase space for the first - order transition of the 3d site - diluted four - states potts model is reduced , as compared with the expectations from simulations in small lattice sizes , a behavior very similar to the above observations . their microcanonical approach @xcite may also be an interesting alternative , not used previously , for the study of the present model . in this subsection , we present the application of the lk method on the numerical data obtained by the one - r wl method . as mentioned in the previous subsection , the one - r approach on both values of the disorder strength ( @xmath2 and @xmath3 ) was applied in a wide energy spectrum and we have conveniently avoided a suitable ground - state neighborhood . the total number of rf realizations simulated ( @xmath82 ) varies from @xmath83 realizations for @xmath84 to @xmath85 realizations for @xmath86 . from a traditional point of view , the nature of a phase transition can be , in principle , determined by examining the finite - size scaling of various thermodynamic quantities , such as the specific heat and susceptibility peaks . these two quantities , as well as others , are expected , from the general theory of first - order transition , to follow an @xmath87 divergence and may be used as indicators of the order of the transition . however , this practice is very often inconclusive even for simple systems and may be seriously questioned for random systems in which fundamental and subtle problems exist concerning the averaging process over disorder . for the present rfim , the lack of self - averaging observed by the present authors @xcite ( see also references @xcite ) , may be of crucial importance especially in trying to construct convenient indicators for the nature of the transition at the strong disorder regime . first - order - like realizations are expected to exhibit sharp specific heat and susceptibility peaks , such as that observed also in the gaussian case studied by wu and machta @xcite . therefore , according to our previous papers @xcite and as pointed out also in reference @xcite the information concerning an individual first - order - like realization will be washed out - as a result of the strong fluctuation in the pseudocritical temperature - in considering for instance the behavior of the average specific heat curve . from the above discussion , it is obvious that in order to avoid problems with the lack of self - averaging property , the first - order - like features of each realization must be computed separately and the disorder average should be applied at the end in the proper first - order indicator . the most convenient approach in this case is to use the free - energy barrier method proposed by lee and kosterlitz @xcite . ( a ) energy pdf at the temperature where the two peaks are of equal height for a first - order - like rf realization of a lattice size @xmath45 at disorder strength @xmath3 . the inset shows the corresponding sharp specific heat peak . ( b ) ratio @xmath88 of realizations showing a dp energy pdf in an ensemble of @xmath82 number of realizations as a function of the linear system size @xmath89 . set a : @xmath90 ( open triangles ) and @xmath91 ( filled squares ) . set b : @xmath92 ( filled circles).,width=16 ] this method has been already successfully applied by chen @xcite for the study of an analogous disordered system , namely the two - dimensional eight - state potts model with quenched bond randomness . thus , we will proceed now to apply the lk method for the identification of the transition of the bimodal rfim at the strong disorder regime . the method is well - known and has been widely applied to several spin models @xcite , so we will proceed giving only the necessary definitions adapted to the present disordered system . figure [ fig:4](a ) illustrates , in the main frame , the typical dp energy probability distribution ( seed @xmath93 ) and in the inset the corresponding sharp specific heat peak of a first - order - like realization on a cubic lattice of linear lattice size @xmath45 at the disorder value @xmath3 . with the help of this figure , let us define the surface tension @xmath94 for each dp realization , where the definition of the lk free - energy barrier is , using the canonical energy pdf @xmath95 , @xmath96}$ ] ( @xmath97 , as in figure [ fig:2 ] ) . therefore , with the help of only the generated dp realizations ( @xmath98 is their number , see also the discussion below ) , we define the disorder average of the surface tension , which is proposed as the relevant indicator representing the ensemble of dp realizations , as @xmath99_{av}=\frac{1}{n_{dp}}\sum_{i=1}^{n_{dp}}\sigma_{i}(l).\ ] ] the second important ingredient in the size development of the observed first - order - like properties of the 3d bimodal rfim is the behavior of the width @xmath100 of the individual dp s , representing the latent heat of the transition , in the case of a first - order transition . again with the help of the illustration in figure [ fig:4](a ) , we define the disorder average over the ensemble of dp realizations of the width of the transition as @xmath101_{av}=\frac{1}{n_{dp}}\sum_{i=1}^{n_{dp}}\delta e_{i}(l).\ ] ] figure [ fig:4](b ) presents the relative number of such first - order - like realizations @xmath98 , in a total number of @xmath82 realizations . set a refers to the straightforward one - r wl approach for the two values @xmath2 and @xmath3 of the disorder strength . the observed increase with lattice size of the probability for such first - order - like realizations , is in qualitative agreement with the general behavior reported by wu and machta @xcite for the gaussian rfim . from table v of reference @xcite one observes a general tendency of the ratio @xmath88 to increase with the system size and the disorder strength and since we are studying the system at a relatively higher disorder strength value , our ratios are quite comparable with those given in table v of reference @xcite , although the latter refer to the gaussian rfim . clearly , at the strong disorder regime and as the lattice size increases , the percentage of realizations showing a dp in the energy probability distribution increases and approaches unity very rapidly . @xmath102-behavior of @xmath103_{av}$ ] ( a ) and @xmath104_{av}$ ] ( b ) at @xmath2 ( open circles ) and @xmath3 ( filled circles ) for @xmath105 . sample - to - sample fluctuations are illustrated by the error bars . the solid and dotted lines are corresponding linear fits for the larger sizes ( @xmath106 ) . in both panels and for both values of the disorder strength , a limiting value very close to zero for the free - energy barrier and the latent heat is obtained , indicating a continuous transition.,width=16 ] thus , in our case for lattice sizes of the order of @xmath107 almost all of the simulated realizations showed a dp energy probability distribution ( in fact for the value @xmath3 the ratio @xmath88 reached unity ) . this observation explains why we have used only the @xmath98 realizations in the disorder averaging ( equations ( [ eq:5 ] ) and ( [ eq:6 ] ) ) , since it implies that only these realizations are of interest , since for large lattices these will dominate the behavior . this practice avoids transient effects , coming from the small lattices , and we have further pushed it by applying here a quite rather strict criterion for the definition of the dp realizations : @xmath108 ( note the normalization of the energy pdf : @xmath109 in figure [ fig:4](a ) ) . the behavior of the disorder average of the surface tension @xmath103_{av}$ ] , for both @xmath2 and @xmath3 , as a function of the inverse lattice size , is shown in figure [ fig:5](a ) . linear fits are applied only for the data corresponding to sizes @xmath106 . from these linear plots ( solid and dotted lines ) it appears that @xmath103_{av}$ ] approaches zero , as expected at a second - order transition . this observation strongly indicates that , what we are observing from these dp realizations for small sizes is a finite - size effect that will disappear in the thermodynamic limit . the solid and dotted lines explicitly illustrate this , using a linear extrapolation of the large size data for @xmath2 and @xmath110 , giving an almost zero surface tension in the limit @xmath76 for both values of the disorder strength : @xmath111 and @xmath112 , respectively . furthermore , figure [ fig:5](b ) depicts an undeniable steady approach to zero of the above representative width , again for both values of the disorder strength @xmath2 and @xmath3 . the linear extrapolation attempts are shown by the solid and dotted lines and give also an almost zero value for the latent heat of the order of @xmath113 and @xmath114 , respectively . this is a further strong manifestation in favor of the continuous phase transition scenario . thus , the evidence presented in this subsection for the 3d bimodal rfim are in agreement with the favored view of most existing theoretical and numerical studies @xcite that the phase transition of the 3d rfim is of second - order . in order to present even stronger numerical evidence of a vanishing ( in the limit @xmath76 ) surface tension we will now attempt to go well beyond the observation of several typical rf realizations . in the next subsection , numerical evidence will be presented for the finite - size behavior of the free - energy barrier and the latent heat of much larger ensembles of rf realizations , obtained via the efficient high - level one - r entropic scheme described in subsection [ sec:2a ] . in this subsection , we will apply the lk method on the numerical data obtained by our second numerical strategy , described in subsection [ sec:2a ] . using the high - level one - r wl approach and its final lee correction , we generated numerical data for large number of rf realizations at the disorder strength value @xmath2 . in this case , the number @xmath82 of realizations varied so that for every lattice size @xmath115 , @xmath48 rf realizations showing a dp structure in the energy pdf ( @xmath116 ) were simulated . for the small sizes @xmath117 and @xmath118 only @xmath85 dp realizations have been identified in respective ensembles of @xmath119 and @xmath120 simulated realizations . set b in figure [ fig:4](b ) refers to the above mentioned large ensembles of realizations simulated at @xmath2 . in the present case , a looser criterion ( @xmath121 ) was applied for the identification of a dp realization . the corresponding ratios @xmath88 in figure [ fig:4](b ) almost coincide for set a and set b , for the larger sizes . this is easily explained by observing that for large sizes almost all dp realizations appear to have a quite deep minimum in the energy pdf . however , as it will be shown below , the scaling of these minima will not support a first - order character of the transition . @xmath102-behavior of @xmath103_{av}$ ] ( a ) and @xmath104_{av}$ ] ( b ) at @xmath2 and for @xmath105 . two set of results are shown , corresponding to the high - level one - r wl approach ( open triangles ) and the lee correction ( filled circles ) . the sample - to - sample fluctuations for the case of the high - level one - r wl results are shown with the larger cap width . the solid lines are the corresponding linear fits for @xmath106 , giving very small , close to zero , negative values for @xmath122_{av}$ ] and @xmath123_{av}$ ] , thus verifying the scenario of figure [ fig:5].,width=16 ] figure [ fig:6 ] presents our results for the disorder averaged surface tension @xmath124_{av}$ ] and latent heat @xmath125_{av}$ ] over set b of realizations at @xmath2 . the open triangles refer to the results obtained by the high - level one - r wl approach , whereas the filled circles to those estimated from the final lee correction . in panel ( a ) the values of @xmath126_{av}$ ] are shown for @xmath105 . although for sizes up to @xmath127 @xmath103_{av}$ ] seems to steadily increase , for sizes @xmath47 a clear approach to zero is observed and this fact is compatible to the behavior of figure [ fig:5](a ) . respectively , panel ( b ) shows the values of @xmath104_{av}$ ] , also for @xmath105 . the final large size decrease , is explicitly illustrated by the solid line ( in both panels of figure [ fig:6 ] ) , revealing the true asymptotic behavior of these quantities . specifically , the solid lines are linear fits performed on the data obtained by the high - level one - r wl approach , for the larger lattice sizes studied ( @xmath106 ) , giving : @xmath128 and @xmath129 for @xmath130_{av}$ ] and @xmath123_{av}$ ] , respectively . as it can be seen from this figure , the data obtained by the lee correction process would practically provide the same limiting values for the surface tension and the latent heat and therefore the corresponding linear fits are not shown . noteworthy that , a comparison between statistical errors and sample - to - sample fluctuations is quite apparent in this figure . the values of @xmath103_{av}$ ] and @xmath104_{av}$ ] estimated over the two sets of realizations ( set a and set b ) for @xmath2 estimated via the two different numerical strategies presented in this paper are of the same order . those of set a are slightly larger as a consequence of the more strict criterion , @xmath108 , used for the identification of the dp rf realizations . the interesting first - order - like properties of the model , reported by hernndez and diep @xcite and hernndez and ceva @xcite for the bimodal rfim and by wu and machta @xcite for the gaussian rfim , have added more complication and novelty to the rfim . the first - order - like characteristics of the gaussian rfim found by wu and machta @xcite revealed that the appearance of these strong finite - size effects are independent of the rf distribution and their existence is related to the value of the disorder strength . this observation is not compatible with mean - field theory , since its first - order prediction for only the bimodal case depends on the existence of a minimum at zero - field of the distribution @xcite . the present study has illustrated that these characteristics are most likely effects complicating the finite - size behavior of the model but not determining its true asymptotic scaling behavior . our results clearly indicate that the interface tension @xmath126_{av}$ ] vanishes and the two peaks of the energy pdf move together in the thermodynamic limit and therefore provide convincing evidence that the transition is continuous and that there in no tcp along the phase transition line . consequently , the coexistence between an ordered phase and a disordered phase will be hardly detectable in the thermodynamic limit . nevertheless , for large but finite systems , the dip represents a considerable barrier between the ordered phase and the disordered one , so that in some sense , one may speak for a phase coexistence for finite systems . the results of figures [ fig:5](a ) and [ fig:6](a ) indicate a linear approach of the interface tension to zero in the limit @xmath131 , and therefore an exponential increase of the ratio @xmath132 in @xmath89 , and point to an unconventional continuous transition , in which the energy pdf will approach two delta functions that move together ( see figures [ fig:5](b ) and [ fig:6](b ) ) in the thermodynamic limit . such an unconventional behavior has been first predicted by eichhorn and binder @xcite , for the the order - parameter pdf of the 3d random - field three - state potts model . these authors have explained such an unusual behavior by presenting in detail the consequences of a scenario ( including leading corrections to scaling ) based on a finite - size scaling statement for the order - parameter universal pdf . according to this scenario , the finite - size scaling behavior in rf systems can be recovered and the relative width of the corresponding order - parameter pdf peaks vanishes in the scaling limit as @xmath133 , where @xmath134 is the critical exponent describing the violation of hyperscaling ( @xmath135 ) . in conclusion , the results of this paper and the observations of references @xcite are strong indications of a similar unusual scenario for a continuous transition , calling for further investigation , such as the determination of the exponent @xmath134 . two entropic sampling numerical strategies have been implemented for the study of the first - order - like properties of the 3d bimodal rfim . our experience and comparative studies , using different numerical approaches , revealed the sensitivity of the double - peak structure of the energy probability density function of the model , especially with increasing the system size . thus , the need for careful implementations of entropic sampling techniques in cases of complex systems has been critically discussed . an efficient high - level one - range wang - landau approach has been proposed as a quite safe alternative , avoiding subtle problems related to the position and the depth of the minima of the double - peak energy probability density functions . reliable data were obtained using this high - level one - range wang - landau approach and the corresponding lee correction for large numbers of random - field realizations and quite large lattice sizes , up to @xmath67 . using these data and by a systematic finite - size analysis , implementing the lee - kosterlitz method , we have studied the nature of the transition at the strong disorder regime . our results for both the free - energy barrier and latent heat for the ensemble of double - peak random - field realizations suggest a behavior in accordance with a continuous transition . these results disclose the open controversy for the existence of a tricritical point in the phase diagram of the 3d bimodal rfim at the strong disorder regime and serve in favor of the unusual scenario for a continuous transition , originally proposed by eichhorn and binder @xcite . it will be interesting to repeat the present investigation for the wide bimodal distribution ( with a gaussian width ) and even for the gaussian distribution , at the strong disorder regime , since this would provide additional confidence to our conclusions .	two numerical strategies based on the wang - landau and lee entropic sampling schemes are implemented to investigate the first - order transition features of the 3d bimodal ( @xmath0 ) random - field ising model at the strong disorder regime . we consider simple cubic lattices with linear sizes in the range @xmath1 and simulate the system for two values of the disorder strength : @xmath2 and @xmath3 . the nature of the transition is elucidated by applying the lee - kosterlitz free - energy barrier method . our results indicate that , despite the strong first - order - like characteristics , the transition remains continuous , in disagreement with the early mean - field theory prediction of a tricritical point at high values of the random - field . _ keywords _ : bimodal random - field ising model , wang - landau sampling , free - energy barrier method	11,837	223
the kuiper belt is a long - lived region of the solar system just beyond neptune where the planetisimals have not coalesced into a planet . it contains about 80,000 objects with radii greater than 50 km ( trujillo , jewitt & luu 2001 ) which have been collisionally processed and gravitationally perturbed throughout the age of the solar system . the short - period comets and centaurs are believed to originate from the kuiper belt ( fernandez 1980 ; duncan , quinn & tremaine 1988 ) . physically , the kuiper belt objects ( kbos ) show a large diversity of colors from slightly blue to ultra red ( @xmath8 to @xmath9 , luu and jewitt 1996 ) and may show correlations between colors , inclination and/or perihelion distance ( jewitt & luu 2001 ; trujillo & brown 2002 ; doressoundiram et al . 2002 ; tegler & romanishin 2003 ) . spectra of kbos are mostly featureless with a few showing hints of water ice ( brown , cruikshank & pendleton 1999 ; jewitt & luu 2001 ; lazzarin et al . the range of kbo geometric albedos is still poorly sampled but the larger ones likely have values between 0.04 to 0.10 ( jewitt , aussel & evans 2001 ; altenhoff , bertoldi & menten 2004 ) . time - resolved observations of kbos show that @xmath10 vary by @xmath11 magnitudes , @xmath12 by @xmath13 magnitudes and @xmath14 by @xmath15 magnitudes ( sheppard & jewitt 2002 ; ortiz et al . 2003 ; lacerda & luu 2003 ; sheppard & jewitt 2004 ) . one object , ( 20000 ) varuna , displays a large photometric range and fast rotation which is best interpreted as a structurally weak object elongated by its own rotational angular momentum ( jewitt & sheppard 2002 ) . a significant fraction of kbos appear to be more elongated than main - belt asteroids of similar size ( sheppard & jewitt 2002 ) . the kbo phase functions are steep , with a median of @xmath16 magnitudes per degree between phase angles of 0 and 2 degrees ( sheppard & jewitt 2002 ; schaefer & rabinowitz 2002 ; sheppard & jewitt 2004 ) . about @xmath17 of the kbos are binaries with separations @xmath18 ( noll et al . 2002 ) while binaries with separations @xmath19 may constitute about @xmath20 of the population ( trujillo 2003 , private communication ) . all the binary kbos found to date appear to have mass ratios near unity , though this may be an observational selection effect . the mechanism responsible for creating kbo binaries is not clear . formation through collisions is unlikely ( stern 2002 ) . weidenschilling ( 2002 ) has proposed formation of such binaries through complex three - body interactions which would only occur efficiently in a much higher population of large kbos than can currently be accounted for . goldreich , lithwick & sari ( 2002 ) have proposed that kbo binaries could have formed when two bodies approach each other and energy is extracted either by dynamical friction from the surrounding sea of smaller kbos or by a close third body . this process also requires that the density of kbos was @xmath21 to @xmath22 times greater than now . they predict that closer binaries should be more abundant in the kuiper belt while weidenschilling s mechanism predicts the opposite . the present paper is the fourth in a series resulting from the hawaii kuiper belt variability project ( hkbvp , see jewitt & sheppard 2002 ; sheppard & jewitt 2002 ; sheppard & jewitt 2004 ) . the practical aim of the project is to determine the rotational characteristics ( principally period and shape ) of bright kbos ( @xmath23 ) in order to learn about the distributions of rotation period and shape in these objects . in the course of this survey we found that 2001 qg@xmath0 had an extremely large light variation and a relatively long period . we have obtained optical observations of 2001 qg@xmath0 in order to accurately determine the rotational lightcurve and constrain its possible causes . 2001 qg@xmath0 has a typical plutino orbit in 3:2 mean - motion resonance with neptune , semi - major axis at 39.2 au , eccentricity of 0.19 and inclination of 6.5 degrees . we used the university of hawaii ( uh ) 2.2 m diameter telescope atop mauna kea in hawaii to obtain r - band observations of 2001 qg@xmath0 on three separate observing runs each covering several nights : ut september 12 and 13 2002 ; august 22 , 26 , 27 and 28 2003 ; september 27 , 28 and 30 2003 . two different ccd cameras were employed . for the september 2002 and september 2003 observations we used a @xmath24 pixel tektronix ccd ( @xmath25 @xmath26 pixels ) camera with a @xmath27 pixel@xmath28 scale at the f/10 cassegrain focus . an antireflection coating on the ccd gave very high average quantum efficiency ( 0.90 ) in the r - band . the field - of - view was @xmath29 . for the august 2003 observations we used the orthogonal parallel transfer imaging camera ( optic ) . optic has two @xmath30 pixel lincoln lab ccid28 orthogonal transfer ccds developed to compensate for real - time image motion by moving the charge on the chips to compensate for seeing variations ( tonry , burke & schechter 1997 ) . howell et al . ( 2003 ) have demonstrated that these chips are photometrically accurate and provide routine sharpening of the image point spread function . there is a @xmath31 gap between the chips . the total field - of - view was @xmath32 with @xmath33 @xmath26 pixels which corresponds to @xmath34 pixel@xmath28 scale at the f/10 cassegrain focus . the same r - band filter based on the johnson - kron - cousins photometric system was used for all uh 2.2 m observations . in addition we used the keck i 10 m telescope to obtain bvr colors of 2001 qg298 at its maximum and minimum light on ut august 30 , 2003 . the lris camera with its tektronix @xmath35 pixel ccd and @xmath25 @xmath26 pixels ( image scale @xmath36 pixel@xmath28 ) was used ( oke et al . 1995 ) with the facility broadband bvr filter set . due to a technical problem with the blue camera side we used only the red side for photometry at bv and r. the blue filter response was cut by the use of a dichroic at 0.460 @xmath26 . all exposures were taken in a consistent manner with the telescope autoguided on bright nearby stars . the seeing ranged from @xmath37 to @xmath38 during the various observations . 2001 qg@xmath0 moved relative to the fixed stars at a maximum of @xmath39 hr@xmath28 corresponding to trail lengths @xmath40 @xmath41 in the longest ( 450 sec ) exposures . thus motion of the object was insignificant compared to the seeing . images from the uh telescope were bias - subtracted and then flat - fielded using the median of a set of dithered images of the twilight sky . data from keck were bias subtracted and flattened using flat fields obtained from an illuminated spot inside the closed dome . landolt ( 1992 ) standard stars were employed for the absolute photometric calibration . to optimize the signal - to - noise ratio we performed aperture correction photometry by using a small aperture on 2001 qg@xmath0 ( @xmath42 to @xmath43 in radius ) and both the same small aperture and a large aperture ( @xmath44 to @xmath45 in radius ) on ( four or more ) nearby bright field stars . we corrected the magnitude within the small aperture used for the kbos by determining the correction from the small to the large aperture using the field stars ( c.f . tegler and romanishin 2000 ; jewitt & luu 2001 ; sheppard & jewitt 2002 ) . since 2001 qg@xmath0 moved slowly we were able to use the same field stars from night to night within each observing run , resulting in very stable relative photometric calibration from night to night . the observational geometry for 2001 qg@xmath0 on each night of observation is shown in table 1 . tables 2 and 3 show the photometric results for 2001 qg@xmath0 . we used the phase dispersion minimization ( pdm ) method ( stellingwerf 1978 ) to search for periodicity in the data . in pdm , the metric is the so - called @xmath46 parameter , which is essentially the variance of the unphased data divided by the variance of the data when phased by a given period . the best - fit period should have a very small dispersion compared to the unphased data and thus @xmath47 1 indicates that a good fit has been found . 2001 qg@xmath0 showed substantial variability ( @xmath48 magnitudes with a single - peaked period near @xmath49 hr ) in r - band observations from two nights in september 2002 . we obtained further observations of the object in 2003 to determine the lightcurve with greater accuracy . pdm analysis of all the apparent magnitude r - band data from the september 2002 and august and september 2003 observations shows that 2001 qg@xmath0 has strong @xmath46 minima near the periods @xmath50 hr and @xmath51 hr , with weaker alias periods flanking these ( figure [ fig : pdmqg ] ) . we corrected the apparent magnitude data for the minor phase angle effects ( we used the nominal 0.16 magnitudes per degree found in sheppard & jewitt 2003 ) and light travel - time differences of the observations to correspond to the august 30 , 2003 observations . we then phased the data to all the peaks with @xmath52 and found only the 6.8872 and 13.7744 hour periods to be consistent with all the data ( figures [ fig : phaseqgsingle ] and [ fig : phaseqgdouble ] ) . through a closer look at the pdm plot ( figure [ fig : pdmqgsmall ] ) and phasing the data we find best fit periods @xmath53 hr ( a lightcurve with a single maximum per period ) and @xmath54 hr ( two maxima per period as expected for rotational modulation caused by an aspherical shape ) the double - peaked lightcurve appears to be the best fit with the minima different by about 0.1 magnitudes while the maxima appear to be of similar brightness . the photometric range of the lightcurve is @xmath55 magnitudes . the keck bvr colors of 2001 qg@xmath0 show no variation from minimum to maximum light within the photometric uncertainties of a few % ( see figures [ fig : phaseqgsingle ] and [ fig : phaseqgdouble ] ) . this is again consistent with a lightcurve that is produced by an elongated shape , rather than by albedo variations . the colors ( @xmath56 , @xmath57 ) show that 2001 qg@xmath0 is red and similar to the mean values ( @xmath58 , @xmath59 , 28 objects ) for kbos as a group ( jewitt and luu 2001 ) . the absolute magnitude of a solar system object , @xmath60 , is the hypothetical magnitude the object would have if it where at heliocentric ( @xmath61 ) and geocentric ( @xmath62 ) distances of 1 au and had a phase angle ( @xmath63 ) of 0 degrees . we use the relation @xmath64 to find the absolute magnitude by correcting for the geometrical and phase angle effects in the 2001 qg@xmath0 observations . here @xmath65 is the apparent red magnitude of the object and @xmath66 is the phase function . using the nominal value of @xmath67 magnitudes per degree for kbos at low phase angles ( sheppard & jewitt 2002 ; sheppard & jewitt 2004 ) and data from table 1 we find that 2001 qg@xmath0 has @xmath68 at maximum light and @xmath69 magnitudes at minimum light . if attributed to a rotational variation of the cross - section , this corresponds to a ratio of maximum to minimum areas of 2.85:1 . the effective radius of an object can be calculated using the relation @xmath70 $ ] where @xmath71 is the apparent red magnitude of the sun ( @xmath72 ) , @xmath73 is the red geometric albedo and @xmath74 ( km ) is the effective circular radius of the object . if we assume an albedo of 0.04 ( 0.10 ) this corresponds to effective circular radii at maximum and minimum light of about 158 ( 100 ) km and 94 ( 59 ) km , respectively . at the mean absolute magnitude of 6.85 mag , the effective circular radius is 122 ( 77 ) km . only three other objects in the solar system larger than 25 km in radius are known to have lightcurve ranges @xmath75 magnitude ( table 4 ) . following jewitt and sheppard ( 2002 ) we discuss three possible models of rotational variation to try to compare the objects from table 4 with 2001 qg@xmath0 . on asteroids , albedo variations contribute brightness variations that are usually less than about @xmath76 ( degewij , tedesco & zellner 1979 ) . rotationally correlated color variations may be seen if the albedo variations are large since materials with markedly different albedos may differ compositionally . as seen in table 4 , saturn s satellite iapetus is the only object in which variations @xmath771 mag . are explained through albedo . the large albedo contrast on iapetus is likely a special consequence of its synchronous rotation and the anisotropic impact of material trapped in orbit about saturn onto its leading hemisphere ( cook & franklin 1970 ) . iapetus shows clear rotational color variations ( @xmath78 0.1 mag . ) that are correlated with the rotational albedo variations ( millis 1977 ) and which would be detected in 2001 qg@xmath0 given the quality of our data . the special circumstance of iapetus is without obvious analogy in the kuiper belt and we do not believe that it is a good model for the extreme lightcurve of 2001 qg@xmath0 . pluto shows a much smaller variation ( about 0.3 magnitudes ) thought to be caused by albedo structure ( buie , tholen & wasserman 1997 ) . pluto is so large that it can sustain an atmosphere which may contribute to amplifying its lightcurve range by allowing surface frosts to condense on brighter ( cooler ) spots . thus brighter spots grow brighter while darker ( hotter ) spots grow darker through the sublimation of ices . this positive feedback mechanism requires an atmosphere and is unlikely to be relevant on a kbo as small as 2001 qg@xmath0 . while we can not absolutely exclude surface markings as the dominant cause of 2001 qg@xmath0 s large rotational brightness variation , we are highly skeptical of this explanation . we measure no color variation with rotation , there appear to be two distinct minima and the range is so large as to be beyond reasonable explanation from albedo alone . since surface markings are most likely not the cause of the lightcurve , the observed photometric variations are probably caused by changes in the projected cross - section of an elongated body in rotation about its minor axis . the rotation period of an elongated object should be twice the single - peaked lightcurve period because of the projection of both long axes ( 2 maxima ) and short axes ( 2 minima ) during one full rotation . if the body is elongated , we can use the ratio of maximum to minimum brightness to determine the projection of the body shape into the plane of the sky . the rotational brightness range of a triaxial object with semiaxes @xmath79 in rotation about the @xmath80 axis and viewed equatorially is @xmath81 where @xmath82 is expressed in magnitudes . this gives a lower limit to @xmath83 because of the effects of projection . using @xmath84 for 2001 qg@xmath0 , we find the lower limit is @xmath85 . this corresponds to @xmath86 and @xmath87 km for the geometric albedo 0.04 case and @xmath88 and @xmath89 km for an albedo of 0.10 . it is possible that 2001 qg@xmath0 is elongated and able to resist gravitational compression into a spherical shape by virtue of its intrinsic compressive strength . however , observations of asteroids in the main - belt suggest that only the smallest ( @xmath70.1 km sized ) asteroids are in possession of a tensile strength sufficient to resist rotational deformation ( pravec , harris & michalowski 2003 ) . observations of both asteroids and planetary satellites suggest that many objects with radii @xmath77 50 to 75 km have shapes controlled by self - gravity , not by material strength ( farinella 1987 ; farinella & zappala 1997 ) . the widely accepted explanation is that these bodies are internally weak because they have been fractured by numerous past impacts . this explanation is also plausible in the kuiper belt , where models attest to a harsh collisional environment at early times ( e.g. davis & farinella 1997 ) . we feel that the extraordinarily large amplitude of 2001 qg@xmath0 is unlikely to be caused by elongation of the object sustained by its own material strength , although we can not rule out this possibility . structurally weak bodies are susceptible to rotational deformation . the 1000-km scale kbo ( 20000 ) varuna ( rotation period 6.3442 @xmath90 0.0002 hr and lightcurve range 0.42 @xmath90 0.02 mag ) is the best current example in the kuiper belt ( jewitt and sheppard 2002 ) . in the main asteroid belt , 216 kleopatra has a very short period ( 5.385 hr ) and large lightcurve range ( 1.18 mag . , corresponding to axis ratio @xmath72.95:1 and dimensions @xmath7 217 @xmath91 94 km , table 4 ) . kleopatra has been observed to be a highly elongated body through radar and high resolution imaging and the most likely explanation is that 216 kleopatra is rotationally deformed ( leone et al . 1984 ; ostro et al . 2000 ; hestroffer et al . 2002 ; washabaugh & scheeres 2002 ) . is rotational elongation a viable model for 2001 qg@xmath0 ? the critical rotation period ( @xmath92 ) at which centripetal acceleration equals gravitational acceleration towards the center of a rotating spherical object is @xmath93 where @xmath94 is the gravitational constant and @xmath95 is the density of the object . with @xmath95 = @xmath96 kg m@xmath97 the critical period is about 3.3 hr . even at longer periods , real bodies will suffer centripetal deformation into triaxial aspherical shapes which depend on their density , angular momentum and material strength . the limiting equilibrium shapes of rotating strengthless fluid bodies have been well studied by chandrasekhar ( 1987 ) and a detailed discussion in the context of the kbos can be found in jewitt and sheppard ( 2002 ) . we briefly mention here that triaxial `` jacobi '' ellipsoids with large angular momenta are rotationally elongated and generate lightcurves with substantial ranges when viewed equatorially . leone et al . ( 1984 ) have analyzed rotational equilibrium configurations of strengthless asteroids in detail ( see figure [ fig : kboperamp ] ) . they show that the maximum photometric range of a rotational ellipsoid is 0.9 mag : more elongated objects are unstable to rotational fission . the 1.14 mag photometric range of 2001 qg@xmath0 exceeds this limit . in addition , the 13.7744 hr ( two - peaked ) rotation period is much too long to cause significant elongation for any plausible bulk density ( figure [ fig : kboperamp ] ) . for these reasons we do not believe that 2001 qg@xmath0 is a single rotationally distorted object . a third possible explanation for the extreme lightcurve of 2001 qg@xmath0 is that this is an eclipsing binary . a wide separation ( sum of the orbital semi - major axes much larger than the sum of the component radii ) is unlikely because such a system would generate a distinctive `` notched '' lightcurve that is unlike the lightcurve of 2001 qg@xmath0 . in addition , a wide separation would require unreasonably high bulk density of the components in order to generate the measured rotational period . if 2001 qg@xmath0 is a binary then the components must be close or in contact . we next consider the limiting case of a contact binary . the axis ratio of a contact binary consisting of equal spheres is @xmath98 , corresponding to a lightcurve range @xmath82 = 0.75 magnitudes , as seen from the rotational equator . at the average viewing angle @xmath99 degrees we would expect @xmath82 = 0.45 mag . the rotational variation of 2001 qg@xmath0 is too large to be explained as a contact binary consisting of two equal spheres . however , close binary components of low strength should be elongated by mutual tidal forces , giving a larger lightcurve range than possible in the case of equal spheres ( leone et al . 1984 ) . the latter authors find that the maximum range for a tidally distorted nearly contact binary is 1.2 magnitudes , compatible with the 1.14 mag . range of 2001 qg@xmath0 ( figure [ fig : kboperamp ] ) . the contact binary hypothesis is the likely explanation of 624 hektor s lightcurve ( hartmann & cruikshank 1978 ; weidenschilling 1980 ; leone et al . 1984 ) and could also explain 216 kleopatra s lightcurve ( leone et al . 1984 ; ostro et al . 2000 ; hestroffer et al . 2002 ) . we suggest that the relatively long double - peaked period ( @xmath100 hr ) and large photometric range ( @xmath101 magnitudes ) of 2001 qg@xmath0 s lightcurve are best understood if the body is a contact binary or nearly contact binary viewed from an approximately equatorial perspective . the large range suggests that the components are of similar size and are distorted by their mutual tidal interactions . using the calculations from leone et al . ( 1984 ) , who take into account the mutual deformation of close , strengthless binary components , we find the density of these objects must be @xmath71000 kg m@xmath97 in order to remain bound in a binary system separated by the roche radius ( which is just over twice the component radius ) . if we assume that the albedo of both objects is 0.04 , the effective radius of each component is about 95 km as found above . using this information we find from kepler s third law that if the components are separated , they would be about 300 km apart . this separation as seen on the sky ( @xmath102 ) is small enough to have escaped resolution with current technology . further , we point out that the maximum of the lightcurve of 2001 qg@xmath0 is more nearly `` u '' shaped ( or flattened ) than is the `` v '' shaped minimum ( figure [ fig : phaseqgdouble ] ) . this is also true for 624 hektor ( dunlap & gehrels 1969 ) and may be a distinguishing , though not unique , signature of a contact or nearly contact binary ( zappala 1984 ; leone et al . 1984 ; cellino et al . 1985 ) . in comparison , ( 20000 ) varuna , which is probably not a contact binary ( see below and jewitt & sheppard 2002 ) , does not show significant differences in the curvature of the lightcurve maxima and minima . in short , while we can not prove that 2001 qg@xmath0 is a contact binary , we find by elimination of other possibilities that this is the most convincing explanation of its lightcurve . the distribution of measured lightcurve properties is shown in figure [ fig : kboperamp ] ( adapted from figure 4 of leone et al . ( 1984 ) ) . there , region a corresponds to the low rotational range objects ( of any period ) in which the variability can be plausibly associated with surface albedo markings . region b corresponds to the rotationally deformed jacobi ellipsoids while region c marks the domain of the close binary objects . plotted in the figure are the lightcurve periods and ranges for kbos from the hkbvp ( jewitt & sheppard 2002 ; sheppard & jewitt 2002 ; sheppard & jewitt 2004 ) . we also show large main belt asteroids ( data from http://cfa-www.harvard.edu/iau/lists/lightcurvedat.html updated by a. harris and b. warner and based on lagerkvist , harris & zappala 1989 ) . once again we note that the measured kbo ranges should , in most cases , be regarded as lower limits to the range because of the possible effects of projection into the plane of the sky . of the 34 kbos in our sample , five fall into region c in figure [ fig : kboperamp ] . of these , 2001 qg@xmath0 is by far the best candidate for being a contact or nearly contact binary system since it alone has a range between the @xmath103 0.9 mag . limit for a single rotational equilibrium ellipsoid and the @xmath1041.2 mag . limit for a mutually distorted close binary ( table 5 ) . it is also rotating too slowly to be substantially distorted by its own spin ( figure [ fig : kboperamp ] ) . both ( 33128 ) 1998 bu@xmath105 and 2000 gn@xmath106 are good candidates which have large photometric ranges and relatively slow periods . kbos ( 26308 ) 1998 sm@xmath107 and ( 32929 ) 1995 qy@xmath108 could be rotationally deformed ellipsoids , but their relatively slow rotations would require densities much smaller than that of water , a prospect which we consider unlikely . we next ask what might be the abundance of contact or close binaries in the kuiper belt . as a first estimate we assume that we have detected one such object ( 2001 qg@xmath0 ) in a sample of 34 kbos observed with adequate time resolution . the answer depends on the magnitude of the correction for projection effects caused by the orientation of the rotation vector with respect to the line of sight . this correction is intrinsically uncertain , since it depends on unknowns such as the scattering function of the surface materials of the kbo as well as on the detailed shape . we adopt two crude approximations that should give the projection correction at least to within a factor of a few . first , we represent the elongated shape of the kbo by a rectangular block with dimensions a @xmath3 b = c. the lightcurve range varies with angle from the equator , @xmath109 , in this approximation as @xmath110.\ ] ] for the limiting case of a highly distorted contact binary with @xmath82 = 1.2 mag . at @xmath109 = 0@xmath111 , eq . ( 3 ) gives @xmath83 = 3 . we next assume that the range must fall in the range 0.9 @xmath112 1.2 mag . in order for us to make an assignment of likely binary structure ( figure [ fig : kbomaghist ] ) . as noted above , only 2001 qg@xmath0 satisfies this condition amongst the known objects . we find , from eq . ( 3 ) with @xmath83 = 3 , that @xmath82 = 0.9 mag is reached at @xmath109 = 10@xmath111 . the probability that earth would lie within 10@xmath113 of the equator of a set of randomly oriented kbos is @xmath114 = 0.17 . therefore , the detection of 1 kbo with 0.9 @xmath115 1.2 mag implies that the fractional abundance of similarly elongated objects is @xmath116 17% . as a separate check on this estimate , we next represent the object as an ellipsoid , again with axes a @xmath3 b = c. the photometric range when viewed at an angle @xmath109 from the rotational equator is given by @xmath117sin^2\theta + 1\right]\ ] ] substituting @xmath83 = 3 , the range predicted by eq . ( 4 ) falls to 0.9 mag at @xmath118 17@xmath111 . given a random distribution of the spin vectors , the probability that earth would lie within 17@xmath113 of the equator is @xmath119 = 0.29 . therefore , the detection of 1 kbo with a range between 0.9 and 1.2 mag in a sample of 34 objects implies , in this approximation , a fractional abundance of similarly elongated objects near @xmath116 10% . given the crudity of the model , the agreement between projection factors from eqs . ( 3 ) and ( 4 ) is encouraging . together , the data and the projection factors suggest that in our sample of 34 kbos , perhaps 3 to 6 objects are as elongated as 2001 qg@xmath0 but only 2001 qg@xmath0 is viewed from a sufficiently equatorial perspective that the lightcurve is distinct . this is consistent with figure 5 , which shows that @xmath120 of @xmath121 kbos ( @xmath122 ) from the hkbvp occupy region c of the period - range diagram . our estimate is very crude and is also a lower limit to the true binary fraction because close binaries with components of unequal size will not satisfy the 0.9 @xmath115 1.2 mag . criterion for detection . the key point is that the data are consistent with a substantial close binary fraction in the kuiper belt . figure [ fig : kboperamp ] also shows that there are no large main - belt asteroids ( radii @xmath123 km ) in region c , which is where similar sized component contact binaries are expected to be . to date , no examples of large binary main - belt asteroids with similar sized components have been found , even though the main belt has been extensively searched for binarity ( see margot 2002 and references therein ) . the main - belt asteroids may have had a collisional history significantly different from that of the kbos . the contact binary interpretation of the 2001 qg@xmath0 lightcurve is clearly non - unique . indeed , firm proof of the existence of contact binaries will be as difficult to establish in the kuiper belt as it has been in closer , brighter populations of small bodies . nevertheless , the data are compatible with a high abundance of such objects . it is interesting to speculate about how such objects could form in abundance . one model of the formation and long term evolution of wide binaries predicts that such objects could be driven together by dynamical friction or three - body interactions ( goldreich et al . 2002 ) . objects like 2001 qg@xmath0 would be naturally produced by such a mechanism . kuiper belt object 2001 qg@xmath0 has the most extreme lightcurve of any of the 34 objects so far observed in the hawaii kuiper belt variability project . the double - peaked lightcurve period is @xmath6 hr and peak - to - peak range is @xmath124 mag . only two other minor planets with radii @xmath77 25 km ( 624 hektor and 216 kleopatra ) and one planetary satellite ( iapetus ) are known to show rotational photometric variation greater than 1 mag . the absolute red magnitude is @xmath125(1,1,0 ) = 6.28 at maximum light and 7.42 mag . at minimum light . with an assumed geometric albedo of 0.04 ( 0.10 ) we derive effective circular radii at maximum and minimum light of 158 ( 100 ) and 94 ( 59 ) km , respectively . no variation in the bvr colors between maximum and minimum light was detected to within photometric uncertainties of a few percent . \4 . the large photometric range , differences in the lightcurve minima , and long period of 2001 qg@xmath0 are consistent with and strongly suggest that this object is a contact or nearly contact binary , viewed equatorially . if 2001 qg@xmath0 is a contact binary with similarly sized components , then we conclude that such objects constitute at least 10% to 20% of the kuiper belt population at large sizes . we thank john tonry and andrew pickles for help with the optic camera and the remote observing system on the university of hawaii 2.2 meter telescope . we also thank henry hsieh for observational assistance and jane luu for comments on the manuscript . this work was supported by a grant to d.j . from the nasa origins program .	extensive time - resolved observations of kuiper belt object 2001 qg@xmath0 show a lightcurve with a peak - to - peak variation of @xmath1 magnitudes and single - peaked period of @xmath2 hr . the mean absolute magnitude is 6.85 magnitudes which corresponds to a mean effective radius of 122 ( 77 ) km if an albedo of 0.04 ( 0.10 ) is assumed . this is the first known kuiper belt object and only the third minor planet with a radius @xmath3 25 km to display a lightcurve with a range in excess of 1 magnitude . we find the colors to be typical for a kuiper belt object ( @xmath4 , @xmath5 ) with no variation in color between minimum and maximum light . the large light variation , relatively long double - peaked period and absence of rotational color change argue against explanations due to albedo markings or elongation due to high angular momentum . instead , we suggest that 2001 qg@xmath0 may be a very close or contact binary similar in structure to what has been independently proposed for the trojan asteroid 624 hektor . if so , its rotational period would be twice the lightcurve period or @xmath6 hr . by correcting for the effects of projection , we estimate that the fraction of similar objects in the kuiper belt is at least @xmath710% to 20% with the true fraction probably much higher . a high abundance of close and contact binaries is expected in some scenarios for the evolution of binary kuiper belt objects .	8,961	394
the ising model is a prototype of a statistical mechanical model for studying order - disorder transitions . it is also the first statistical model which has led to exact solution in one and two dimensions . since its inception@xcite this model has triggered an intense effort in investigation of other models resulting in an extensive literature on the subject @xcite . there are now a large library of statistical and quantum mechanical models , differing in their degrees of freedom , interaction type , the type and dimensions of lattices , and of course methods of solution @xcite . there are also models which can be called integrable , meaning that they allow sufficient number of conserved quantities , leading to a full determination of their spectra and other observable quantities . among the well - known classical models , we can mention particularly the @xmath0-state potts model @xcite , which is the simplest generalization of the ising model in that the degrees of freedom take @xmath0 instead of two different values and interact according to the hamiltonian [ hpot ] h=-_i , j ( s_i , s_j ) , where @xmath1 means that interactions are between nearest neighbors and @xmath2 takes @xmath0 different values . this is a classical model in that every configuration of the so - called spin variables @xmath2 is an energy eigenstate . the importance of this model , like ising model , is that it can be mapped to many other important models in science , i.e. the famous four - color problem in the case of potts model , @xcite . the ising model has a @xmath3 symmetry , meaning that the configurations @xmath4 and @xmath5 both have the same energy . ] in the potts model , this symmetry has been elevated to the @xmath6 group , where now due to the form of the hamiltonian ( [ hpot ] ) , one can shift all the spin variables by any integer value @xmath7 $ ] and the energy remains the same . this symmetry can be seen in a better way if we use an identity and write the local energy term as ( s , s)=_n=0^d-1 ^ns^-ns , where @xmath8 is a @xmath9th root of unity . + it is the purpose of this work to generalize the potts and indeed ising model to the case where the symmetry group is a finite ( non - abelian ) group . such a model , if properly defined will certainly have a very rich structure and will certainly deepen and widen our perspective of exactly solvable models of statistical mechanics and integrable models . moreover it has the potential of enriching our knowledge of order - disorder transitions in statistical mechanics . of course the model as defined now is mainly of theoretical interest and it may be difficult to make concrete connections with specific physical problems . we only hope with hindsight and in view of the experience with other more or less abstract models in statistical mechanics like the face models and vertex models @xcite , this new model will also find a proper place in the library of exactly solvable models , possibly with new applications in the future . + we will focus our attention to a one dimensional lattice and show that such a natural generalization is indeed possible and will define the non - abelian potts model . we determine the full spectrum and show that the ground states and indeed many of the excited states of this model are entangled . we will determine the amount of this entanglement both for a single site and for a block of finite length . we also determine the entanglement between two different sites of the lattice . in this sense the model is a quantum mechanical model , in contrast to classical models where there is no entanglement in their energy eigenstates . it turns our that the properties of irredudcible representations play an important role in the nature of the spectrum and its entanglement properties . we will also calculate the partition function of the model in two different ways , that is we follow an algebraic approach where we calculate the trace of the thermal state and a combinatorial approach where we also count the degeneracy of all energy levels . the results of the two approaches , which correspond respectively to the high and low temperature expansions , agree as they should . in all aspects the results pertaining to this model reduce to the potts model when the symmetry group @xmath10 reduces to the abelian group @xmath6 . + the structure of this paper is as follows : in section [ pre ] we review the preliminary material from theory of finite groups and their representations . in section [ model ] , we define the non - abelian model on a one dimensional lattice , which as we see , is a special case of the kitaev quantum double model @xcite . in two dimensions this correspondence in no longer true , since as is well known the kitaev model entails four - body interactions while our model entails two - body interactions , as it should as a generalization of potts model . in fact generalization of the potts model to two dimensions is a non - trivial problem as we will discuss in the discussion . in section [ spec ] we derive the ground states and determine their entanglement properties , and in section [ excited states ] we determine all the excited states , hence the full spectrum . in section [ part ] , we calculate the partition function in closed form , by summing up the terms in both high and low temperature expansions . in all our study we make contact with the simpler and special case where the group we are considering is the abelian group @xmath6 , and the model is the d - state potts model . let @xmath10 be a finite group of size @xmath11 . two elements @xmath12 and @xmath13 of the group @xmath10 are said to be conjugate if there is a @xmath14 , such that @xmath15 . this is an equivalence relation which partitions the group into conjugacy classes . the conjugacy class of an element @xmath12 is denoted by @xmath16 . the number of conjugacy classes is denoted by @xmath17 . this equals the number of irreducible representations of the group . it is customary to denote the former by latin indices and the latter by greek indices , hence we have @xmath18 as a conjugacy class and @xmath19 as an irreducible representation . the number of elements in @xmath18 is denoted by @xmath20 , and the dimension of an irreducible representation @xmath19 is denoted by @xmath21 . we then have @xmath22 . let @xmath23 be a vector space spanned by orthonormal vectors @xmath24 . on this vector space , two regular representations called left and right actions respectively , are defined as l(g ) \|h= \|gh , r(g)\|h=\|hg^-1 . obviously the two kinds of actions commute with each other . for an abelian group , @xmath25=[r(g),r(g')]=0 $ ] leading to a simple disentangled spectrum as we will see in sections [ model ] , while for a non - abelian group this is no longer the case . + both regular representations decompose into a sum of irreducible representations , each representation @xmath19 occurring with a multiplicity equal to its dimension @xmath21 , hence we have the relation @xmath26 the matrix entries of an irreducible representation @xmath27 of an element @xmath14 are denoted by @xmath28 . for each set of labels @xmath29 , a @xmath11 dimensional vector @xmath30 is defined as : [ basis ] \|d_mn^:= _ gg d^_mn(g ) \|g . this is in fact the fourier transform on the finite group @xmath10 . it is well known that these vectors provide an orthonormal and complete basis for @xmath31 @xcite . the orthonormality implies that [ orthonormal ] d^_m , n\|d^_p , q=^,_m , p_n , q , or [ orthonormality ] _ g g d^_mi(g ) = ^ _ mn _ ij . here @xmath32 denotes complex conjugate of @xmath33 . we will later use a graphical representation for @xmath34 which we depict in figure [ contraction ] . + for each representation @xmath19 , the character of a conjugacy class @xmath18 , is defined as @xmath35 , where @xmath36 . the significance of characters of irreducible representations is that they are orthonormal and complete in the sense that if we define the @xmath17 dimensional formal vectors @xmath37 , where @xmath38 s form a computational basis , then these new vectors form a complete orthonormal basis . the orthonormality implies that [ ccort ] _ g g ^(g)=\|g\| ^ , . finally note that the product of irreducible representations decomposes into sum of them in the form d^(g ) d^(g ) = _ f^,_d^(g ) , where @xmath39 are the ( clebsh - gordan ) or fusion coefficients of this group . taking the trace of both sides , this leads to [ chichi ] ^(g ) ^(g ) = _ f^,_^(g ) . we are now equipped with almost all the necessary ingredients of group theory to define and study the non - abelian potts model . s and their contraction : a ) a representation for @xmath34 . b ) a representation for @xmath40 . ] consider a periodic chain of length @xmath41 . let @xmath10 be a finite ( abelian or non - abelian ) group of size @xmath11 . to each site of the chain , a @xmath11 dimensional hilbert space @xmath23 is assigned with orthonormal basis vectors @xmath42 . therefore we have @xmath43 and @xmath44 . the dimensional of the full hilbert space @xmath45 is given by @xmath46 . + the hamiltonian of the model is the reduction of the quantum double hamiltonian @xcite , in which only the vertex operators are retained and all the plaquette operators are ignored . since in a one - dimensional lattice , there is a one to one correspondence between vertices and links , we have used this liberty to put the dynamical ( generalized spin ) variables on the vertices instead of the links . this makes the model much more akin to the other models of statistical mechanics , like the ising , potts and heisenberg models . to each pair of neighboring sites @xmath47 , we assign the following local hamiltonian [ localh ] h_i=_gg r_i(g ) l_i+1(g ) , where the indices @xmath48 and @xmath49 show that @xmath50 and @xmath51 act on these two sites respectively . it is easily verified that @xmath52 is a projector , namely @xmath53 and @xmath54=0,\ \ \forall \ \ i , j$ ] . therefore the eigenvalues of @xmath52 are restricted to @xmath55 and @xmath56 . the total hamiltonian is the sum of all these local hamiltonians , i.e. [ hamiltonian ] h=- _ i=1^n h_i , where periodic boundary condition @xmath57 is implied in all states and operators . + consider the special case where @xmath10 is an abelian group , and in particular @xmath6 the finite cyclic group of order @xmath0 , @xmath58 where the group operation is addition modulo @xmath0 . then we have + l(g ) \|h= \|h+g , r(g)\|h=\|h - g h , gz_d , + leading to l(g ) x^g , r(g)x^-g , where @xmath59 is the shift operator or the @xmath9 dimensional generalization of the pauli matrix @xmath60 , and @xmath61 is @xmath62 to the power @xmath63 . therefore in the @xmath6 case , the local hamiltonian [ localh ] can be written as h_i= _ g=0^d-1 x_i^-g x_i+1^g . to see the relevance of this local hamiltonian to the potts model , we note that in this abelian case and only in this case , all the operators @xmath64 can be diagonalized in the same basis , since @xmath25=0 $ ] for an abelian group . this is the basis in which the shift operator @xmath62 is diagonal . this operator has the property @xmath65 which means that its eigenvalues are @xmath66 , where @xmath8 is the @xmath9th root of unity , @xmath67 . the eigenvectors of @xmath62 are easily found [ dstates ] \|d^s:=_j=0^d-1 ^-js\|j , x\|d^s=^s\|d^s . now any product state of the form \|^s:=\|d^s_1\|d^s_2\|d^s_i\|d^s_i+1\|d^s_n , is an eigestate of all @xmath52 s and hence the total @xmath68 . in such a basis , @xmath52 can be replaced with its eigenvalues . then we have h_i & & _ g=0^d-1^(s_i+1-s_i)g= _ s_i , s_i+1 . hence in the special case @xmath69 , we will have the classical @xmath9state potts model described by [ pottsdelta ] h=-_i=1^n _ s_i , s_i+1 , where @xmath2 s take @xmath0 different values . we now turn to the spectrum when @xmath10 is a finite non - abelian group . first we find the ground states and then determine all the excited states . in order to find the ground states of the hamiltonian [ hamiltonian ] , we first consider the potts model which will act as a guideline for the more complicated non - abelian case . as equation [ pottsdelta ] indicates , the ground state of the potts model is when all the labels @xmath2 are equal to each other , so there are @xmath0 different ground states , namely @xmath70 in [ dstates ] , we can rewrite this as \|^s=_j_1 , j_2 , j_n^-j_1s^-j_2s^-j_ns\|j_1 , j_2,j_n . this can be written as a matrix product state ( mps ) as \|^s=_j_1 , j_2 , j_n^s(j_1 , j_2 , j_n)\|j_1 , j_2,j_n , where [ mpspotts ] ^s(j_1 , j_2 , j_n)= ( d^s(j_1)d^s(j_2)d^s(j_n ) ) , in which @xmath71 is the @xmath72 representation of element @xmath73 . + we now go on to consider the spectrum of the non - abelian case . motivated by the mps representation of the abelian ( potts ) model in [ mpspotts ] , we form the following states : \|^&:= & ( d^(g_1)d^(g_2)d^(g_n))\|g_1 , g_2 , g_n&=&^(g_1g_2g_n)\|g_1 , g_2 , g_n , where a sum over all group elements @xmath74 is implied . here @xmath75 is a normalization factor and @xmath76 is the matrix of @xmath63 in the irreducible representation @xmath19 . it is then obvious that these states are eigenstates of the operators @xmath52 . to see this we note that h_i\|^&= & _ ggr_i(g)l_i+1(g)\|^&= & _ gg^(g_1g_2g_n)\|g_1 , g_2 g_ig^-1 , gg_i+1 , g_n=\|^ , where in the last line we have relabeled the group elements @xmath77 to @xmath78 and @xmath79 to @xmath80 and have used the fact that a sum over all group elements is performed . therefore for each irreducible representation of the group we find one ground state . thus the ground state is @xmath17-fold degenerate , where @xmath17 is the number of in - equivalent irreducible representations , or equivalently the number of conjugacy classes . to find the normalization , we use the orthogonality relation of characters which reads from [ ccort ] @xmath81 or @xmath82 . therefore we find ^\|^=_g_1 , g_n \|(g_1g_n)\|^2=\|g\|^n-1_gg\|(g)\|^2=\|g\|^n , giving the normalized ( in fact orthogonal ) ground states [ gs ] \|^=^(g_1g_2g_n)\|g_1 , g_2 , g_n . note that in contrast to the abelian case , the ground states are generally entangled . moreover consider the following string operators t^=_g_1 , g_2 , g_n ^(g_1g_2g_n ) \|g_1 , g_2 , g_ng_1 , g_2 , g_n\| . we may ask what is the effect of this string operator on a given ground state . in view of the relation ( [ chichi ] ) , it turns out that t^\|^=_f^,_\|^ , implying that the ground space carries a representation of the fusion algebra of the group representation . + we now study the entanglement properties of the ground states . first we determine how much a given site is entangled with the rest of the lattice , when the whole system is in a given ground state . then we calculate the entanglement of a given block of length @xmath83 . third we determine the entanglement of two sites which are near each other or are far apart . before proceeding to the proofs , we present the main results . when the lattice is in a given ground state @xmath84 , + i ) the entanglement of any given site with the rest of the lattice , measured by the von - neumann entropy of the reduced density matrix of this site is @xmath85 . + ii ) the entanglement of any given block of length @xmath83 is given by @xmath86 } ) = \log(n^2_\mu)$ ] which is independent of the length of the block and verifies the area law . + iii ) any two sites which are not nearest neighbors are separable , their reduced density matrices is a product of those of individual sites . + iv ) any two neighboring sites are entangled and their entanglement measured by the negativity of the reduced density matrix of the two sides is given by @xmath87 . + these results are proved in appendix a. using the conventions of figure [ contraction ] and the form of the ground state in [ myground ] , we can graphically depict a ground state as in figure [ gsfig ] . this is again a generalization of the abelian potts model , where all the sites have been colored by one single color , here a color represents a label of an irreducible representation . each line between two bulbs means that the two adjacent indices of the matrices have been contracted ( i.e. made equal and summed over ) . to obtain the excited states of [ hamiltonian ] , consider now a cut in such a configuration as shown in figure [ exs ] . we will see that these cuts which separate connected regions of different colors ( i.e. labels of irreducible representations ) produce excited states . to prove this we remind the reader that local hamiltonians , being projectors , have eigenvalues only equal to @xmath55 or @xmath56 , therefore the exited states are those in which one or more of the vertex operators have @xmath55 eigenvalue . therefore we want to find states which are common eigenstates of all the local hamiltonians where one or more of these local hamiltonians have zero eigenvalues . to proceed , consider local states like @xmath88 . we first obtain the action of left and right operators on these states , which is easily verified to be as follows : [ l ] l(h)\|d^_m , n= d^_m , p(h^-1)\|d^_p , n and [ r ] r(h)\|d^_m , n= \|d^_m , pd^_p , n(h ) where summation over repeated latin indices are implied . from these two relations we find the action of @xmath52 on the two neighboring elementary states , @xmath89 . then from equations [ l ] and [ r ] we obtain : h_i\|d^_m , n_i\|d^_p , q_i+1 = \|d^_m , r_i ( _ g d^_r , n(g)d^_p , s(g^-1))\|d^_s , q_i+1 . using the orthonormality relation [ orthonormality ] , we find [ local action ] h_i\|d^_m , n_i\|d^_p , q_i+1 = _ n , p^,_r\|d^_m , r_i\|d^_r , q_i+1 . let us first consider the special case @xmath90 , where we have : [ exs1 ] h_i\|d^_m , n_i\|d^_p , q_i+1 = 0 . and @xmath91 are different . c ) an excited state of the kind [ exs3 ] , where both the colors and matrix indices are the same . ] this means that if the two colors @xmath92 and @xmath93 on the two sides of the cut are different , then the local hamiltonian @xmath52 annihilates the state and so the energy is raised by one unit . this kind of excited state is depicted in figure [ exs]a . another special case is when the two colors ( i.e. representation labels ) are the same , but the middle indices are different : [ exs2 ] h_i\|d^_m , n_i\| d^_p , q_i+1 = 0 np , these kinds of excited states are depicted in figure [ exs]b . finally we learn from [ local action ] that if @xmath94 and @xmath95 , then the right hand side of [ local action ] , although non - zero , is independent of @xmath96 , this means that for any two matrix indices @xmath97 [ exs3 ] h_i(\|d^_m , n_i\|d^_n , q_i+1-\|d^_m , p_i\|d^_p , q_i+1 ) = 0 . hence the third kind of excitations are depicted as in figure [ exs]c . the number of these new kinds of excitations for a single cut in @xmath84 which are independent is @xmath98 . using the simple notation @xmath99 for brevity , they can be chosen to be @xmath100 ( @xmath101 ) or if we want to make them orthogonal we can choose them to be @xmath102 , @xmath103 , @xmath104 and so on which are obviously orthogonal . note that if we form their uniform summation as @xmath105 we will get a state whose eigenvalue for @xmath52 is @xmath56 , which is not an excitation any more . + all the excited states are derived by inserting one or more of these different kinds of cuts or excitations in a given ground state . this again generalizes a feature of the potts model , where only one type of cut exists which separates two different values of spins or colors . in the non - abelian case , where the representations are not one dimensional , the domain walls are more complex . they are designated by a pair of indices @xmath48 and @xmath91 . this completes our analysis of the spectrum of the non - abelian potts model . in this section we calculate the partition function which captures the statistical properties of the model . we follow two different approaches for calculating the partition function , an algebraic approach in which we calculate the trace of @xmath106 and a combinatorial approach in which we count the number of excited states of any given energy . the two approaches correspond respectively to high and low temperature expansions . the reason for this correspondence will be made clear in the derivations . since the local hamiltonians commute with each other we have z_(g)= e^-h = _ i=1^n e^h_i . noting that the local hamiltonians are projectors @xmath53 , we can write the exponentials in the following linear form z_(g)= , where @xmath107 . this parameter , which for high temperatures is small and for @xmath108 approaches @xmath55 , will act as an expansion parameter . at high temperatures the first few terms of the expansion will be a good approximation of the partition function . however in the present case , we can exactly determine the partition function by calculating all the terms . we can write z_(g ) = _ n=0^n ^n z^(n)_(g ) , where @xmath109 is the trace of product of @xmath96 local hamiltonians . in appendix b we prove that @xmath110 and @xmath111 . using these two results , we obtain the exact form of the partition function to be [ zg ] z_(g)= ( e^-1+\|g\|)^n + ( e^-1)^n(k-1 ) . it is instructive to first consider the abelian potts model . this will then teach us how to go about the non - abelian case . in the limit @xmath112 , only the ground states contribute to the partition functions . at very low temperatures both the ground states and the low lying excited states contribute to the partition function . as we will see the low lying states corresponds to few domain walls in the configuration of spins . as the temperature rises more domain walls contribute and the low temperature expansion necessitates the counting of contribution of various configurations of domain walls . again in the present context , we can sum all the contributions exactly and obviously the result should be equal to that of the high - temperature expansion . more precisely , we note that the ground state energy of the abelian potts model is equal to @xmath113 ( when all the spins are the same , or when all the spins are colored by one single color ) and the degeneracy of the ground state is equal to @xmath0 . coming to the excited states , for each neighboring pair where the two labels @xmath2 and @xmath114 are unequal , i.e. for each domain wall , an excited state is created . so for @xmath115 domain walls , the energy will be @xmath116 . the degeneracy of this energy is then equal to the total number of different configurations with @xmath115 domain walls . this degeneracy comes from the different positions of the @xmath115 walls and the different possible spin configurations ( which we call colors ) for each configuration of walls . the first factor is easy to calculate . with a periodic boundary condition , it is simply given by @xmath117 . note that @xmath115 should be greater than @xmath56 , since in a periodic lattice , there can not be one single wall . calculation of the second factor , which we denote by @xmath118 , is done as follows . consider figure [ walls ] which shows @xmath115 different domains , separated by @xmath115 walls . for domain @xmath56 we have @xmath0 colors to choose . for the second domain we are left with @xmath119 walls , since the color of this domain has to be different from the first one . for the third domain we have again @xmath119 choices , since the color of this domain has to be different from the second one . so going around the circle and coming to the last domain @xmath115 we have @xmath120 different choices of colors , but in so counting , we have over - counted the number of colors , since the color of the last region @xmath115 should be different from that of region @xmath56 . the number of such configurations is simply @xmath121 , since in this case the two domains @xmath115 and @xmath56 merge into one single domain . thus we arrive at the recursion relation [ rr ] c_k(d ) = d ( d-1)^k-1 - c_k-1(d ) , c_0(d ) = d , the solution of which is [ cd ] c_k(d)=(d-1)^k+(d-1)(-1)^k . therefore the degree of degeneracy of each energy level @xmath122 is given by , leading to the partition function of the @xmath9 level potts model z _ ( ) = _ k=0^n e^(n - k ) nk = ( e^-1+d)^n + ( e^-1)^n(d-1 ) . ( for abelian group ) is characterized by the position of the walls and different possible colors . for a configuration with @xmath115 fixed walls , there are @xmath123 different eigenstates as given in [ cd ] . ] we now go on to the quantum case . here the structure or the labels of a domain wall is more complex , since as in figure [ circle ] , each domain is characterized not only by a color ( a representation label @xmath124 ) but also by two matrix indices @xmath48 and @xmath91 . consider the first domain in figure [ circle ] . the total number of choices for this domain is thus @xmath125 . what is the number of possible choices for the next domain ? from the structure of excited states in [ exs1 ] , [ exs2 ] and [ exs3 ] , we see that from all the combinations of labels for this wall , only one combination does nt lead to an excitation at the position of the wall , hence there are @xmath126 possible choices for this second domain . the rest of reasoning is exactly the same as for the abelian case , namely we again have the same recursion relation as in [ rr ] , but with a different initial condition : c_k(g ) = \|g\| ( \|g\|-1)^k-1 - c_k-1(g ) , c_0(g ) = k , where we remind the reader that @xmath17 is the number of different irreducible representations or the number of conjugacy classes . this leads to [ cg ] c_k(g ) = ( \|g\|-1)^k + ( -1)^k ( k-1 ) , and @xmath127 $ ] and finally to the partition function [ pg ] z_(g)= ( e^-1+\|g\|)^n + ( e^-1)^n(k-1 ) , in accordance with [ zg ] . ( for a general finite group ) is characterized by the position of the walls and the type of domains separated by them . for a configuration with @xmath115 fixed walls , there are @xmath128 different eigenstates as given in [ cg ] . ] in the limit @xmath112 , where @xmath129 , we find from [ pg ] that @xmath130 , as it should be , since in this limit only the ground states contribute to the partition function and there are @xmath17 ground states , each with energy @xmath113 . conversely in the high temperature limit , where @xmath131 , we find @xmath132 , meaning that all the states contribute equally to the partition function and the thermal density matrix is a completely mixed one . + we have introduced the non - abelian potts model , and have made a rather detailed study of its properties . in the same way that ising and potts model have led to a large number of applications in physics , it may also be the case that the non - abelian case , in view of its richer structure , may find such applications . on the theoretical side , we believe that there are many avenues of research which are opened by this study . here are a few examples : + i ) the definition of the model on 2d lattices remains to be done . although in one dimensional lattices the local hamiltonian [ localh ] is the same as the one pertaining to the vertex operators of the quantum double model of kitaev @xcite , one can not simply carry this to 2d lattices since it would lead to a four - body interaction , while a natural generalization of the potts model should include two - body interactions . it may still be the case that a 2d model with two - body interaction as we have defined , is solvable , in the sense that its ground state and low - lying states can be determined in closed form , although in 2d lattice not all the local hamiltonians will commute with each other anymore . if this is the case , then the structure of the 2d model will be even more interesting and challenging to determine . one may even be able to establish a duality relation between the high and low temperature expansions in this case which will then enable one to determine the critical point without an exact solution . to find an exact solution for the 2d case , the first step may be to define the model on a one - dimensional ladder , where despite the incommutability of the operators , there is still some simplicity in the geometry of the lattice . this has already been exploited to find the explicit form of the full spectrum of the kitaev model on spin ladders @xcite . + ii ) one can also generalize this to lie groups on discrete lattices , or to lie groups on the real line , where a suitable definition of the model hamiltonian should be made . in view of the matrix product structure of the ground states , it may be the case that the generalization of matrix product formalism for quantum field theories as developed in @xcite will be a natural framework for studying this type of generalization . + iii ) one can add terms which pertain to an external magnetic field in the ising model . in the same way that adding an external transverse field to the ising model leads to a rich many - body system which undergoes a quantum phase transition , this may also happen for the non - abelian potts model . in such case , one anticipates that the solution will be highly non - trivial and technical , if an exact solution exists at all . moreover the meaning of transversality of the external field should also be clarified in this case . + iv ) it will also be interesting to study possible phase transitions between this type of order and other possibly topological orders in models where suitable interaction terms are added to the hamiltonian @xcite . + v ) finally it will be interesting to investigate the relation between this 1d quantum mechanical system and the 2d classical model . once a 2d classical possibly exactly solvable model is at hand , one can obtain the transfer matrix which represents a 1d quantum mechanical model . in the present case , knowledge of the complete spectrum of the 1d model means that the 2d classical model , once properly defined , is exactly solvable . these studies are now underway by the authors . + ising , ernst . `` beitrag zur theorie des ferromagnetismus . 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neumann entropy , as a measure of bipartite entanglement between this site and the rest of the system , is equal to : s(^_1 ) = -_n _ n ( _ n)= - n^2 _ ( ) = ( n^2 _ ) . this means that only the one dimensional representations lead to product ground states and the larger the dimension of a representation , the larger will be the entanglement of the corresponding ground state . + to obtain bipartite entanglement between a block and the rest of the system one should obtain the reduced density matrix of a block with length @xmath83 , say the block consisting of sites @xmath56 to @xmath83 . we denote this density matrix by @xmath143}$ ] and the calculation is done as follows : ^_[1,l ] = _ \| ^^\| , where @xmath144}}$ ] means trace over all of the spins that are not in this block . using the same arguments as above one finds that : [ densityl ] ^_[1,l ] & = & _ [ i_1,i_n ] , [ j_1,j_n ] + & = & _ [ i_1,i_n ] , [ j_1,j_n ] _ i_1 , j_1_i_l+1,j_l+1 + & & \|d^_i_1i_2d^_i_1j_2\| \|d^_i_li_l+1d^_j_l , i_l+1\| + & = & _ i_1,i_l+1 \|^_i_1,i_l+1(l ) ^_i_1,i_l+1(l)\| , where \|^_i_1,i_l+1(l ) = _ i_2 , i_3 , i_l\|d_i_1 , i_2\|d_i_2,i_3\|d_i_l , i_l+1 . these states are not normalized and in fact they satisfy @xmath145 . the reduced density matrix is diagonal in terms of these new states and has @xmath140 eigenvectors with nonzero eigenvalues equal to @xmath141 and @xmath146 eigenvectors with 0 eigenvalue . therefore the von - neumann entropy of the reduced density matrix is equal to : s(^_[1,l ] ) = - n^2 _ ( ) = ( n^2 _ ) . therefore the entanglement of a block of length @xmath83 is independent of the length of the block , depending only on its two end points , which is nothing but a manifestation of area law in this exactly solvable quantum mechanical model . + finally we come to the entanglement of two different sites with each other . first we consider the entanglement of two non - adjacent sites . following exactly the same calculation which led to the above results , it is not hard to see that the density matrix of two distant sites , say sites @xmath56 and @xmath115 , will be given by ^_(1,k)=_i , j\|d^_ijd^_ij\|_m , n\|d^_mnd^_mn\|^_1^_k , which means that there is no entanglement between the two sites . if there is any entanglement , it is between adjacent sites . this is indeed true as we can verify . in fact we have already found the density matrix of a block of length @xmath83 in [ densityl ] . specializing [ densityl ] to @xmath147 , we find the density matrix of two neighboring sites , say @xmath56 and @xmath148 . ^_(1,2)= _ i , k \|^_i , k(2 ) ^_i , k(2)\| , where \|^_i , k(2 ) = _ j\|d^_i , j\|d^_j , k . we can find the negativity of @xmath149 by calculating its negativity in the form @xmath150 , where @xmath151 means partial transpose with respect to second subsystem and @xmath152 means the trace - norm of @xmath62 which is equal to @xmath153 . the partial transpose with respect to the second subsystem is equal to : ( ^_(1,2))^t_2 = _ j , j^ \|d^_i , j d^_i , j^\| \|d^^_j^,k d^^_j , k \| , where @xmath154 and the negativity of this reduced density matrix is equal to : ( ^_(1,2))= , which means for the higher representations of the group the ground state contains more entanglement between the nearest neighbor spins . in this appendix we present details of the calculations of the partition function in the high temperature expansion . we have z_(g ) = _ n=0^n ^n z^(n)_(g ) , where @xmath109 is the trace of product of @xmath96 local hamiltonians . obviously we have @xmath155 . to calculate the first order term , consider the trace of a single local hamiltonian like @xmath156 on the local hilbert spaces of the first two sites @xmath56 and @xmath148 , where it acts nontrivially . we write _ 12(h_1)&= & = _ g , h , kg h , k\| r(g)l(g)\|h , k&=&_g , h , k h , k\|hg^-1 , gk=_g , h , k ( g , e)=\|g\| . this leads to @xmath157 , hence @xmath158 . in second order we face two types of terms , those which are adjacent connected like @xmath159 ( which we call connected segments ) or those which are far apart like @xmath160 with @xmath161 , terms like @xmath162 and so forth ( which we call disconnected ) . consider a disconnected terms first . for a disconnected term like @xmath162 we note that @xmath163 and therefore @xmath164 . for a connected term like @xmath159 we have _ 123(h_1h_2)&= & & = & _ g_1,g_2,h , k , lg h , k , l\| r(g_1)l(g_1)r(g_2)l(g_2)\|h , k , l&=&_g_1,g_2,h , k , l h , k , l\|hg_1 ^ -1 , g_1kg_2 ^ -1,g_2 l. therefore _ 123(h_1h_2)=_g_1,g_2,h , k , l ( g_1,e)(g_2,e)=\|g\| , leading to @xmath165 . therefore both types of connected and disconnected terms have the same trace which leads to a great simplification . this gives @xmath166 . this feature is true for all the terms up to and including the @xmath167-th term . therefore we have @xmath168 . for the last terms where we have a full cycle of terms , i.e. a closed loop , the situation is a little bit different . in this case we have @xmath169 or ( h_1h_2h_n)= _ g_1 , g_n_h_1(g_2 g_3g_n g_1 h_1g_1 ^ -1 g_n^-1g_2 ^ -1,h_1 ) . using the change of variable @xmath173 and performing the sum over all the other group elements @xmath174 to @xmath175 , we find where z_(g)=_k=0^n-1 ^k nk \|g\|^n - k + ^n k or [ zg ] z_(g)= ( e^-1+\|g\|)^n + ( e^-1)^n(k-1 ) . equation ( [ zg ] ) gives the final expression for the partition function of the non - abelian potts model on a ring of @xmath41 sites for a finite group @xmath10 . apart from the obvious dependence on @xmath41 and @xmath178 , it depends on the size of the group and the number of its conjugacy classes or irreducible representations .	we generalize the classical one dimensional potts model to the case where the symmetry group is a non - abelian finite group . it turns out that this new model has a quantum nature in that its spectrum of energy eigenstates consists of entangled states . we determine the complete energy spectrum , i.e. the ground states and all the excited states with their degeneracy structure . we calculate the partition function by two different algebraic and combinatorial methods . we also determine the entanglement properties of its ground states . * the quantum ( non - abelian ) potts model and its exact solution * 4em * razieh mohseninia * and * vahid karimipour * 1em department of physics , + sharif university of technology , p.o . box 111555 - 9161 , tehran , iran . + pacs numbers : 05.50.+q , 64.60.de , 75.10.-b , 75.10.hk , 03.65.ud	14,095	266
the @xmath8-state potts model can be represented as the correlated site - bond percolation in terms of fortuin - kasteleyn clusters @xcite . at the critical point of the second order phase transition , the infinite cluster is formed . this cluster crosses the system connecting the opposite sides of the square lattice . in the last decade the study of the shape of the crossing probability was performed by conformal methods @xcite as well as numerically @xcite . according to refs . @xcite the distribution function of the percolation thresholds is gaussian function . following the number of works @xcite the tails of the distribution function are not gaussian ones . the authors of the recent work ref . @xcite are still uncertain to distinguish a stretched exponential behavior from a gaussian . the aim of this paper is to investigate the shape of the probability of a system to percolate only in horizontal direction @xmath0 . we perform numerical simulation of correlated site - bond percolation model for @xmath1 ( the percolation model @xmath9 , the ising model @xmath10 and the potts model @xmath11 ) for lattice sizes @xmath12 . the scaling formulas for a body of the crossing probability at criticality and for tails of the crossing probability were obtained . the final result for the representative case @xmath10 , @xmath13 is immediately presented in fig . [ fig1]a ) . details of fitting procedure are described in section [ secapp2 ] . in this figure we plot ( by crosses ) the numerical data for the absolute value of the logarithm of crossing probability @xmath0 for the ising model ( @xmath10 ) on the lattice @xmath13 as a function of the absolute value of the scaling variable @xmath14 . here @xmath5 is a probability of a bond to be closed , @xmath15 is the inverse temperature , @xmath4 is the correlation length scaling index . the critical point in the @xmath16 scale for the @xmath8-state potts model is @xmath17 see ref . @xcite and we get @xmath18 . we can see from fig . [ fig1]a ) that the function @xmath19 consists of two parts : the body @xmath20 and the tails @xmath21 . the negative logarithm of the body of the crossing probability as a function of @xmath22 is well described by function @xmath23 ( solid line on the fig . [ fig1]a ) ) . here @xmath7 is some scaling index . the value of the crossing probability at the critical point @xmath24 may be computed ( at least for percolation ) by conformal field methods @xcite . the negative logarithm of the tails of the crossing probability have shape @xmath25 ( dashed line in the fig . [ fig1]a ) ) . this line is tangent to the body at the point @xmath26 . this point is marked by the horizontal line . let us note that in fig . [ fig1]a ) we plot two branches of the crossing probability ( for @xmath27 and @xmath28 ) . the coincidence of this two branches indicate the remarkable symmetry of the crossing probability with respect to the variable @xmath22 . in fig . [ fig1]b ) we plot the crossing probability by crosses ( bottom ) and the magnetic susceptibility by triangles ( top ) as a functions of the inverse temperature @xmath15 with logarithm scale for the ordinate axis . in fig . [ fig1]b ) we indicate the position of crossover region of @xmath0 by horizontal solid line on a level @xmath29 . for the magnetic susceptibility we mark the region with critical behavior @xmath30 by horizontal dashed lines . we see from fig . [ fig1]b ) that tails of the crossing probability directly correspond to the critical region of the magnetic susceptibility . in this critical region the correlation length @xmath31 is smaller than the sample size @xmath32 . as the temperature approaches to the critical point , the correlation length reaches the sample size . at that point the magnetic susceptibility on the finite lattice deviates from the critical behavior eq . ( [ eq1 ] ) and becomes smooth see the region over the top dashed horizontal line in the fig . [ fig1]b ) . at the same point the crossing probability crosses over from tails to body the region over the solid horizontal line in fig . [ fig1]b ) ( and the region _ under _ the horizontal line in fig . [ fig1]a ) ) . at the critical point @xmath33 both the magnetic susceptibility and the crossing probability reach a maximum . the detailed description of the fitting procedure as well as numerical data for @xmath1 are presented below . the main numerical result of this paper is the proving of the formula @xmath34l^{{\nu}})$ ] for the tails of the crossing probability . therefore , we pay special attention to fitting procedures . the paper is organized as follows : in the second section , we describe details of the numerical simulation . in the third section , the method for determining the pseudocritical point @xmath35 on the finite lattice is described . we use @xmath35 to perform the approximation of the tails . in section [ secapp1 ] we approximate the double logarithm of the crossing probability @xmath36 tails as a function of the logarithm of deviation from the critical point @xmath37 by the linear function @xmath38 . we get @xmath39 for this approximation procedure . in section [ secapp2 ] we describe new fitting procedure using the scaling variable @xmath14 . results of approximation are discussed in section [ secres ] . we perform the massive monte - carlo simulation on the square lattice of size @xmath40 to obtain the high - precision data for @xmath0 . we use the dual lattice shown in fig . [ fig2 ] . on such lattice the critical point of the bond percolation ( @xmath9 ) is exactly equal @xmath41 and is not dependent on the lattice size @xcite . to produce the pseudorandom numbers we use the r9689 random number generator with four taps @xcite . we close each bond with a probability @xmath16 and leave it open with a probability @xmath42 . then we split the lattice into clusters of connected sites by using the hoshen - kopelman algorithm @xcite . after that we check the percolation through this configuration . we average the crossing probability over @xmath43 random bond configurations . to investigate the potts model @xmath44 we use the wolff @xcite cluster algorithm to generate a sequence of thermally equilibrated spin configurations . for each particular inverse temperature @xmath45 we flip 20000 wolff clusters to equilibrate the system . for the spin model on the finite lattice the deviation of the pseudocritical point from the position of the critical point on the infinite lattice is smaller for periodic boundary condition ( pbc ) rather than the open boundary condition ( obc ) @xcite . for this reason we use the pbc for the wolff algorithm . the monte - carlo algorithm generates spin configurations on a torus . for a generated spin configuration we create a configuration of bonds . each bond between sites with equal spin variable @xmath46 is closed with the probability @xmath5 and is open with probability @xmath47 . bonds between sites with different values of @xmath46 are always open in accordance with fortuin - kasteleyn rule @xcite . then we split the particular spin and bond configurations into different clusters . here we use obc . it means that for each generated configuration we cut the torus and check the crossing on the square with open boundary conditions . we fix the obc for crossing only in horizontal direction @xmath0 because it implies the vertical crossing is absent and the top and bottom rows must be disjointed . but we take into consideration the additional raw and column of bonds , as shown in fig . we check the percolation through an obtained cluster configuration , generate new spin configuration and so on . we average the crossing probability over @xmath43 configurations for each value of the inverse temperature @xmath15 . so the resolution of our computations is about @xmath48 . by this way we perform numerical simulation and get a set of data for @xmath49 for the lattice sizes @xmath12 and @xmath44 . the formal definition of @xmath49 as a sum over different cluster configurations is described in @xcite . for the potts model we use the dual lattice as we do for the percolation . it means , that we take into account additional bonds attached to the bottom row of spins . in the same way we take into account additional bonds attached to the right column of the spins . on the lattice with pbc these bonds have to connect the right and the left columns . we cut the torus ( because we use obc for crossing probability ) but we keep these additional bonds and take into account the checking of the crossing . in fig . [ fig2 ] contact points are shown by arrows . the left contact points are attached to the left column of sites . the right contact points are attached to additional bonds . in fig . [ fig2 ] the bond configuration with the horizontally spanning cluster is shown . then we check the percolation through the obtained cluster configuration . after that we flip three wolff clusters , check spanning for a new spin configuration and so on . we investigate the crossing probability as a function of deviation from the critical point . therefore , we perform the preliminary approximations to obtain the critical points for the finite samples . namely , we : * obtain the pseudocritical point and the shape of the central part of the crossing probability , * determine the shape of the tails of the crossing probability * combine together the information for body and tails , and reconstruct the total shape of the crossing probability . we need to recall that we consider the crossing probability as a function of the variable @xmath5 ( probability of a bond to be closed ) . it is easy to understand that we must take the pseudocritical point on the finite lattice @xmath35 as the reference point . the crossing probability is a symmetric function of the variable @xmath50 . this fact implies that the high temperature tail @xmath51 ( @xmath52 ) and the low temperature tail @xmath53 coincide @xmath54 . for the bond percolation on the dual lattice the position of the percolation point does not depend on the lattice size @xmath55 @xcite . for the determination of @xmath35 for the ising and the potts model we use the following procedure . we can assume @xcite that in the the fitting formula is true @xmath56^{\zeta(l , q)}).\ ] ] therefore we fit the logarithm of the crossing probability @xmath57 by the function @xmath58 namely @xmath59^{\zeta(l , q)},\ ] ] where @xmath60 . we plot the data for @xmath61 as a function of @xmath16 in fig . [ fig3]a ) , fig . [ fig3]b ) , fig . [ fig3]c ) , fig . [ fig3]d ) for @xmath1 respectively . the errorbars in these figures are about the symbol size . it seems , that behavior of @xmath61 near @xmath2 is parabolic . results of the approximation are plotted in the same figures by lines . we shall see that there is a good agreement between the numerical data and the results of the approximation . but we see deviation at the point @xmath2 especially for @xmath11 . in the vicinity of @xmath2 real graphs are more smooth than fitting functions . finally for each pair of numbers @xmath62 we obtain four fitting parameters @xmath63 . here @xmath64 defines the crossing probability in the critical point , @xmath65 is a scaling variable , @xmath35 is the position of the pseudocritical point on the lattice @xmath40 , and @xmath66 is a scaling index . in table [ tab1 ] we collect data for the logarithm of critical amplitude @xmath60 . the fitting parameter @xmath67 defines a vertical shift of graphs in fig . [ fig3]a)-[fig3]d ) from the zero level . .results of the approximation for the fitting parameter @xmath60 . [ cols="^,^,^,^,^",options="header " , ] [ tab11 ] using the dual lattice ( see fig . [ fig2 ] ) allows us to avoid finite size shift of the critical point for the bond percolation and to diminish it for spin models . the accuracy of definition of the critical point on the finite lattice play the principal role for the investigation of the tails scaling . the high quality of our approximation is proved by remarkable symmetry of the crossing probability with respect to the critical point @xmath2 . in fig . [ fig5 ] and fig . [ fig7 ] we can observe that the two branches @xmath68 and @xmath69 practically coincide . the two different scaling region of the crossing probability clearly seen in fig . [ fig1]a ) can explain the long time uncertainty about its shape . in ref.@xcite the scaling index for the percolation threshold for @xmath70 percolation model was found @xmath71 . this result coincides with our approximation of the body of the crossing probability for percolation @xmath72 . in more recent works @xcite the tails region for percolation was investigated which is described by the scaling formula @xmath73 . the crossover form to gaussian - like behavior to slope @xmath74 is observed in figures of ref.@xcite . it seems , near the critical point the behavior of the crossing probability is parabolic . the rounding happens in the interval @xmath75 . this interval is relatively small in comparison with regions of the body @xmath76 and tails @xmath77 as it can be seen in fig . [ fig7 ] . we have five fitting parameters @xmath78 , @xmath79 , @xmath80 , @xmath81 , @xmath82 in expressions ( [ eq10 ] ) and ( [ eq11 ] ) . in fig . [ fig7]a)-fig . [ fig7]d ) we see the crossover region between the body and the tails of the crossing probability . in this region the function @xmath83 touched the line @xmath84 . it means , that in some point @xmath26 the values of the functions are equal , therefore @xmath85 and first derivatives of this functions are equal too @xmath86 substituting @xmath81 from eq . ( [ eq16 ] ) in eq . ( [ eq15 ] ) we obtain expression for @xmath79 @xmath87 if the crossing probabilities at the critical points @xmath88 ( and logarithms @xmath78 ) can be calculated analytically by conformal field methods ( at least for the percolation it is possible ) @xcite then only four independent parameters @xmath79 , @xmath89 and @xmath80 , @xmath78 remain for the crossing probability . * in accordance with scaling theory the finite size scaling of the crossing probabilities may be eliminated by introducing the scaling variable @xmath90 . the crossing probability as a function of @xmath22 does not depend on the lattice size @xmath40 . * the body of the crossing probability scales @xmath91 . * the tails of the crossing probabilities scales @xmath92 . the value of parameter @xmath93 is about 1 . * the finite size scaling for @xmath94 does not describe by the analytical value of the correlation length index @xmath95 . we obtain some scaling index @xmath96 . this index @xmath97 for the tails of the crossing probability ( see table [ tab5 ] ) or @xmath98 for the body of the crossing probability ( see table [ tab2 ] ) .	the scaling of the tails of the probability of a system to percolate only in the horizontal direction @xmath0 was investigated numerically for correlated site - bond percolation model for @xmath1 . we have to demonstrate that the tails of the crossing probability far from the critical point @xmath2 have shape @xmath3^{\nu})$ ] where @xmath4 is the correlation length index , @xmath5 is the probability of a bond to be closed . at criticality we observe crossover to another scaling @xmath6^{\nu } \right\}^{z } \right)$ ] . here @xmath7 is a scaling index describing the central part of the crossing probability .	4,157	175
type ia supernovae ( sne ia ) have proven to be effective standardizable candles and their high peak luminosity has made them excellent cosmological probes . an empirically determined relation between light - curve properties and peak absolute magnitude @xcite has yielded improved estimates of the global hubble constant @xcite . although the earliest attempts to measure the matter density of the universe preceded these light - curve shape methods @xcite , and the initial efforts by the supernova cosmology project ( scp ) led to incorrect conclusions about the cosmic matter density @xcite , early work by both the scp and the high - z supernova search team @xcite found evidence for low matter density @xcite . subsequent publications by @xcite and @xcite came to the surprising conclusion that the universe is expanding at an accelerating rate , driven by a mysterious dark energy ; see @xcite , @xcite , or @xcite for reviews . this surprising result has been confirmed by more - recent supernova observations @xcite and by complementary measurements of the cosmic microwave background ( cmb ) anisotropies ( e.g. , * ? ? ? * ) and large - scale structure ( e.g. , * ? ? ? focus has now shifted from demonstrating the existence of dark energy to constraining its properties @xcite and sharpening the tools available for its study ( e.g. , * ? ? ? * ; * ? ? ? . analyses of their spectra and light curves have led to the consensus that sne ia are well characterized by a thermonuclear disruption of a carbon - oxygen white dwarf ( wd ) . the transition from a dynamically stable wd to an explosion capable of outshining an entire galaxy is precipitated by the wd mass approaching the chandrasekhar mass limit through matter accretion from a close binary companion . the nature of this accretion process is still uncertain , though two scenarios have become the focus of sn ia progenitor modeling . the first is the single - degenerate ( sd ) model in which a wd accretes matter from a non - degenerate companion @xcite , while the second is the double - degenerate ( dd ) model involving the merger to two wds @xcite . though promising , our understanding of these models remains incomplete with questions persisting about the nature of the binary companion ( main sequence , asymptotic giant branch , etc . ) and accretion flow in the sd scenario , the apparent likelihood that a wd merger will result in an accretion - induced collapse rather than a sn ia , and the potential for multiple stellar - population - dependent channels leading to sn ia explosions . differentiating among dark - energy models requires mapping the expansion history of the universe with high precision and a small systematic error . applying sne ia to this problem is complicated by the uncertain nature of the explosion physics and progenitor stars ( e.g. , * ? ? ? * ; * ? ? ? even with the most basic assumption , that a sn ia results from the thermonuclear disruption of a near - chandrasekhar mass carbon - oxygen wd , there remain potential dangers in blindly applying sne ia to the dark - energy problem . theoretical models @xcite suggest that population age and metallicity may have an effect on the wd composition and influence the sn ia peak luminosity . since the average age and metal content of stars have evolved over cosmic time , the ensemble character of sn ia explosions may be a function of lookback time . but the local universe contains a range of stellar ages and metallicities , so an empirical calibration of these effects may be possible . @xcite were the first to note a correlation between sn ia decline rate and host - galaxy morphology . the results showed that intrinsically faint events occur in early - type ( e / s0 ) galaxies , while luminous events are often hosted by late - type galaxies . this correlation with morphology has since been confirmed in larger sn ia samples ( e.g. , * ? ? ? * ) , but the cause of the trend remains unknown . the existence of a correlation is important because it demonstrates that the sn ia environment imprints itself on some aspect of the progenitor . on average , spiral and elliptical galaxies differ in population age , star - formation rate ( sfr ) , and metal content , with significant overlap in these properties across the morphological types . @xcite measured the metallicity of a small sample of star - forming hosts to test if age or metal content was the factor influencing sn ia luminosity , but the results were ambiguous . @xcite expanded the number of observed hosts to test the impact of stellar environment on the photometric properties of nearby sne ia . potential dependencies of sn decline rate on host - galaxy absolute magnitude and star - formation history were investigated in a sample of galaxies with a broad range of properties and across the full hubble sequence . metallicity was measured from emission lines , limiting this aspect of the study to star - forming hosts . no significant correlation between metallicity and decline rate was detected , though a tenuous correlation was found between the metal content and hubble residuals . furthermore , only galaxies without significant star formation were found to host faint , fast - declining sne ia . the division by star - formation history was more stark than by morphology alone , suggesting that the observed luminosity spread in sne ia results from the influence of population age on the wd progenitors . the luminosity / star - formation relation has also been seen in a high - redshift sample of sne ia studied by the supernova legacy survey ( snls ) @xcite . the specific sn ia rate is strongly dependent on the amount of current star formation @xcite . actively star - forming galaxies host approximately 10 times as many sne ia per unit mass than do their low star - forming counterparts . @xcite find that the sn ia rate is well described by a combination of two distinct populations of sne ia : a prompt " ( shorter delay time ) sn ia component dependent upon the recent star formation in the galaxy and an delayed " or `` tardy '' ( longer delay time ) component dependent on the number of low - mass stars . assuming the light - curve shape correction methods are perfect , then this will not have an effect on the recent cosmological results . however , if the corrected photometric properties of these two sn populations differ ( e.g. , a variation due to age or metallicity ) , then it could have implications for derived cosmological parameters because the short delay time sne are expected to dominate at high redshift whereas the long delay time events are expected to dominate at low redshift . here we present our study on the luminosity - weighted ages and metal abundances of early - type host galaxies . early - type galaxies , unlike star - forming spirals , have been shown to host both faint and bright sne ia , thereby allowing us to sample the full diversity of sne ia . consequently , we focus our study on absorption - line spectra of host galaxies and utilize single age , single metallicity stellar population models to characterize the stellar populations @xcite . the impact of both age and metallicity on sn peak magnitude is studied and the sample age and metallicity distributions will be compared to absorption - line spectra of field galaxies obtained from the sloan digital sky survey ( sdss ) . furthermore , we will infer from our data a measurement of the relative sn ia rate as a function of galaxy age . finally , we will search for second - order dependencies at the level of the sn ia intrinsic scatter by comparing host - galaxy ages and metallicities to the sn residuals from the hubble diagram . we detail our observational techniques and data - reduction pipeline in 2 of this paper . in 3 we report the methods for performing emission - line corrections , and for obtaining age and metallicity estimates of our host sample . our results are presented in 4 , and the summary and conclusions are in 5 . the objective of this study was to obtain absorption - line spectra of sn ia host galaxies with the intent of estimating their ages and metallicities through a comparison with the simple stellar population ( ssp ) models of alexandre vazdekis @xcite . to this end , we obtained spectra of a sample of sn ia host galaxies with the 1.5 m tillinghast telescope located at the f. l. whipple observatory . we employed the fast spectrograph @xcite fitted with the 300 line mm@xmath3 reflection grating and a @xmath4 slit . the setup gave a resolution of @xmath5 full width at half maximum intensity ( fwhm ) . the spectra were obtained over the course of 6 nights during april and july of 2005 . our set was compiled from the host galaxies of sne ia from the samples of @xcite , @xcite , and @xcite . suitable targets were those classified as having early type ( s0e ) , as well as spiral galaxies containing strong absorption lines in their spectra ( and minimal emission lines ) as determined by @xcite . in addition to the host galaxies , we also obtained spectra for a set of comparison elliptical galaxies having age and metallicities estimates determined by @xcite . the slit was aligned along the galaxy s major axis with position angles ( pas ) determined via the digital sky survey ( dss ) plates . in a few cases the pa was altered to prevent a nearby star from falling on the slit . exposure times were varied depending on the galaxy brightness , with a target signal - to - noise ratio ( s / n ) of 1020 . for each target we obtained three spectra that we subsequently combined to improve the s / n and remove cosmic rays . bias , dark , and dome - flat exposures were taken at the beginning and the end of each night , and a comparison spectrum of a he - ne - ar lamp was acquired before each target for wavelength calibration . finally , flux standard star spectra were taken each night with the slit oriented along the parallactic angle @xcite . our full sample of galaxies can be seen in table 1 . columns ( 1 ) and ( 2 ) give the host - galaxy name and hosted sn , respectively . column ( 3 ) provides the peak magnitudes of the hosted sn ia , determined from the sn light - curve data and employing mlcs2k2 @xcite . column ( 4 ) gives the morphological classification of each galaxy , while column ( 5 ) reports the pa for the each observation . column ( 6 ) shows the width of the extracted aperture encompassing all the light out to approximately the effective radius , @xmath6 . finally , column ( 7 ) gives the source of the sn light - curve : ( 1 ) the cfa sample @xcite , ( 2 ) the katzman automatic imaging telescope ( kait ) @xcite , and ( 3 ) sn 2005bl @xcite . the majority of data reduction for this study was performed using standard techniques within iraf . the data were bias and dark - subtracted , and flat - fielded to remove the pixel - to - pixel variations in the ccd ; also , the bad rows and columns of the ccd were interpolated across using the routine fixpix . following preprocessing , the task was to combine the set of three spectral images that we had for each target . the first step was the removal of cosmic rays . in the interest of preserving every image in a set , thereby increasing our statistics , we removed each cosmic ray individually using the imedit routine in iraf . cosmic rays that fell on the galaxy nucleus itself were left alone until after the aperture extraction . one - dimensional aperture extraction was performed using the apall routine with extraction radii equal to the effective radius ( @xmath6 ) of each galaxy . the effective radii were determined through an idl code written to integrate over the average profile of 400 centrally located columns along the spatial axis of each image . each spectrum was then individually checked for cosmic rays landing on one of our relevant absorption - line indices or the surrounding continua . in the event that an interfering cosmic ray was found , steps were taken to either remove it interactively within splot , or to remove the image altogether if the cosmic ray was too disruptive . the average of each set of images was generated using im - combine , and the resultant spectra were wavelength and flux calibrated . finally , the spectra were corrected for galactic extinction using the iraf routine deredden @xcite . a well known and much reviled puzzle in the study of stellar populations is the observed similarity between the effects of age and metallicity on the integrated light of stellar populations ( e.g. , * ? ? ? astronomers attempt to unravel this age / metallicity degeneracy through the development of stellar population models . the simplified goal of stellar population synthesis modeling is to find a combination of stars for which the integrated spectrum of the stars matches the observed spectrum of the population under study . early empirical techniques such as quadratic programming devised by @xcite have given way to evolutionary population synthesis models which have been improving since the 1970s @xcite . whereas empirical models are constructed from a combination of stellar spectra or the spectra of stellar clusters , evolutionary synthesis models utilize theoretical stellar evolutionary isochrones as the primary constituent . a single isochrone on the hertzsprung - russell ( hr ) diagram gives the locus of luminosities and temperatures at a single moment in time for stars of all masses . together with an assumed initial mass function ( imf ) , an isochrone can be built to accurately model a single - age stellar population . the final model is obtained by converting isochrone parameters to observed spectra and finally integrating along the isochrone . the populations that are modeled using the above techniques are known as simple stellar populations , or ssps . ssps are , by definition , a population of stars created during a single burst event ( i.e. , all the stars have the same age ) and possessing a single global metallicity . however , early - type galaxies are often comprised of multiple - component stellar populations . galaxy ages derived through a comparison with these single - component ssps are luminosity weighted , and are typically sensitive to the youngest component in a real early - type galaxy . consequently , it is appropriate to view the ages derived in studies such as ours as lower - limit estimates @xcite . the determination of age and metallicity via galactic spectra is fairly simple , in principle . we can exploit predictions made by population synthesis models to relate physical properties , such as age and metallicity , to observables , such as the absorption - line strengths for the strongest atomic and molecular absorption features in the optical range . measuring the relative strengths of these absorption features , we can estimate the age and metallicity of the stellar population that created the spectrum . for the age and metallicity estimates of our galaxies , we employ the single age , single metallicity stellar population models described by @xcite , @xcite , @xcite , @xcite , and @xcite . the models make use of the empirical stellar spectral library miles @xcite . miles is a marked improvement over previous libraries ; it has 985 stars , a spectral resolution of 2.3 ( fwhm ) , and a wavelength range of 35257500 . one advantage of empirical libraries over theoretical stellar libraries is that they are based on actual stellar spectra and are not dependent on the potential inaccuracies and underlying assumptions inherent in any theoretical model . however , in order for an empirical stellar library to sufficiently cover the parameter space in temperature , gravity , abundance , and [ @xmath7/fe ] , the stellar observations must be of the highest quality . consequently , empirical stellar libraries have traditionally been restricted to the nearest stars , thereby introducing a bias of atmospheric parameters toward those which are seen in the solar neighborhood . the miles library acquired atmospheric parameters from the literature and calibrated the set through a subset of reference field stars from @xcite ; for full details see @xcite . in this way , miles has optimized its stellar atmosphere coverage and improved upon previous libraries . nevertheless , because early - type galaxies are generally @xmath7 enhanced and metal rich , the potential bias toward solar abundance ratios should be kept in mind . the stellar population synthesis models derived from miles consist of single age , single metallicity spectral energy distributions ( seds ) that adopt a standard salpeter imf @xcite . the seds possess a fwhm resolution of 2.3 and a spectral range similar to our data at 35407410 . there are 276 models with ages ranging from 0.10 to 17.78 gyr and metallicities ranging from [ m / h ] = @xmath8 to 0.20 . in the past , the most widely used method of estimating stellar population age and metallicity has been through a comparison of absorption - line indices with those of the lick / ids system @xcite . since the stars of the lick / ids stellar library are not flux calibrated , data must be converted to the instrumental response curve of the original data set @xcite . with the advent of improved , flux - calibrated stellar libraries , it became possible to generate complete seds of a single stellar population @xcite , thereby eliminating the need for such a correction . furthermore , in order to use the lick population models , it is necessary to transform the observational data to match the resolution of the lick spectrograph ( @xmath910 fwhm ) . for our analysis , the model seds are at a higher resolution than the data ; consequently , the models were required to be broadened to the resolution of our data rather than the other way around . thus , the first step in the analysis was to characterize the wavelength - dependent resolution of our spectra . this was accomplished by measuring the line widths of emission lines in the he - ne - ar lamp across our full spectral range . a polynomial fit to the resulting scatter plot provided a wavelength - dependent resolution function for the fast spectrograph . we derived a wavelength - dependent broadening term with a square equal to the difference between the square of the host - galaxy resolution and that of the resolution of the vazdekis models , given as 2.3 . the next broadening term we considered was from the velocity dispersion of the host galaxies . to determine the velocity dispersion for each galaxy , we obtained spectra of velocity standard stars hd12623 and hd52071 ; these were taken within a few months of our observing runs on the tillinghast telescope using the fast spectrograph . in total , we obtained 6 spectra of hd12623 and 14 spectra of hd52071 . cross - correlation analysis was then performed between these templates and the host - galaxy spectra using the iraf routine fxcor . a velocity dispersion and redshift estimate were obtained using each template - galaxy combination , with the final result for a given host being the average of the measurements made with each of the twenty templates . our two broadening terms were summed in quadrature , yielding the final width of our gaussian kernel used in the convolution with the host galaxy . each host galaxy had a unique final broadening term and the vazdekis model seds were broadened independently according to this term . in this way , we created separate age / metallicity grids for each host galaxy . the velocity dispersions and heliocentric velocities resulting from this analysis are shown in table [ tbl-2 ] . these heliocentric velocities were converted to cmb rest frame for the subsequent analysis using the ned velocity calculator . the age dependent absorption index chosen for this study was h@xmath10 . h@xmath10 is known to be metallicity insensitive and older stellar populations are known to have relatively strong h@xmath10 absorption , eliminating the need for high s / n @xcite . although our sample was comprised mostly of older stellar populations , there was a chance for h@xmath10 emission contamination . consequently , our data were emission - line corrected with the procedure discussed in the next section . we chose two age - insensitive fe absorption lines for our abundance analysis , fe@xmath115270 and fe@xmath114383 . plotting both of these iron lines against h@xmath10 produces a strongly orthogonal grid well suited for untangling the age - metallicity degeneracy . our chosen lick / ids indices are summarized in table [ tbl-3 ] . the presence of h@xmath10 emission superimposed on h@xmath10 absorption poses a significant problem for our index measurements . the contamination of even weak h@xmath10 emission lines in our galaxy spectra could significantly decrease the strengths of our measured h@xmath10 absorption indices , thus systematically biasing our age distribution toward greater ages . consequently , an emission correction was performed on our host - galaxy spectra . the basic procedure is outlined by @xcite . the method is based on two main assertions . first , although the underlying absorption - line spectrum at h@xmath10 is highly sensitive to age , the sensitivity to both age and metallicity is negligible at h@xmath7 . second , the observed hn / fe index @xmath12h@xmath13/fe@xmath114045 + h@xmath14/fe@xmath114325 + h8/fe@xmath113859@xmath15 . ] represents an effective indicator of the true underlying absorption - line spectrum , implying that two galaxies with similar hn / fe indices will likewise possess similar h@xmath7 absorption strengths . conservatively , we assumed that each galaxy in our sample was contaminated with balmer emission from h ii regions not included in the vazdekis models . therefore , we predicted the underlying h@xmath7 absorption spectrum for a given contaminated galaxy spectrum by matching that galaxy s hn / fe index with a vazdekis model galaxy displaying a similar hn / fe index . the matching vazdekis model sed was then treated as our model absorption spectrum . next , we smoothed this spectrum to the appropriate resolution and velocity dispersion , normalized it to zero , and subtracted it from the host - galaxy spectrum . this left the h@xmath7 emission free of any contaminating absorption . the emission - line flux was measured and converted into h@xmath10 emission flux assuming a case b balmer decrement @xcite . we used a flux - calibrated spectrum of the orion nebula taken with the same instrumental setup to model the emission spectrum intended to be subtracted off the contaminated galaxy . the orion spectrum was smoothed to the appropriate velocity dispersion and scaled so that the h@xmath10 flux matched the calculated value . the continuum was removed from the modified orion spectrum , and the result was subtracted from the contaminated host - galaxy spectrum . in order to measure the h@xmath10 , fe@xmath115270 , and fe@xmath114383 indices , we employed the fortran77 code lector provided by alexandre vazdekis . the code measures the indices of a one - dimensional input spectrum and calculates an error estimate based on photon statistics @xcite . with the line strengths we generated our index - index diagrams , in which each vertex represents a combination of line - index measurements for a single model of given age and metallicity . a qualitative comparison between several models and a representative host galaxy are given in figure [ example_specs ] , showing a comparison of the spectrum for host galaxy ngc 4786 ( @xmath16 ) to model seds with similar metallicity ( @xmath16 ) but with varying population age . quantitative age and metallicity estimates are found through a comparison with these model seds by way of an interpolation of the index - index diagnostic grids . figure [ fig1 ] shows the fe@xmath115270h@xmath10 and fe@xmath114383h@xmath10 grids . in each case the non - emission - corrected host - galaxy indices are plotted in the upper frames and the emission - corrected indices are plotted in the lower frames . although for each galaxy metallicity and age interpolation we broadened the models according to its corresponding velocity dispersion , for the sake of presentation we show the whole sample plotted on a model grid broadened to a velocity dispersion of @xmath17 = 200 km s@xmath3 . index measurements . the series shows how the models change with age at a fixed metallicity and gives an alternate illustration of the technique used to realize our galaxy age / metallicity estimates . in this case , the age - sensitive index , h@xmath10 , for ngc 4786 most closely matches that measured for the sed at solar metallicity and a single population age of 7.08 gyr . [ example_specs ] ] computing ages and metallicities for our galaxy samples required the interpolation , and on occasion extrapolation , of an irregular grid . this was accomplished using bivariant polynomial transformations of the following form @xcite : @xmath18 where @xmath19 is the order of the polynomial and _ p@xmath20 _ and _ q@xmath20 _ are the coefficients that are solved for a set of nearest - neighbor points ( _ indx_,_indy _ ) indexed by _ i_. we performed a second - degree approximation , thus requiring us to compute at least 12 coefficients by solving two systems of 6 linear equations . an idl code was written which allowed us to interactively select the nearest 612 grid points ( the number was dependent on the quality of the grid around the host galaxy s grid position ) and the coefficients were computed by solving the above system of equations using the method of least squares . the upper and lower limits for age and metallicity were determined directly from the absorption - line index limits outputted by lector . the extent of the index limits can be seen from the error bars in figure [ fig1 ] . the upper and lower limits were treated as unique points for which age and metallicity estimates were made . in the case of large index errors , the grid points chosen by our code were different from those used to determine the age / metallicity of the galaxy . an assumption was made that the upper and lower h@xmath10 index limits corresponded to the lower and upper limits , respectively , of our galaxy age estimate . similarly , the upper and lower limit of the fe@xmath115270 index corresponded to the upper and lower limits of the galaxy metallicity estimate . the final uncertainty was recorded as the magnitude of the difference between the age and metallicity of the limits and those of the galaxy . furthermore , due to the irregular nature of the grid , a symmetric error in an index measurement translated into an asymmetric error in age and metallicity . the results from our analyses are shown in table [ tbl-4 ] . column ( 1 ) gives the galaxy name . columns ( 2)(7 ) give the ages , metallicities , and corresponding errors determined via the fe@xmath115270h@xmath10 diagram , while columns ( 8)(13 ) give the same results ascertained via the fe@xmath114383h@xmath10 diagram . , is plotted against the metallicity - sensitive line indices ew , fe@xmath115270 and fe@xmath114383 , before ( top ) and after ( bottom ) emission correction . the underlying grid reflect the vazdekis model seds broadened to a velocity dispersion , @xmath17 = 200.0 km s@xmath3 . dashed lines ( near vertical ) are models of common metallicity , and dotted lines ( near horizontal ) are those of common age . the bold dashed line identifies models of solar metallicity , and the bold dotted line marks models of age 7.08 gyr.[fig1 ] ] aspiring to test the uniqueness of sn ia early - type host galaxies relative to random early - type galaxies , we compared the global properties of our sn ia sample of elliptical host galaxies to those of a general sample of elliptical field galaxies from sdss . the following steps were taken to generate a comparative sample of field galaxies . the sdss catalog archive server was queried for galaxy spectra within the redshift range 0 @xmath21 that had a velocity dispersion measured by sdss . the positions of these galaxies were then cross - referenced against galaxies in the nasa / ipac extragalactic database ( ned ) , and only those galaxies characterized as e / s0 by ned were accepted . the galaxy spectra of these elliptical field galaxies were then obtained from the sdss data archive server . the spectra were emission - line corrected using a synthetic orion spectrum created via the iraf artificial data generation package , artdata . furthermore , rather than broadening the models to the specific velocity dispersion of each galaxy , we broadened the set of models to velocity dispersions in 30 km s@xmath3 intervals from 100 km s@xmath3 to 340 km s@xmath3 . the set of models used in the analysis for a given sdss galaxy spectrum was the set broadened for a velocity dispersion that most closely matched that of the galaxy . finally , to speed up the calculation , the grid points to perform the diagnostic interpolations were automatically chosen for the sdss sample by an algorithm designed to mimic the grid - point selection criteria used for the interactive point selection in the host - galaxy analysis . the quality of our data is assessed via three checks , the results of which are shown in figure [ fig2 ] . first , we compare the respective age and metallicity estimates obtained from our two index - index diagnostics . figure [ fig2]a shows the fe@xmath114383 age estimate plotted against the age estimate of fe@xmath115270 . given that both ages were determined using the same age - sensitive index , namely h@xmath10 , we would expect these data to be consistent with unity . as the top - left panel shows , we find moderate scatter about unity by virtue of our error estimates . nevertheless , the data are consistent with unity , with a @xmath22@xmath23 of 0.88 for the fit . figure [ fig2]b shows the metallicity comparison between our two index - index diagnostics . in this case we see more significant dispersion ; however , the data are still consistent with unity given the calculated uncertainty . consequently , as a matter of preference , the galaxy ages and metallicities used in the following analysis will be those determined from the h@xmath10fe@xmath115270 diagnostic . , respectively . in both cases we find minor variations that are nonetheless consistent with unity . panels ( b ) and ( d ) present a comparison showing reasonable agreement between our results and those of @xcite.[fig2 ] ] our second quality check is a comparison of our age and metallicity estimates to those of control galaxies studied by @xcite ( hereafter t00 ) . our data are plotted against the data of t00 in figure [ fig2]c and [ fig2]d . t00 made age and metallicity measurements through a central @xmath6/2 aperture and we are reporting the preferred " non - solar abundance ratio ( nsar ) model results ( model 4 ) . the age - sensitive line index used was h@xmath10 , while the two metallicity - sensitive indices were @xmath12fe@xmath15fe@xmath15 @xmath24 ( @xmath25)(fe@xmath115270 + fe@xmath115335 ) . ] and mg b. figure [ fig2]c shows the age comparison for the two studies with the dotted line shows a model one - to - one relation . the plot shows fairly good agreement between the two studies . the @xmath22@xmath23 of the one - to - one fit is 8.55 , though it is improved if we throw out ngc 5813 at ( 8.69 , 24.30 ) , reducing @xmath22@xmath23 to 2.72 . the agreement is less encouraging for the metallicity comparison in figure [ fig2]d . ignoring the few outlier points , the t00 galaxies seem to be systematically more metal rich than those measured in our study . this is likely due to the fact that we did not account for nsar in our analysis . studies have shown that the abundance ratios in early - type galaxies are often non - solar . in particular , the mg / fe abundance ratio has been shown to be larger in more luminous early - type galaxies @xcite . given that both mg and fe lines strongly factored into the metallicity measurements of t00 , we should not expect to see a straight 1:1 comparative ratio . nevertheless , throwing out the points with the two greatest dispersions ( ngc 5813 and ngc 4489 ) , we seem to be measuring the relative metallicities consistently with t00 , thereby enabling us to find potential trends between age / metallicity and sn ia properties . we also compare our results to those published in @xcite . in their study , they seek to set constraints on the epochs of early - type galaxy formation through an analysis of 124 early - type galaxies in both high and low - density regions . we have 15 galaxies in common with their study and compare our h@xmath10 index , age , and metallicity measurements to those from their analysis . similar to the comparison with t00 , we see general agreement between our respective results . however , the agreement is far from one to one , with broad scatter greater than that seen in our t00 comparison . once again , this not entirely unexpected given our lower s / n and the fact that we did not accounted for potential @xmath7 enhancement in our galaxies . moreover , they were sampling a significantly smaller fraction of the galactic light with an aperture radius of @xmath6/10 . overall , our comparisons with both t00 and @xcite show reasonable agreement and confirm the ability of our analysis to measure acceptable relative population ages and metallicities for our sn ia host - galaxy sample . finally , we compare the @xmath26 colors for the host galaxy and sdss data to the galaxy age ; see figure [ fig3 ] . the expected average @xmath26 color for a sample of elliptical galaxies is 4.0 @xcite , and we can see that both our host galaxies and the sdss galaxies meet this expectation . the figure also shows that both of the samples fall within the range predicted by the evolutionary synthesis models of @xcite . the center line is the predicted trend of @xmath26 color with age for a population of solar metallicity , the top line is the prediction for a metallicity of [ m / h ] @xmath27 , and the bottom line is the prediction for a metallicity of [ m / h ] @xmath28 . color vs. age . we see that both of the samples fall within the range predicted by the evolutionary synthesis models of @xcite . the center line is the predicted trend of @xmath26 color with age for a population of solar metallicity , the top line is the prediction for a metallicity of [ m / h ] @xmath27 , and the bottom line is the prediction for a metallicity of [ m / h ] @xmath28.[fig3 ] ] the host - galaxy age and metallicity vs. sn ia peak @xmath29-band magnitude are plotted in figure [ fig4 ] . the peak magnitudes were derived from measurements of the light - curve shape parameter , @xmath30 , fitted using mlcs2k2 by the authors of @xcite(peak ) = @xmath31 + 0.182@xmath30@xmath32 + 5 log ( h@xmath33 / 65 km s@xmath3 mpc@xmath3 ) mag . ] . in figure [ fig4]a , the data show that the youngest populations tend to host the brightest sne ia , but with a wide spread . however , the eye has a tendency to gravitate toward large error bars , thereby placing an unwanted amount of attention to those points . consequently , we have replotted the data in figures [ fig4]b and [ fig4]d with point sizes that are inversely proportional to their uncertainty . in this way , the data with the smallest uncertainty in age will be represented by the largest points . when points with the largest uncertainty are de - emphasized , the trend between age and luminosity is very clear , but it is nevertheless difficult to distinguish whether the effect is a smooth transition with age or the result of two distinct populations . = 72 km@xmath3 s@xmath3 mpc@xmath3 ) vs. ( a ) luminosity - weighted host - galaxy age and ( c ) metallicity . panels ( b ) and ( d ) show the same data with the point size reflecting the average error along the abscissa . point size is inversely proportional to the average error along the abscissa , with the exception of our young hosts in ( b ) where an upper limit is imposed on the point size . the data in panel ( b ) show a clear trend between sn ia peak magnitude and host - galaxy age . the behavior could be evidence of either a smooth transition of sn magnitude with age or of two distinct populations of sne ia . panel ( d ) shows a less convincing correlation likely arising from the degeneracy between age and metallicity . [ fig4 ] ] @xcite found that only luminous sne ia occur in strongly star - forming hosts while e / s0 galaxies show a wide range of sn ia luminosities . this is confirmed in figure [ magsfr ] , where we have plotted sn ia peak magnitude vs. host - galaxy specific star formation , i.e. , the sfr per stellar mass . the sfrs were calculated from the h@xmath7 emission flux using the relation of @xcite , with the distance to each galaxy coming from the sn ia luminosities . solid circles represent the host galaxies from the current sample of early - type galaxies , while open circles are from the sample studied by g05 . we eliminated galaxies from g05 if they were in our current sample since the h@xmath7 flux from g05 was uncorrected for underlying absorption . the points with arrows represent star formation upper limits in which h@xmath7 emission was buried within the noise . this trend , combined with figure [ fig4]b , suggests that the age of the dominant population in e / s0 galaxies determines the resulting sn ia peak luminosity . a correlation between sn ia peak luminosity and sfr has also been seen in the high - redshift snls data @xcite , but their type of search is biased against very low - luminosity events . our `` young '' e / s0 galaxies correspond to their `` passive '' hosts , and the extension to very old populations ( with low specific sfr ) confirms that population age is the major factor that determines the sn ia peak luminosity . . closed circles are the current sample of early - type hosts while open circles represent the high - sfr hosts of g05 . the points marked by arrows indicate upper limits . on average , our specific sfr distribution is lower than that seen in the sample of high-@xmath34 passive " hosts of @xcite . however , aside from the expected decrease of sfr over cosmological time , it should be noted that our data only sample half of the galactic light given our extraction radius ( @xmath6 ) , and the sfrs for the passive " galaxies of @xcite were randomly assigned a rate of @xmath00.005 m@xmath35 yr@xmath3.[magsfr ] ] the plots of host metallicity vs. sn ia peak luminosity ( figures [ fig4]c and [ fig4]d ) also show a mild correlation . this may be due to the `` age / metallicity degeneracy '' shown in figure [ fig5 ] . there we present measured galaxy age vs. metallicity for both the hosts and sdss field galaxies , showing that the galaxies are not evenly distributed in the diagram but instead concentrated to the lower - left half . this is simply due to the evolution of the universe which started metal poor , meaning that there were few galaxies that are both old and metal rich and occupy the upper - right of the diagram . thus , any correlation between age and a supernova property will result in some correlation with metallicity as well . here , the relationship between host age and sn ia luminosity is so clear that we claim it represents a physical connection . we attribute mild effect in the metallicity relation to the lack of galaxies in the lower - right quadrant of figure [ fig4]d , which comes from age correlation combined with the fact that there are no old galaxies having high metal abundance in our sample . ] the distributions of age and metallicity for the two samples are also shown in figure [ fig5 ] , with the cumulative fraction plots shown in figure [ fig6 ] . a kolmogorov - smirnov test reveals that there is a 63% chance that the host age distribution is drawn from the sdss field galaxy age distribution , and a 44% chance that the host metallicity distribution is drawn from the field galaxy metallicity distribution . the results suggest that , given the size of the sn ia sample , the abundance and age of early - type galaxies that host sne ia are similar on average to normal early - type field galaxies . assuming this , the probability of a sn ia going off in a given early - type galaxy does not strongly depend on the age or metallicity of the galaxy . and 44@xmath36 probability that the sdss age and metallicity distributions , respectively , represent acceptable parent populations from which the host metallicities were drawn.[fig6 ] ] the relation correcting sn ia peak luminosity by the light - curve shape reduces the scatter in the low-@xmath34 hubble diagram to @xmath0 0.18 mag @xcite . if this residual scatter is purely random , then the evolution of age and metallicity in the universe will not induce a bias in cosmological measurements with sne ia . however , recent studies such as timmes , brown , & truran ( 2003 ; hereafter tbt03 ) and @xcite have predicted that sn calibration methods should be affected by a cosmic evolution in metallicity . in order to investigate this possible effect for sne ia hosted by early - type galaxies , we plot our measurements of age and metallicity against post - corrected sn ia magnitude residual from the best fit to the hubble diagram . the hubble residual ( hr ) is defined as @xmath37 where @xmath38 is the distance modulus determined from the host - galaxy redshift to decrease uncertainty due to peculiar velocities . ] , and @xmath39 is the distance modulus determined via the mlcs2k2-corrected sn magnitude . hr is defined in the conventional manner such that overluminous sne ia have a negative hubble residual . the primary component to the uncertainty in hr comes from @xmath40 . the distance errors estimated from mlcs2k2 were found to be smaller than the scatter about the hubble diagram . the authors of @xcite attributed this to an intrinsic scatter in sn ia magnitudes and accordingly imposed a constant additive correction to these distances . given that we are probing for systematic changes in this intrinsic scatter with age and metallicity , we have not applied this correction to @xmath40 . the top - left panel of figure [ fig7 ] shows the hubble residual vs. host - galaxy metallicity along with a least - squares fit to the data ( solid line ) . the data and the linear fit show a correlation between host metallicity and deviation from the hubble diagram such that supernovae in metal - rich galaxies are fainter than average and metal - poor galaxies host brighter than average events . the dashed line is a representation of the tbt03 relation ( m@xmath41vs@xmath42 metallicity ) showing that , at least qualitatively , the observed trend is consistent with that by expected by tbt03 . with negligible significance . [ fig7 ] ] to map the tbt03 prediction onto the plot , we converted the @xmath43ni yields from tbt03 into v - band peak magnitude via the models of @xcite as was done in gallagher et al . ( 2005 ) . to avoid an over - dependence on models , we also converted ni masses to peak magnitudes through an empirically determined relation using ni mass estimated by @xcite and corresponding v - band luminosities given in @xcite . mapping tbt03 onto figure [ fig7 ] using this conversion yields a qualitatively similar trend to both our data and the tbt03-hflich model . the top - right panel shows the results of the significance analysis for our linear fit . for each point ( [ m / h]@xmath44,hr@xmath44 ) , the hubble residual was replaced by a randomly selected hr from a gaussian distribution centered on the actual hr with a standard deviation equal to @xmath30hr . the same was done for each metallicity using the corresponding [ m / h]@xmath44 and @xmath30[m / h]@xmath44 . we performed 100,000 iterations , measured the best - fit slope ( @xmath10 ) to the new data each time , and generated the @xmath10 distribution for the test . the results are seen in the top - right panel and show that a no - correlation result ( @xmath10 = 0 ) is ruled out at the 98% confidence level . a similar correlation between dust / extinction and hubble residual is not seen for our sample , thereby removing the possibility of this correlation being a secondary dust effect . assuming that the least - squares fit to the top - left panel of figure [ fig7 ] adequately represents the trend of hr with metallicity , we would predict a 0.26 mag increase in intrinsic sn ia magnitude per unit decline in progenitor ( galaxy ) metallicity . we have looked at several studies , two theoretical and one observational , that seek to characterize cosmic chemical evolution . the theoretical studies by @xcite and @xcite predict an approximate 0.05 dex and 0.25 dex drop in metallicity per unit redshift , respectively . the observational study conducted by @xcite find a preliminary result of 0.15 dex/@xmath34 . taking the average of these three studies ( 0.15 dex/@xmath34 ) for the true decline in metallicity with redshift would translate into a 0.039 mag decrease in intrinsic sn ia magnitude per unit redshift . assuming an ideal sn ia data set out to @xmath45 , a @xmath22@xmath32 minimization of three parameters ( @xmath46@xmath47 , @xmath48 , and h@xmath33 ) applied to a set of model sne ia subject to this 4% systematic error would induce an approximate 9% systematic error on the measurement of the equation - of - state parameter @xmath2 . @xcite showed that the scatter of sn magnitudes about the hubble diagram is lower in e / s0 galaxies ( @xmath0 0.13 mag ) than for sample as a whole ( @xmath0 0.18 mag ) . our sample of e / s0 hosts show a scatter of approximately 0.14 mag . by applying a correction to sn magnitudes consistent with our least - squares fit , we reduce the scatter about the hubble diagram to approximately 0.11 mag , illustrating the potential for such a correction to significantly improve sn ia distance measurements . the bottom - left panel of figure [ fig7 ] shows the age of the parent galaxy vs. the hubble residual of the sn ia . the plot does not reveal a significant trend of hr with host - galaxy age . indeed , the probability of a zero - correlation result is only ruled out at the 56% confidence level , suggesting that sn ia magnitudes corrected for light - curve shape are likewise being corrected for any age bias . it should be noted that two galaxies ( cgcg 016 - 058 and mcg+07 - 41 - 001 ) were removed from the age plot because their positions on their respective diagnostic grids required a difficult extrapolation that rendered their age estimates extremely uncertain . since the late 1970s , observations have shown that sne ia are more prevalent in star - forming late - type galaxies than in early - type galaxies @xcite . this fact has been confirmed again and again with new studies showing that the sn ia rate per unit mass is significantly higher in blue / late - type galaxies than in red / early - type galaxies @xcite . the current explanation for these observations is that there are prompt " and delayed " ( `` tardy '' ) sn ia explosions . the prompt component is dependent on the rate of recent star formation , and the delayed component is dependent on the total number of low - mass stars . the combination of these two components is believed to form the overall observed sn ia rate @xcite . @xcite investigated the parameters shaping the overall sn ia rate by observing the relative presence of sne ia in high-@xmath34 host galaxies of differing mass and sfr . our direct age measurement means that we can do a similar analysis at low redshifts by estimating the relative sn ia rate as a function of host - galaxy age . figures [ fig8 ] and [ fig9 ] summarize the details of our analysis on the relative sn ia rate in elliptical host galaxies of various ages . -band luminosity . the left panel shows numbers of host galaxies , or sne , per age bin along with the expected number of stars per age bin derived using the sdss elliptical field galaxies . the right panel gives the expected relative sn ia rate per age bin , showing a consistency with the dual - component " model . [ fig8 ] ] ] the panel on the left of figure [ fig8 ] contains two unique histograms . the dotted - line histogram represents the number of host galaxies , or equivalently the number of sne ia , per age bin ( identical to the distribution in figure [ fig5 ] with decreased bin size ) . the solid - line histogram is the expected number of stars per age bin . each of the distributions has been normalized such that the bin encompassing 8 gyr contains exactly one element . the number of stars is derived from the @xmath49-band sdss galaxy distribution , the @xmath49 band being a relatively accurate tracer of stellar mass . @xmath49-band apparent magnitudes for the sdss sample were compiled from ned and the distances to the galaxies in the sample were calculated from the sdss redshift measurements , where care was taken to exclude those galaxies with @xmath50 km s@xmath3 . the @xmath49-band absolute magnitudes , @xmath51 , were derived from the distances and apparent magnitudes . finally , the number of stars per galaxy was approximated by the expression @xmath52}.\ ] ] the variable @xmath53 is the corresponding absolute magnitude of the sun in the @xmath49 band = 3.28 . ] . the number of stars per age bin was then calculated by summing the stars in each galaxy in each age bin . the relative sn ia rate per unit stellar mass is defined as the ratio of the number of sne per age bin to the expected number of stars per age bin . the panel on the right of figure [ fig8 ] shows the expected sn ia rate per unit stellar mass as a function of age . the plot predicts a rate that is high for young galaxies , falls for intermediate - age galaxies , and increases and moderates for older populations . this result is consistent with the `` dual - component model , '' given that the sn rate at low age could arise from a short delay time population of sne ia predicted to occur in galaxies with high sfr , and the rise in sn rate at late age could likely result from the predicted rise in sn rate with stellar mass @xcite . however , we take the less general route and suggest the possibility that the rate at low population age is proportional to the rate of wd production following an initial burst of star formation . this function can be seen in figure [ fig9 ] . similarly , the late - time sn rate can be traced by the cumulative number of wds present in the galaxy ( figure [ fig9 ] ) . this value is determined for an age , @xmath54 , by integrating the wd production rate from time @xmath55 to @xmath56 . these components of the sn ia rate are related to those of @xcite , and yet they specifically target the population of stars believed to be the progenitors of sn ia explosions . an important caveat to this analysis is that we are only sampling the dominant stellar population within a given host galaxy . it has been shown that a significant portion ( 1530% ) of elliptical and lenticular ( s0 ) galaxies show evidence of recent star formation @xcite , which has the potential to compromise our conclusion that sne ia exploding in old stellar populations possess longer delay times . we have analyzed a sample of early - type galaxies that have hosted sne ia . comparing the data with stellar population synthesis models , we have measured the global age and metallicity of the sn ia elliptical host galaxies and a general sample of elliptical galaxies from the sdss . our results indicate that there is likely a significant correlation between age or metallicity and sn ia absolute magnitude . given the mounting evidence of a sn ia rate dependence on specific sfr , we find it most likely that the sn ia peak magnitudes are correlated with age , and the observed trend with metallicity is merely an artifact brought about by the evolutionary entanglement of age and metallicity . we find the global distributions of age and metallicity to be similar to those of our sdss elliptical galaxy sample , suggesting that the presence , or absence , of a sn ia in an elliptical galaxy is not dependent on either age or metallicity . moreover , we detect a trend between early - type host - galaxy metallicity and the residuals from the hubble diagram at the 98% confidence level . this trend is consistent with the predictions of tbt03 and with a trend observed for late - type galaxies by @xcite ; it suggests that metal - rich galaxies produce underluminous sne ia , even after correcting for their light - curve shapes . furthermore , we conclude that the failure to apply a metallicity correction to sn ia magnitudes could potentially introduce a 9% error into current and future measurements of @xmath48 . finally , we find the predicted sn ia rate as a function of age for our sample of early - type galaxies , and determine it to be moderately consistent with the `` dual - component model '' of @xcite . also , we describe our preferred model governing the sn ia rate in which the prompt " component is proportional to the wd production rate and the delayed " component scales with the cumulative wd population . partial funding for this work came through nasa ltsa grant nag5 - 9364 . supernova research at harvard university is supported by nsf grant ast06 - 06772 . a.v.f.s supernova group at the university of california , berkeley is supported by nsf grant ast0607485 , as well as by the tabasgo foundation . kait was made possible by generous donations from sun microsystems , inc . , the hewlett - 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024 & 2002 g & -19.347 & e & 10 & 4.16 & 2 + ugc 04322 & 2002he & -19.01 & e & 40 & 5.32 & 2 + cgcg 141 - 044 & 2001bf & -19.514 & & 100 & 5.04 & 2 + anon & 2002aw & -19.35 & sb ( f ) & 80 & 13.94 & 2 + ngc 4786 & 2002cf & -17.82 & e+pec & -20 & 6.47 & 2 + ngc 6702 & 2002cs & & e & 55 & 6.31 & + mcg+07 - 41 - 001 & 2002do & -18.48 & e1 & 0 & 20.85 & 2 + ngc 7761 & 2002ef & -19.30 & s0 & 90 & 4.61 & 2 + ngc 6986 & 2002el & -19.11 & sb0- & 5 & 5.89 & 2 + mcg-01 - 25 - 009 & 2003d & -18.31 & e1 & 10 & 5.18 & 2 + ic 0522 & 2003y & & s0 & -20 & 7.34 & + ugc 03787 & 2003ch & & e - s0 & 0 & 4.47 & + ngc 4059 & 2005bl & -17.62 & e & 40 & 4.61 & 3 + ngc 3608 & & & e2 & 80 & 6.19 & + ngc 4472 & & & e2/s0 & -20 & 12.80 & + ngc 4478 & & & e2 & 40 & 6.17 & + ngc 4489 & & & e & 110 & 4.61 & + ngc 4552 & & & e;hii & -30 & 6.31 & + ngc 4649 & & & e2 & 90 & 13.09 & + ngc 5638 & & & e1 & -20 & 8.04 & + ngc 5813 & & & e1 - 2 & -20 & 6.17 & + ngc 5846 & & & e0 - 1 & 50 & 11.35 & + ngc 6127 & & & e & 90 & 5.32 & + ngc 6703 & & & sa0- & 90 & 5.17 & + lccc anon & 7932.43 & 19.66 & 136.91 + cgcg 016 - 058 & 10411.31 & 17.79 & 147.99 + cgcg 141 - 044 & 4665.13 & 23.01 & 109.73 + cgcg 189 - 024 & 10113.14 & 21.01 & 188.70 + ic 0522 & 5092.26 & 19.05 & 170.76 + mcg+07 - 41 - 001 & 4584.86 & 68.68 & 344.72 + mcg-01 - 25 - 009 & 6623.73 & 21.58 & 251.97 + ngc 2841 & 633.49 & 25.43 & 244.72 + ngc 2962 & 1967.99 & 19.01 & 186.61 + ngc 2986 & 2310.81 & 33.56 & 283.67 + ngc 3608 & 1237.80 & 21.24 & 209.54 + ngc 4059 & 7195.99 & 33.21 & 210.88 + ngc 4374 & 1020.48 & 30.57 & 316.56 + ngc 4472 & 957.20 & 30.95 & 309.86 + ngc 4478 & 1352.18 & 17.67 & 183.09 + ngc 4489 & 948.10 & 11.75 & 106.89 + ngc 4493 & 6957.26 & 20.98 & 214.39 + ngc 4526 & 604.65 & 24.90 & 235.75 + ngc 4552 & 316.64 & 27.18 & 282.82 + ngc 4649 & 1087.03 & 37.32 & 344.45 + ngc 4786 & 4611.97 & 33.49 & 288.86 + ngc 5005 & 932.64 & 23.29 & 207.73 + ngc 5061 & 2059.50 & 22.15 & 218.96 + ngc 5308 & 2014.72 & 22.08 & 241.36 + ngc 5440 & 3697.03 & 27.35 & 238.53 + ngc 5490 & 4965.75 & 37.73 & 338.21 + ngc 5638 & 1640.05 & 20.15 & 172.96 + ngc 5813 & 1948.27 & 28.57 & 248.58 + ngc 5846 & 1692.00 & 27.71 & 238.42 + ngc 6038 & 9399.72 & 32.83 & 183.91 + ngc 6127 & 4715.52 & 29.99 & 264.50 + ngc 6411 & 3726.29 & 23.52 & 186.15 + ngc 6702 & 4721.40 & 32.71 & 209.44 + ngc 6703 & 2384.69 & 20.62 & 199.69 + ngc 6986 & 8534.95 & 25.52 & 267.16 + ngc 7131 & 5409.40 & 36.77 & 185.25 + ngc 7761 & 7192.42 & 19.90 & 211.39 + ugc 11149 & 16137.64 & 92.35 & 146.10 + ugc 11198 & 4518.02 & 19.91 & 145.28 + ugc 03787 & 8612.81 & 57.54 & 140.66 + ugc 04322 & 7376.56 & 23.98 & 248.08 + lccc h@xmath10 & 4847.8754876.625 & 4827.8754847.875 , 4876.6254891.625 & h , ( mg ) , ( cr ) , c + fe5270 & 5245.6505285.650 & 5233.1505248.150 , 5285.6505318.150 & fe , c , ( mg ) , ca + fe4383 & 4369.1254420.375 & 4359.1254370.375 , 4442.8754455.375 & fe , c , ( si ) + lcccccccccccc anon & 0.081 & 0.523 & 0.323 & 1.338 & 0.194 & 0.644 & 0.158 & 0.405 & 0.350 & 1.338 & 0.150 & 0.644 + cgcg 016 - 058 & -0.374 & 0.278 & 0.266 & 23.454 & 13.895 & 17.898 & -0.532 & 0.271 & 0.246 & 26.083 & 15.953 & 22.769 + cgcg 141 - 044 & 0.472 & 0.244 & 0.318 & 1.273 & 0.101 & 0.101 & 0.121 & 0.118 & 0.129 & 1.256 & 0.078 & 0.061 + cgcg 189 - 024 & 0.187 & 0.295 & 0.270 & 2.611 & 0.700 & 0.694 & 0.031 & 0.183 & 0.190 & 2.891 & 0.829 & 2.027 + ic 0522 & -0.072 & 0.112 & 0.110 & 8.200 & 1.858 & 2.849 & 0.157 & 0.089 & 0.082 & 6.583 & 2.828 & 3.035 + mcg+07 - 41 - 001 & -0.185 & 0.058 & 0.072 & 32.651 & 5.932 & 6.483 & -0.432 & 0.043 & 0.132 & 36.121 & 2.281 & 1.886 + mcg-01 - 25 - 009 & -0.174 & 0.116 & 0.107 & 15.698 & 5.170 & 6.809 & 0.041 & 0.100 & 0.081 & 11.856 & 3.307 & 4.690 + ngc 2841 & 0.037 & 0.069 & 0.067 & 12.558 & 1.739 & 2.606 & 0.036 & 0.048 & 0.050 & 12.578 & 1.837 & 3.073 + ngc 2962 & 0.261 & 0.093 & 0.092 & 5.180 & 1.993 & 2.185 & 0.385 & 0.066 & 0.106 & 4.210 & 1.572 & 1.818 + ngc 2986 & 0.052 & 0.086 & 0.076 & 9.652 & 2.003 & 2.108 & 0.014 & 0.058 & 0.059 & 9.667 & 1.988 & 1.998 + ngc 3608 & -0.013 & 0.067 & 0.069 & 11.591 & 2.072 & 2.745 & 0.011 & 0.050 & 0.052 & 10.448 & 1.235 & 3.472 + ngc 4059 & -0.350 & 0.158 & 0.151 & 7.567 & 1.664 & 3.247 & 0.090 & 0.108 & 0.109 & 5.514 & 2.517 & 2.934 + ngc 4374 & 0.102 & 0.055 & 0.052 & 8.667 & 1.195 & 1.389 & 0.081 & 0.045 & 0.048 & 8.851 & 1.369 & 1.700 + ngc 4472 & 0.179 & 0.044 & 0.045 & 8.752 & 1.273 & 1.466 & 0.154 & 0.041 & 0.038 & 9.360 & 1.325 & 1.346 + ngc 4478 & -0.088 & 0.079 & 0.077 & 9.027 & 1.641 & 1.964 & -0.031 & 0.051 & 0.057 & 8.838 & 1.882 & 1.812 + ngc 4489 & 0.428 & 0.186 & 0.200 & 2.539 & 0.719 & 0.753 & 0.167 & 0.099 & 0.104 & 3.350 & 0.808 & 2.099 + ngc 4493 & 0.014 & 0.221 & 0.224 & 7.165 & 3.964 & 5.585 & -0.077 & 0.161 & 0.160 & 7.561 & 4.088 & 6.693 + ngc 4526 & 0.400 & 0.061 & 0.062 & 1.669 & 0.215 & 0.197 & 0.488 & 0.036 & 0.031 & 1.122 & 0.279 & 0.312 + ngc 4552 & -0.040 & 0.024 & 0.063 & 13.481 & 1.762 & 2.532 & 0.056 & 0.047 & 0.041 & 11.834 & 1.136 & 2.511 + ngc 4649 & -0.076 & 0.051 & 0.055 & 21.565 & 2.507 & 2.533 & 0.089 & 0.047 & 0.048 & 16.931 & 2.295 & 2.219 + ngc 4786 & 0.000 & 0.099 & 0.097 & 7.100 & 1.684 & 1.740 & 0.092 & 0.077 & 0.067 & 6.215 & 2.690 & 2.219 + ngc 5005 & & & & & & & & & & & & + ngc 5061 & 0.359 & 0.035 & 0.056 & 1.923 & 0.185 & 0.054 & 0.427 & 0.026 & 0.026 & 1.429 & 0.271 & 0.202 + ngc 5308 & -0.177 & 0.063 & 0.063 & 11.087 & 1.923 & 1.809 & 0.025 & 0.054 & 0.053 & 9.578 & 1.758 & 1.934 + ngc 5440 & 0.057 & 0.107 & 0.102 & 7.415 & 1.957 & 2.536 & 0.248 & 0.076 & 0.080 & 5.453 & 2.151 & 2.645 + ngc 5490 & 0.012 & 0.124 & 0.120 & 9.840 & 2.867 & 3.296 & 0.155 & 0.099 & 0.097 & 8.113 & 3.096 & 3.338 + ngc 5638 & -0.105 & 0.079 & 0.081 & 13.743 & 3.146 & 3.677 & 0.175 & 0.054 & 0.060 & 9.903 & 2.322 & 2.165 + ngc 5813 & 0.027 & 0.086 & 0.074 & 8.692 & 1.891 & 1.996 & 0.369 & 0.068 & 0.050 & 4.921 & 0.737 & 2.135 + ngc 5846 & -0.147 & 0.099 & 0.047 & 14.421 & 2.525 & 4.724 & -0.038 & 0.045 & 0.053 & 13.536 & 2.046 & 3.490 + ngc 6038 & -0.302 & 0.194 & 0.191 & 12.286 & 5.161 & 10.610 & -0.482 & 0.186 & 0.167 & 13.933 & 6.198 & 8.528 + ngc 6127 & -0.178 & 0.121 & 0.066 & 14.917 & 3.711 & 6.099 & -0.122 & 0.097 & 0.050 & 13.975 & 3.287 & 5.596 + ngc 6411 & -0.108 & 0.092 & 0.091 & 14.833 & 3.788 & 5.096 & -0.442 & 0.067 & 0.067 & 19.433 & 4.681 & 5.155 + ngc 6702 & 0.442 & 0.121 & 0.126 & 1.965 & 0.379 & 0.530 & 0.157 & 0.082 & 0.068 & 2.845 & 0.478 & 0.548 + ngc 6703 & 0.078 & 0.069 & 0.063 & 8.186 & 1.451 & 1.695 & 0.213 & 0.052 & 0.052 & 6.829 & 1.805 & 1.727 + ngc 6986 & 0.602 & 0.264 & 0.320 & 1.289 & 0.111 & 0.067 & 0.744 & 0.225 & 0.122 & 1.020 & 2.512 & 0.179 + ngc 7131 & -0.281 & 0.134 & 0.130 & 23.653 & 7.315 & 8.609 & -0.327 & 0.112 & 0.107 & 24.228 & 7.778 & 9.103 + ngc 7761 & 0.164 & 0.186 & 0.192 & 2.907 & 0.613 & 0.803 & 0.268 & 0.128 & 0.085 & 2.670 & 0.652 & 0.864 + ugc 11149 & & & & & & & -0.480 & 0.251 & 0.245 & 9.768 & 5.339 & 12.562 + ugc 11198 & -0.238 & 0.108 & 0.112 & 18.449 & 6.004 & 5.519 & -0.204 & 0.085 & 0.083 & 17.861 & 5.797 & 5.749 + ugc 03787 & 0.446 & 0.291 & 0.419 & 1.284 & 0.076 & 0.082 & 0.422 & 0.186 & 1.729 & 1.172 & 0.235 & 0.190 + ugc 04322 & -0.155 & 0.155 & 0.137 & 11.844 & 3.818 & 6.540 & -0.116 & 0.108 & 0.102 & 11.527 & 3.927 & 6.212 +	we have obtained optical spectra of 29 early - type ( e / s0 ) galaxies that hosted type ia supernovae ( sne ia ) . we have measured absorption - line strengths and compared them to a grid of models to extract the relations between the supernova properties and the luminosity - weighted age / composition of the host galaxies . such a direct measurement is a marked improvement over existing analyses which tend to rely on general correlations between the properties of stellar populations and morphology . our galaxy sample ranges over a factor of ten in iron abundance and shows both old and young dominant population ages . the same analysis was applied to a large number of early - type field galaxies selected from the sdss spectroscopic survey . we find no difference in the age and abundance distributions between the field galaxies and the sn ia host galaxies . we do find a strong correlation suggesting that sne ia in galaxies whose populations have a characteristic age greater than 5 gyr are @xmath0 1 mag fainter at @xmath1 than those found in galaxies with younger populations . however , the data can not discriminate between a smooth relation connecting age and supernova luminosity or two populations of sn ia progenitors . we find that sn ia distance residuals in the hubble diagram are correlated with host - galaxy metal abundance , consistent with the predictions of @xcite . the data show that high iron abundance galaxies host less - luminous supernovae . we thus conclude that the time since progenitor formation primarily determines the radioactive ni production while progenitor metal abundance has a weaker influence on peak luminosity , but one not fully corrected by light - curve shape and color fitters . this result , particularly the secondary dependence on metallicity , has significant implications for the determination of the equation - of - state parameter , @xmath2 , and could impact planning for future dark - energy missions such as jdem . assuming no selection effects in discovering sne ia in local early - type galaxies , we find a higher specific sn ia rate in e / s0 galaxies with ages below 3 gyr than in older hosts . the higher rate and brighter luminosities seen in the youngest e / s0 hosts may be a result of recent star formation and represents a tail of the prompt " sn ia progenitors .	24,239	626
the _ egret _ experiment aboard the _ compton gamma ray observatory _ has detected more than 50 blazars extending out to redshifts greater than 2 ( thompson , _ et al . _ it is expected that @xmath0-rays from blazars with energies above the threshold energy for electron - positron pair production through interactions with low energy intergalactic photons will be annihilated , cutting off the high energy end of blazar spectra . such absorption is strongly dependent on the redshift of the source ( stecker , de jager & salamon 1992 ) . stecker & de jager ( 1997 ) have calculated the absorption of extragalactic @xmath0-rays above 0.3 tev at redshifts up to 0.54 and presented a comparison with the spectral data for the low redshift blazar mrk 421 . the study of blazar spectra at energies below 0.3 tev is a more complex and physically interesting subject . in addition to intergalactic absorption , one must be able to distinguish and to separate out the effects of intrinsic absorption and natural cutoff energies in blazar emission spectra . initial estimates of intergalactic absorption of 10 to 300 gev @xmath0-rays in blazar spectra at higher redshifts have been given by stecker ( 1996 ) , stecker & de jager ( 1996 ) and madau & phinney ( 1996 ) . however , in order to calculate such high - redshift absorption properly , it is necessary to determine the spectral distribution of the intergalactic low energy photon background radiation as a function of redshift as realistically as possible . this calculation , in turn , requires observationally based information on the evolution of the spectral energy distributions ( seds ) of ir through uv starlight from galaxies , particularly at high redshifts . conversely , observations of high - energy cutoffs in the @xmath0-ray spectra of blazars as a function of redshift , which may enable one to separate out intergalactic absorption from redshift - independent cutoff effects , could add to our knowledge of galaxy formation and early galaxy evolution . in this regard , it should be noted that the study of blazar spectra in the 10 to 300 gev range is one of the primary goals of a next generation space - based @xmath0-ray telescope , _ glast ( gamma - ray large area space telescope ) _ ( bloom 1996 ) as well as a number of ground - based @xmath0-ray telescopes currently under construction . our main goal is to calculate the opacity of intergalactic space to high energy @xmath0-rays as a function of redshift . this depends upon the number density of soft target photons ( ir to uv ) as a function of redshift , whose production is dominated by stellar emission . to evaluate the sed of the ir - uv intergalactic radiation field we must integrate the total stellar emissivity over time this requires an estimate of the dependence of stellar emissivity on redshift . previous work has either assumed that all of the background was in place at high redshifts , corresponding to a burst of star formation at the initial redshift ( stecker 1996 ; stecker & de jager 1996 ; macminn and primack 1996 ) or strong evolution ( similar to a burst ) , or no evolution ( madau and phinney 1996 ) . in this paper , we use a more realistic model which is consistent with recent observational data . pei & fall ( 1995 ) have devised a method for calculating stellar emissivity which bypasses the uncertainties associated with estimates of poorly defined luminosity distributions of evolving galaxies . the core idea of their approach is to relate the star formation rate directly to the evolution of the neutral gas density in damped ly@xmath2 systems , and then to use stellar population synthesis models to estimate the mean co - moving stellar emissivity @xmath3 ( erg / s-@xmath4-hz ) of the universe as a function of frequency @xmath5 and redshift @xmath6 ( fall , charlot & pei 1996 ) . our calculation of stellar emissivity closely follows this elegant analysis , with minor modifications as described below . damped ly@xmath2 systems are high - redshift clouds of gas whose neutral hydrogen surface density is large enough ( @xmath7 @xmath8 ) to generate saturated ly@xmath2 absorption lines in the spectra of background quasars that happen to lie along and behind common lines of sight to these clouds . these gas systems are believed to be either precursors to galaxies or young galaxies themselves , since their neutral hydrogen ( hi ) surface densities are comparable to those of spiral galaxies today , and their co - moving number densities are consistent with those of present - day galaxies ( wolfe 1986 ; see also peebles 1993 ) . it is in these systems that initial star formation presumably took place , so there is a relationship between the mass content of stars and of gas in these clouds ; if there is no infall or outflow of gas in these systems , the systems are `` closed '' , so that the formation of stars must be accompanied by a reduction in the neutral gas content . such a variation in the hi surface densities of ly@xmath2 systems with redshift is seen , and is used by pei & fall ( 1995 ) to estimate the mean cosmological rate of star formation back to redshifts as large as @xmath9 . pei & fall ( 1995 ) have estimated the neutral ( hi plus hei ) co - moving gas density @xmath10 in damped ly@xmath2 systems from observations of the redshift evolution of these systems by lanzetta , wolfe , & turnshek ( 1995 ) . ( here @xmath11 is the critical mass density of the universe . the deceleration parameter is assumed throughout to be @xmath12 , with cosmological constant @xmath13 . ) lanzetta , _ et al . _ have observed that while the number density of damped ly@xmath2 systems appears to be relatively constant over redshift , the fraction of higher density absorption systems within this class of objects decreases steadily with decreasing redshift . they attribute this to a reduction in gas density with time , roughly of the form @xmath14 , where @xmath15 is the current gas density in galaxies . pei & fall ( 1995 ) have taken account of self - biasing effects to obtain a corrected value of @xmath16 ; we have reproduced their calculations to obtain @xmath16 under the assumptions that the asymptotic , high redshift value of the neutral gas mass density is @xmath17 , where @xmath18 km / s - mpc ) . in a `` closed galaxy '' model , the change in co - moving stellar mass density @xmath19 , since the gas mass density @xmath10 is being converted into stars . this determines the star formation rate and consequent stellar emissivity ( pei & fall 1995 ) . to determine the mean stellar emissivity from the star formation rate , an initial mass function ( imf ) @xmath20 must be assumed for the distribution of stellar masses @xmath21 in a freshly synthesized stellar population . to further specify the luminosities of these stars as a function of mass @xmath21 and age @xmath22 , fall , charlot , & pei ( 1996 ) use the bruzual - charlot ( bc ) population synthesis models for the spectral evolution of stellar populations ( bruzual & charlot 1993 , charlot & bruzual 1991 ) . in these population synthesis models , the specific luminosity @xmath23 ( erg / s - hz ) , of a star of mass @xmath21 and age @xmath22 is integrated over a specified imf to obtain a total specific luminosity @xmath24 per unit mass ( erg / s - hz - g ) for an entire population , in which all stellar members are produced simultaneously ( @xmath25 ) . following fall , charlot , and pei ( 1996 ) , we have used in our calculations the bc model corresponding to a salpeter imf , @xmath26 , where @xmath27 . the mean co - moving emissivity @xmath28 is then obtained by convolving over time @xmath29 the specific luminosity @xmath30 with the mean co - moving mass rate of star formation , @xmath31 : @xmath32 note that the star mass formation rate @xmath33 that appears in this equation is not the same as @xmath34 , the change in total stellar mass density . this is because @xmath35 is the rate at which mass is _ permanently _ being converted into stars ; since some stellar mass is continuously being returned to the interstellar medium ( ism ) , the _ instantaneous _ mass rate of star formation @xmath36 is larger than @xmath35 , the two being related by @xmath37 where @xmath38 , provided by the bc models , is the fraction of the initial mass of a generation of stars formed at @xmath25 that has been returned to the ism . the bc models specific luminosities @xmath24 are calculated assuming that the metallicity content _ z _ during star formation is fixed at our current solar metallicity value ( @xmath39 ) . however , the metallicity content of the universe is not static , but evolves with redshift as early populations of stars return freshly synthesized metals to the interstellar medium during their various phases of mass loss . for example , in a survey of 1/3 of the known damped lyman - alpha absorbers , pettini _ ( 1994 ) found that the typical metallicity is 0.1 that of the present solar value at a redshift of @xmath40 . since the specific luminosity of a star of a given mass is also a function of its metallicity content ( lower metallicities give bluer spectra ) , the metallicity of a stellar population must be taken into account when integrating the mean emissivity over redshift . the effect of metallicity content in stellar population models has been examined by worthey ( 1994 ) . using the imf @xmath41 with @xmath42 , worthey has calculated the mass - to - light ratios @xmath43 as a function of population age @xmath22 and metallicity @xmath44 , for the color bands @xmath45 through @xmath21 . we have plotted his @xmath43 values for the @xmath45 and @xmath46 bands in figures 1 and 2 respectively . one can see that for a fixed metallicity , the logarithm of the luminosity decreases approximately linearly with the logarithm of population age , and that for a fixed age the @xmath45 and @xmath46 luminosities decrease as the metallicity increases . we have made a linear fit to each fixed - metallicity @xmath47 computed data set , obtaining a metallicity correction factor factor @xmath48 , where @xmath49 designates the color band . > from the parallel linear fits made to the computed data for each @xmath50 value , it is seen that a common correction factor , @xmath51 , applied to each @xmath52 data set will bring these data into rough agreement with the @xmath53 values of @xmath47 . these correction factors are plotted in the inset figures , whose abscissa is @xmath54 and whose ordinate is the correction factor . our fit to worthey s computed data in figure 1 gives a continuous correction factor @xmath55 for @xmath56 @xmath57 m , the center of the @xmath45 band . similar fits to the @xmath46 band data ( figure 2 ) and to the @xmath58 , @xmath59 , and @xmath60 band data ( not shown ) result in the relation @xmath61\approx\left[0.33-\frac{0.30}{\lambda } \right]\log\left(\frac{z}{z_{\odot}}\right ) + \left[0.066-\frac{0.063}{\lambda}\right]\left [ \log\left(\frac{z}{z_{\odot}}\right)\right]^{2},\ ] ] for @xmath62 m @xmath63 m . outside of this wavelength region we take @xmath64 and @xmath65 . note that increased metallicity gives a redder population spectrum ( bertelli _ et al . _ 1994 ) . limitations to this correction factor include the fact that worthey s calculations only apply to stars with ages greater than @xmath66 gyr , and that the upper mass limit of his imf ( @xmath67 ) is much lower than that of the bc model which we employ ( @xmath68 ) . additional uncertainty exists below 0.3 @xmath57 m since worthey s calculations extend only to the @xmath45 band . we have chosen to assume a constant enhancement factor below @xmath69 m . for all of the above reasons , our enhancement factor @xmath51 is really a conservative lower limit to the corrections to the bc models in the ultraviolet . population synthesis models in which varying metallicity is included do exist ( bertelli _ et al . _ 1994 ) , and efforts to reconcile differences in computed spectra generated by these various models have been made ( charlot , worthey , and bressan , 1996 ) . the emissivity @xmath70 given in eq . [ emissivity.eq ] assumes that all stellar emission escapes from the gas system which contains the stars . however , some absorption of stellar radiation occurs both by dust and gas within the larger damped ly@xmath2 systems . above the lyman limit , this absorption is dominated by dust , while below the lyman limit , absorption by neutral hydrogen and singly - ionized helium dominates . defining the mean transmission fractions , averaged over the optical depths of damped ly@xmath2 systems , by @xmath71 and @xmath72 , the final expression for the effective stellar emissivity is @xmath73.\ ] ] the distribution of optical depths @xmath74 of ly@xmath2 clouds due to dust is can be adequately represented by @xmath75 , where @xmath76 , @xmath77 being a characteristic ( redshift dependent ) cloud dust opacity , and @xmath78 ( fall _ et al . _ 1996 ) . under the assumption that both dust and stars are uniformly distributed throughout each ly@xmath2 cloud , the fraction of radiation @xmath79 produced by stars in a given cloud of optical depth @xmath74 that escapes dust absorption is given by @xmath80,\ ] ] where @xmath81 , and @xmath82 is the average albedo of dust , taken to be the same value as in our galaxy ( @xmath83 to 0.6 ; whittet 1992 ) . ( we have calculated eq . [ twostream.eq ] using the 2-stream approximation [ chandrasekhar 1950 ] ) . we note that the dust opacity @xmath74 in eq.[twostream.eq ] is assumed to be proportional to the hi surface column density @xmath84 and metallicity @xmath44 , @xmath85 where @xmath86 is the normalized galactic interstellar dust extinction curve ( savage and mathis 1979 ) . integrating eq.[twostream.eq ] over the ly@xmath2 opacity distribution function @xmath87 of pei & fall ( 1995 ) , we obtain @xmath71 , and find it to have a minor effect on the emissivity , @xmath71 being typically of order unity . below the lyman limit ( @xmath88 m ) , the opacity is dominated by neutral gas absorption : @xmath89 , where @xmath90 and @xmath91 are the hi and hei photoionization cross sections ( osterbrock 1989 ) . with the @xmath92 and @xmath93 distributions of the ly@xmath2 systems being related to the dust opacity distribution @xmath94 through eq.[c ] , the distribution for @xmath95 can be obtained . integrating eq.[twostream.eq ] ( now with @xmath96 ) , weighted with the @xmath95 distribution , gives @xmath72 . figure 3 shows the calculated stellar emissivity as a function of redshift at 0.28 @xmath57 m , 0.44 @xmath57 m , and 1.00 @xmath57 m , both with and without the metallicity correction factor @xmath97 . we have also plotted the observations of the cosmic emissivity by the canada - french redshift survey ( lilly , le fevre , hammer , & crampton 1996 ) at these rest - frame wavelengths for comparison . with a lower mass cutoff of @xmath98 in the imf , we obtain emissivities which are roughly a factor of 2 higher than those obtained by lilly , _ ( 1996 ) . to bring our emissivities down to the observed values requires that we reduce the lower mass limit in the imf to @xmath99 , which puts a fraction ( 0.45 ) of the mass into effectively nonluminous compact objects . we note that a similar reduction was achieved by fall , _ et al . _ ( 1996 ) by modifying the power law index in the imf ; a higher index results in a lower emissivity ( pei 1996 , personal communication ) . overall , our emissivities , both with and without the metallicity corrections , are in reasonable agreement with the data at lower redshifts ( lilly , _ et al . _ although the differences for @xmath70 between the no - metallcity and metallicity cases for @xmath100 are not great , _ they become substantial at larger redshifts for both optical and uv wavelengths_. this has notable effects on the opacity of the radiation background to high energy @xmath0-rays , as will be seen in section 4 . we note that our dotted - line curves in figure 3 ( no metallicity correction ) are essentially a reproduction of the emissivities calculated by fall , _ et al . _ ( 1996 ) . in all cases as shown in figure 3 , the stellar emissivity in the universe peaks at @xmath101 , dropping off at both lower and higher redshifts . indeed , madau , _ et al . _ ( 1996 ) have used observational data from the hubble deep field to show that metal production has a similar redshift distribution , such production being a direct measure of the star formation rate . ( see also the review by madau ( 1996 ) . ) the co - moving radiation energy density @xmath102 ( erg/@xmath4-hz ) is the time integral of the co - moving emissivity @xmath3 , @xmath103 where @xmath104 and @xmath105 is the redshift corresponding to initial galaxy formation . the extinction term @xmath106 accounts for the absorption of ionizing photons by the clumpy intergalactic medium ( igm ) that lies between the source and observer ; although the igm is effectively transparent to non - ionizing photons , the absorption of photons by hi , hei and heii can be considerable ( madau 1995 ) . the presence of damped ly@xmath2 and lyman - limit systems ( lanzetta , _ et al . _ 1995 ) and the lyman - alpha forest , coupled with the absence of a hi gunn - peterson effect ( gunn & peterson 1965 ; steidel & sargent 1987 ) indicates that essentially all of the hi , hei , and heii exists within intergalactic clouds whose measured hi column densities range from approximately @xmath107 to @xmath108 @xmath8 . the effective optical depth @xmath109 between a source at redshift @xmath110 and an observer at redshift @xmath6 owing to poisson - distributed intervening lyman - alpha clouds is given by ( paresce , mckee , & bowyer 1980 ) @xmath111 where @xmath112 $ ] , @xmath113 , and @xmath114 is the distribution function of clouds in redshift @xmath6 and column density @xmath92 . as pointed out by madau & shull ( 1996 ) , when @xmath115 , @xmath109 is just the mean optical depth of the clouds ; when @xmath116 , @xmath109 becomes the number of optically thick clouds between the source and observer , so that the poisson probability of encountering no thick clouds is @xmath106 , as required . for the distribution function of lyman - alpha clouds we use the parameterization of madau ( 1995 ) ( see also miralda - escud & ostriker , 1990 , model a2 ) : @xmath117 using eqs . [ e ] and [ f ] and the stellar emissivity @xmath3 in eq.[b ] , we obtain the background energy density @xmath102 , shown in figures 4 and 5 , calculated with and without the metallicity correction , @xmath51 , respectively . these also include the contribution to the uv background from qsos ( madau 1992 ) , which are believed to dominate the diffuse background radiation below the lyman limit and to be responsible for the early ( @xmath118 ; see schneider , schmidt , & gunn 1991 ) reionization of the igm . although it is possible that uv emission from qsos alone may be able to account for the nearly complete reionization of the igm ( meiksin & madau 1993 ; fall & pei 1993 ; madau & meiksin 1994 ) , it has been argued that additional sources of of ionizing radiation are required ( miralda - escud & ostriker 1990 ) , these perhaps being young galaxies which leak a fraction ( up to @xmath119 ) of their ionizing radiation through hii `` chimneys '' ( dove & shull 1994 ; madau & shull 1996 ) . we have therefore assumed in our calculations that 15% of the stellar emission escapes from the galaxies ( protogalaxies ) through these chimneys , unattentuated by dust or gas . ( we note , however , that recent observations of four starburst galaxies by the hopkins uv telescope ( leitherer _ et al . _ 1995 ) indicate that less than 3% of lyman continuum photons escape from these sources . ) figures 4 and 5 indicate that in our calculation the @xmath120 m background is indeed dominated by qsos , so that the actual value of the escape fraction we choose is not too significant . the intergalactic energy densities given in figures 4 and 5 are quite consistent with the present upper limits in the uv ( martin & bowyer 1989 ; mattila 1990 ; bowyer 1991 ; vogel , weymann , rauch & hamilton 1995 ) . it should be noted that our results as shown in figures 4 and 5 give emissivities _ from starlight only _ and do not include dust emissivities in the mid - infrared and far - infrared . with the co - moving energy density @xmath102 evaluated , the optical depth for @xmath0-rays owing to electron - positron pair production interactions with photons of the stellar radiation background can be determined from the expression ( stecker , _ et al . _ 1992 ) @xmath121\sigma_{\gamma\gamma}[s=2e_{0}h\nu x(1+z)^2],\ ] ] where @xmath122 is the observed @xmath0-ray energy , @xmath123 is the redshift of the @xmath0-ray source , @xmath124 , @xmath125 being the angle between the @xmath0-ray and the soft background photon , @xmath126 is planck s constant , and the pair production cross section @xmath127 is zero for center - of - mass energy @xmath128 , @xmath129 being the electron mass . above this threshold , @xmath130,\ ] ] where @xmath131 . figures 6 and 7 show the opacity @xmath132 for the energy range 10 to 500 gev , calculated with and without the metallicity correction . extinction of @xmath0-rays is negligible below 10 gev . above 500 gev , interactions with photons with wavelengths of tens of @xmath57 m become important , so that one must include interactions from infrared photons generated by dust reradiation ( stecker & de jager 1997 ) , which we have neglected here . for 300 gev @xmath0-rays , at redshifts below 0.5 , our opacities agree with the with the opacities obtained by stecker & de jager ( 1997 ) . our calculated opacity , even with the metallicity correction , is probably somewhat low in the 10 to 30 gev energy range , because we have underestimated the value of @xmath51 in the uv ( see previous discussion ) . note that these calculated opacities are _ independent _ of the value chosen for @xmath133 , as seen in eqs . 4 , 7 , and 10 . the emissivity @xmath70 in eq . 4 scales as @xmath134 , since neither @xmath30 , @xmath51 nor @xmath135 depends on @xmath133 , while @xmath136 scales as @xmath134 . eq . 7 shows then that @xmath137 scales as @xmath133 , and in eq . 10 this @xmath133 factor is cancelled by the integration over time @xmath29 . with the @xmath0-ray opacity @xmath132 calculated out to @xmath138 , the cutoffs in blazar @xmath0-ray spectra caused by extragalactic pair production interactions with stellar photons can be predicted . figure 8 shows the effect of the intergalactic radiation background on a few of the @xmath0-ray blazars ( `` grazars '' ) observed by _ egret _ , _ viz . _ , 1633 + 382 , 3c279 , 3c273 , and mrk 421 . we have assumed that the mean spectral indices obtained for these sources by _ egret _ extrapolate out to higher energies attenuated only by intergalactic absorption . observed cutoffs in grazar spectra may be intrinsic cutoffs in @xmath0-ray production in the source , or may be caused by intrinsic @xmath0-ray absorption within the source itself . whether cutoffs in grazar spectra are primarily caused by intergalactic absorption can be determined by observing whether the grazar cutoff energies have the type of redshift dependence predicted here . figure 8 indicates that the next generation of satellite and ground - based @xmath0-ray detectors , both of which will be designed to explore the energy range between 10 and 300 gev , will be able to reveal information about low - energy radiation produced by galaxies at various redshifts and at different stages in their evolution . our opacity calculations have implications for the determination of the origin of @xmath0-ray bursts , if such bursts are cosmological . as indicated in figure 6 , @xmath0-rays above an energy of @xmath139 15 gev will be attenuated if they at emitted at a redshift of @xmath139 3 . on 17 february 1994 , the _ egret _ telescope observed a @xmath0-ray burst which contained a photon of energy @xmath139 20 gev ( hurley , _ et al._1994 ) . if one adopts the opacity results which include our conservative metallicity correction ( figure 6 ) , this burst would be constrained to have originated at a redshift less than @xmath1392 . ( an estimated redshift constraint of @xmath139 1.5 was given by stecker and de jager ( 1996 ) , based on a simpler model . ) future detectors may be able to place redshift constraints on bursts observed at higher energies . in a previous paper ( stecker & salamon 1996 ) , we presented a model for calculating the extragalactic @xmath0-ray background ( egrb ) due to unresolved grazars . we gave results for @xmath0-ray energies up to 10 gev ( where there is effectively no @xmath0-ray absorption ) which were compared to preliminary _ egret _ data ( kniffen _ et al . _ 1996 ) using the intergalactic @xmath0-ray opacities calculated here , we can now extend the results of this egrb model out to an energy of 0.5 tev . our egrb model assumes that the grazar luminosity function is related to that of flat spectrum radio quasars ( fsrq ) , so that we can use fsrq luminosity and redshift distributions ( dunlop and peacock , 1990 ) to obtain a grazar luminosity function . the effects of grazar flaring states , @xmath0-ray spectral index variation , and redshift dependence are also been included in this model ; see stecker and salamon ( 1996 ) for details . by integrating the grazar luminosity function weighted by our new opacity results , we obtain a grazar background spectrum up to 500 gev which properly includes the effect of @xmath0-ray absorption . figure 9 shows this egrb spectrum compared with the preliminary _ egret _ data . note that the spectrum is concave at energies below 10 gev , reflecting the dominance of hard - spectrum grazars at high energies and softer - spectrum grazars at low energies ; it then steepens above 20 gev , owing to extragalactic absorption by pair - production interactions with radiation from external galaxies , particularly at high redshifts . both the concavity and the steepening are signatures of a blazar dominated @xmath0-ray background spectrum . because the extragalactic @xmath0-ray background in our model is made up of a superposition of _ lower - luminosity , unresolved _ grazars , its intensity is determined by the number of sources in the universe which are below the detection threshhold of a particular telescope . a telescope with a superior point source sensitivity gives a higher source count , thereby reducing the number of unresolved sources which constitute the diffuse @xmath0-ray background . in figure 9 , the upper spectra which are close to the _ egret _ data are obtained using the _ egret _ threshold ; the lower curves correspond to the projected sensitivity of the proposed next generation _ glast _ satellite detector , which is expected to have a detection threshhold of @xmath140 @xmath8s@xmath141 above 0.1 gev . it should also be noted that above 10 gev , blazars may have natural cutoffs in their source spectra ( stecker , de jager & salamon 1996 ) and intrinsic absorption may also be important in some sources ( protheroe & biermann 1996 ) . thus , above 10 gev our calculated background flux from unresolved blazars , shown in figure 9 , may actually be an upper limit . the nature of the dark matter in the universe is one of the most important fundamental problems in astrophysics and cosmology . the non - baryonic mixed dark matter model with a total @xmath142 ( shafi & stecker 1984 ) gave predictions for fluctuations in the cosmic background radiation ( schaefer , shafi & stecker 1989 ; holtzman 1989 ) which were found to be in good agreement with the later cobe measurements . the best agreement appears to be found for @xmath139 20% hot dark matter , of which massive neutrinos are the most likely candidates , and @xmath139 80% cold dark matter ( pogosian & starobinsky 1993 , 1995 ; ma & bertchinger 1994 ; klypin , _ et al._1995 ; primack , _ et al . _ 1995 ; liddle , _ et al . _ 1996 ; babu , schaefer & shafi 1996 ) . the most popular cold dark matter particle candidates are the lightest sypersymmetric particles ( lsps ) , the neutralinos ( hereafter designated as @xmath143 particles ) . cosmologically important @xmath143 particles must annihilate with a weak cross section , @xmath144 10@xmath145 @xmath146s@xmath141 ; calculations show that such cross sections lead to a value for @xmath147 with @xmath148@xmath149@xmath150 . the fact that supersymmetry neutralinos are predicted to have such weak annihilation cross sections is an important reason why they have become such popular dark matter candidates . preliminary lep 2 results give a lower limit on the mass of the @xmath143 of @xmath151 gev ( ellis , falk , olive & schmitt 1996 ) . in the minimal supersymmetry model ( mssm ) , @xmath143 can be generally described as a superposition of two gaugino states and two higgsino states . grand unified models with a universal gaugino mass generally favor states where @xmath143 is almost a pure b - ino ( @xmath152 ) ( _ e.g. _ diehl , _ et al . _ 1995 ) , but other states such as photinos and higgsinos are generally allowed by the theory . kane & wells ( 1996 ) have presented possible accelerator evidence from cdf that @xmath143 may be a higgsino of mass @xmath139 40 gev . dark matter neutralinos will produce @xmath0-rays by mutual pair annihilation . this process is expected to occur because neutralinos are majorana fermions , _ i.e. _ , they are their own antiparticles . indeed , most of this mutual annihilation would have occured in the very early universe , a process which determines the present ( `` freeze out '' ) value of @xmath148 and leads to the relation @xmath153 , where the bracketed quantity is the thermal - averaged annihilation cross section times velocity ( ellis , _ et al . _ 1984 this leads to the relation that the @xmath0-ray flux from neutralino annihilation is inversely proportional to @xmath148 . thus , the annihilation @xmath0-ray flux is limited from below by cosmological constraints on the maximum value of @xmath148 . there are two types of @xmath0-ray spectra produced by @xmath154 annihilations , _ viz . _ , ( 1 ) @xmath0-ray continuum spectra from the decay of secondary particles produced in the annihilation process , and ( 2 ) @xmath0-ray lines , produced primarily from the process @xmath155 ( _ e.g. _ , rudaz 1989 ) . the cosmic @xmath0-ray flux from @xmath154 annihilation is proportional to the line - of - sight integral of the _ square _ of the @xmath143 particle density times @xmath156 . the continuum @xmath0-ray production spectra from @xmath154 annihilation can be calculated for different types of neutralinos by starting with the appropriate branching ratios for annihilation into fermion - antifermion pairs which produce hadronic cascades leading to the subsequent production and decay of neutral pions ( rudaz & stecker 1988 ; stecker 1988 ; stecker & tylka 1989 ) . stecker & tylka ( 1989 ) discuss in detail the various channels involved in continuum @xmath0-ray production via @xmath154 annihilation and give the resulting spectra for some lower mass @xmath143 particles . such continuum fluxes from @xmath154 annihilations would be difficult to observe above the extrapolated cosmic background which we show in figure 9 . however , with good enough sensitivity and energy resolution , it might be possibile to observe a two - photon annihilation line from @xmath154 annihilation . the general considerations for observability of this line were discussed by rudaz & stecker ( 1991 ) . we update this discussion here , using ( 1 ) our new calculation of the @xmath0-ray background flux from blazars shown in figure 9 , ( 2 ) recent accelerator limits on supersymmetric particle masses , and ( 3 ) the proposed sensitivity and energy resolution of a next generation space based @xmath0-ray telescope taken from the _ glast _ proposal ( elliot 1996 ) . the energy of the @xmath155 decay line is @xmath157 . the line width is given by doppler broadening . for galactic halo particles , this width is roughly @xmath158 @xmath159 , much smaller than the energy resolution proposed for any future @xmath0-ray telescope . upper and lower limits on @xmath160 yield lower and upper limits on the @xmath0-ray line flux respectively ( see above ) . other limits can be obtained in flux - energy space ( rudaz & stecker 1991 ) . accelerator determined lower limits on @xmath161 give lower limits on the line energy . lower limits on the mass of the sfermion exchanged in the annihilation process give upper limits on @xmath162 since @xmath162 @xmath163 @xmath164 . in fact , since the particle density @xmath165 and @xmath162 @xmath163 @xmath166 , the predicted annnihilation line flux @xmath167 . further limits are obtained from the inequality @xmath168 , which is the tautology following from the condition that @xmath143 be the lsp . if we assume that annihilations occur mainly through slepton exchange , _ i.e. _ , @xmath169 , we can obtain an upper limit on the 2@xmath0 line flux . this is because lep 1.5 gives a lower limit of @xmath139 70 gev on the slepton mass ( de boer , miquel , pohl & watson 1996 ) , whereas the substantially higher squark mass lower limit of @xmath139 150 gev would imply much lower fluxes , since @xmath170 @xmath171 . the lower limit on the slepton mass implies an upper limit on the line flux from @xmath172 annihilation such that the event rate for a next generation @xmath0-ray telescope with an aperture of 1 m@xmath173sr would be about 5 photons per year for a line in the energy range between 20 and 100 gev . ( if the @xmath143 particles are higgsinos , the event rate would be much lower . ) using all of these constraints , the allowed region for a neutralino annihilation line in flux - energy space is plotted in figure 10 . in constructing figure 10 , we have used the _ glast _ proposed estimate of the point source sensitivity after a one - year full sky survey to estimate the background from unresolved faint blazars ( see figure 9 and the discussion in the previous section ) . we then obtain the background photon number for an appropriate exposure factor of 1 m@xmath173yr - sr and energy resolution of 10% , and plot the square root of this number , which represents the natural background fluctuations above which a line must be observed . of course , a higher exposure factor would reduce the point source background and increase the sensitivity to a line flux , as would a better energy resolution . it should also be noted that the background above 10 gev shown in figures 9 and 10 may be overestimated ( see previous section ) . another possible way in which dark matter may produce @xmath0-rays and neutrinos is if the lsp is allowed to decay to non - supersymmetric , ordinary particles . supersymmetry theories involve a multiplicative quantum number called _ r - parity _ , which is defined so that it is even for ordinary particles and odd for their supersymmetric partners . thus , if r - parity is conserved , as is usually assumed , the lsp is completely stable , making it a potential dark matter candidate . however , such may not be the case . r - parity may be very weakly broken , allowing the lsp to decay with branching ratios involving @xmath0-rays and neutrinos ( _ e.g. _ , berezinsky , masiero & valley 1991 ) . for @xmath143 particles to be the dark matter , their decay time should be considerably longer than the age of the universe . the possible radiative decay @xmath174 will give a @xmath0-ray line with energy @xmath175 . such a line has the potential of being more intense than the annihilation line . whereas the @xmath154 annihilation rate and consequent line flux is cosmologically limited by requiring @xmath148 to be a significant fraction near 1 ( see previous discussion ) , the decay - line flux is limited only by the particular physical supersymmetry model postulated and constraints from related accelerator and astrophysical data . thus , invocation of @xmath143 decay involves a higher order of particle theory model building and speculation . we only wish to mention here that there is a possibility that a decay line may be sufficiently intense to be observable above the background . we have calculated the @xmath0-ray opacity as a function of both energy and redshift for redshifts as high as 3 by taking account of the evolution of both the sed and emissivity of galaxies with redshift . in order to accomplish this , we have adopted the recent analysis of fall , _ et al . _ ( 1996 ) and have also included the effects of metallicity evolution on galactic seds . we have then considered the effects of the @xmath0-ray opacity of the universe on @xmath0-ray bursts , blazar spectra , and on the extragalactic @xmath0-ray background from blazars . in particular , we find that the 17 feb . 1994 _ egret _ burst probably originated at @xmath176 . because the stellar emissivity peaks between a redshift of 1 and 2 , the @xmath0-ray opacity which we derive shows little increase at higher redshifts . this weak dependence indicates that the opacity is not determined by the initial epoch of galaxy formation , contrary to the speculation of macminn and primack ( 1996 ) . the extragalactic @xmath0-ray background , which can be accounted for as a superposition of spectra of unresolved blazars , and which we have predicted to be concave between 0.03 and 10 gev ( stecker & salamon 1996 ) , should steepen significantly above 20 gev owing to our estimates of extragalactic @xmath0-ray absorption at moderate to high redshifts . both the predicted concavity and steepening may be too subtle to detect with present data from _ egret_. however , next generation @xmath0-ray telescopes which are presently being designed , such as _ glast _ , may be able to observe these features and thereby test the blazar background model . we also discuss the possible observability of dark matter lines in the multi - 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[ atten_emiss.eq ] , for three wavelength values , @xmath1880.28 , 0.44 , and 1.0 @xmath57 m , for a hubble constant value of @xmath189 . note that the emissivity scales as @xmath134 ( _ cf . _ eq . 4 and section 3 ) . solid line curves are for the case where the metallicity correction factor ( @xmath51 from eq . [ worthey.eq ] ) is used ; dashed lines give the emissivity when this correction factor is _ not _ included . the data from the canada - french redshift survey ( lilly _ et al . _ 1996 ) are also plotted . figure 4 : the intergalactic radiation energy density from stars and qsos as a function of wavelength for redshifts @xmath6 of 0 , 1 , 2 , and 3 , for a hubble constant value of @xmath189 ( the energy density scales as @xmath133 ; see eq . 7 and section 3 ) these densities are calculated with the metallicity correction factor , @xmath51 , included . figure 6 : the opacity @xmath190 of the universal soft photon background to @xmath0-rays as a function of @xmath0-ray energy and source redshift . these curves are calculated with the metallicity correction factor included in the expression for stellar emissivity . as discussed in the text , these results are independent of the value chosen for @xmath133 . figure 8 : the effect of intergalactic absorption by pair - production on the power - law spectra of four prominent grazars : 1633 + 382 , 3c279 , 3c273 , and mrk 421 . the solid ( dashed ) curves are calculated with ( without ) the metallicity correction factor . figure 9 : the extragalactic @xmath0-ray background energy spectrum from unresolved grazars . the top and bottom sets of curves correspond to point - source sensitivities of @xmath191 and @xmath192 @xmath8s@xmath141 , respectively , for @xmath0-ray energies above 0.1 gev , corresponding to the approximate point - source sensivities of the _ egret _ and _ glast _ detectors respectively . because the fsrq luminosity fuction that we employ scales as @xmath193 ( dunlop and peacock , 1990 ) , our calculated egrb spectrum scales as @xmath134 ( see eq . 10 in stecker and salamon , 1996 ) . figure 10 . the dot - dash polygon shows the allowed region of expected @xmath0-ray photon counts calculated for a bino ( @xmath152 ) annihilation line as a function of @xmath194 . present accelerator and cosmological constraints are indicated by the labels on the sides of the polygon ( see text ) . in the figure labels , the letter `` b '' represents the bino ( @xmath152 ) and the letter `` l '' represents the slepton ( @xmath195 ) . as an illustration of how to read the figure , the arrow within the polygon indicates the line flux upper limit for a bino of mass 100 gev . an exposure factor of 1 m@xmath173sr yr and an energy resolution of 10% are assumed . we also show the background fluctuation count rate appropriate to these parameters for the lower set of flux curves ( _ i.e. _ with and without the metallicity correction ) shown in figure 9 ( see text ) .	in this paper , we extend previous work on the absorption of high energy @xmath0-rays in intergalactic space by calculating the absorption of 10 to 500 gev @xmath0-rays at high redshifts . this calculation requires the determination of the high - redshift evolution of the intergalactic starlight photon field , including its spectral energy distribution out to frequencies beyond the lyman limit . to estimate this evolution , we have followed a recent analysis of fall , charlot & pei , which reproduces the redshift dependence of the starlight background emissivity obtained by the canada - france redshift survey group . we also include the uv background from quasars . we give our results for the @xmath0-ray opacity as a function of redshift out to a redshift of 3 . we also give predicted @xmath0-ray spectra for selected blazars and extend our calculations of the extragalactic @xmath0-ray background from blazars to an energy of 500 gev with absorption effects included . our results indicate that the extragalactic @xmath0-ray background spectrum from blazars should steepen significantly above 20 gev , owing to extragalactic absorption . future observations of a such a steepening would thus provide a test of the blazar origin hypothesis for the @xmath0-ray background radiation . we also note that our absorption calculations can be used to place limits on the redshifts of @xmath0-ray bursts ; for example , our calculated opacities indicate that the 17 feb . 1994 burst observed by _ egret _ must have originated at @xmath1 . finally , our estimates of the high - energy @xmath0-ray background spectrum are used to determine the observability of multi - gev @xmath0-ray lines from the annihilation of supersymmetric dark - matter particles in the galactic halo .	15,367	480
cosmic bursts of gamma - rays are some of the most elusive and mysterious events in the universe . typical burst lasts less than a minute and disappears thereafter . however , since the _ bepposax _ s first discovery on 1997 february 28 ( costa et al . 1997 ) it is known that gamma - ray bursts ( grbs ) also produce x - ray afterglows , which can be detected with modern x - ray observatories for several hours and sometimes several days after the burst . more than a few x - ray afterglows have been observed in last five years by _ bepposax , asca and chandra _ satellites . several afterglow observations were carried out with _ xmm - newton_. in this _ letter _ we report on two of the latter observations . grb 011211 was detected and localized in _ bepposax _ wfc1 on 2001 december 11 , 19:09:21 ut(gandolfi 2001 ) . the distinguishing features of grb 011211 were its long duration ( the longest event localized with _ bepposax _ ) and its faintness both in x- and @xmath0-rays ( frontera et al . 2002 ) . grav et al . ( 2001 ) discovered a new point source in optical r - band ( r@xmath719 ) at r.a.=11@xmath815@xmath917@xmath10.98 , dec=@xmath1121@xmath1256@xmath1356@xmath14.2 ( j2000 , @xmath61@xmath14 ) identified as the afterglow of grb 011211 . fruchter et al . ( 2001 ) and gladders et al . ( 2001 ) measured the optical spectrum and found an absorption line system corresponding to a redshift of z=2.14 . later the optical transient was found to be superposed on an apparent host galaxy with r = 25.0@xmath60.3 ( burud et al . 2001 ) . grb 001025 was detected by the _ rxte all - sky monitor _ on 2000 october 25 at about 03:10:05 ut ( smith et al . . lasted approximately 15 - 20 s and reached a peak 5 - 12 kev flux of @xmath154 crab . optical afterglow of grb 001025 has never been detected down to r=24.5 for any significantly variable object ( fynbo et al . the observation of grb 011211 afterglow was started on 2001 december 12 at 06:16:56 ut ( santos - lleo et al . a source was initially located so close to the edge of ccd#7 in epic - pn detector that some of the source photons were falling in the inter - ccd gap . therefore , the telescope was re - pointed the re - pointing slew started at 08:31:16ut and finished at 08:40:50ut . useful exposure of total observation is @xmath1527 ks . xmm - newton observed the location of grb 001025 from oct.27.003 to oct.27.46 ut , starting @xmath151.9 days after the burst . two x - ray sources were detected in the error box ( altieri et al . below we discuss the brighter source ( r.a.=8@xmath836@xmath935@xmath10.92 , decl.=-13@xmath1204@xmath1309@xmath14.9 , j2000 ) . we analyzed data products from the two _ xmm - newton _ observations . in all observations the epic instruments ( turner et al . 2001 , strueder et al . 2001 ) were operated in the _ full window mode _ ( @xmath16 diameter fov ) . the medium optical blocking filter was used with mos detectors for grb 011211 observation . thin filter was used with pn detector for grb 011211 and with mos detectors in grb 001025 observation . no pn data were available for grb 001025 . we reduced epic data with the _ xmm - newton _ science analysis system ( sas v.5.2 ) . epic detectors data were fit together with common model parameters . the count rates were converted into energy fluxes using analytical fits to the spectra . we have used galactic absorption values provided by the heasarc `` nh '' tool ( dickey & lockman , 1990 ) . central part of mos1 detector ( converted to celestial coordinates ) is shown in fig . the bright source in the middle of the image corresponds to the position of grb 011211 optical afterglow . significant x - ray flux decline during the observation ( fig . 1b ) provides an additional evidence for an identification of the x - ray source with the grb . x - ray light curve can be fit with f@xmath17t@xmath18 ( @xmath19=1.5 - 1.7 ) or with f@xmath17e@xmath20 ( @xmath21=30.@xmath60.5 ks ) . average measured flux in 0.2 - 10 kev band was equal to 1.7@xmath2210@xmath23 erg / s/@xmath24 . extrapolating back to t=100 s since the initial detection of the burst we get an x - ray flux f@xmath25=5@xmath2210@xmath26 erg / s/@xmath24 or x - ray afterglow luminosity l@xmath25=2@xmath2210@xmath27 erg / s ( 0.6 - 30 kev ) in the rest frame at z=2.14 and a @xmath28=65 km s@xmath3mpc@xmath3 , @xmath29 , @xmath30 cosmology . spectrum of the grb 011211 afterglow can be described as a simple power - law with photon index @xmath5=2.16@xmath60.03 modified by the galactic absorption only ( fig . no significant spectral evolution has been found in a sequence of 5-ks intervals within the observation ( fig . 2b ) . we built the spectrum of the brightest source in error box of grb 001025 ( fig . combined spectrum of two mos detectors can be readily approximated by a power - law with photon index @xmath5=2.01@xmath60.09 and galactic absorption value n@xmath31=6@xmath2210@xmath32 @xmath2 . average flux from the source during the observation was 5.3@xmath2210@xmath33erg / s/@xmath24 . no significant flux variability was detected , but the spectrum is very typical for x - ray afterglows of grb , supporting the identification of this source with grb 001025 . it is interesting to consider an alternative identification of the x - ray source with a background quasar . an x - ray bright quasar 3c273 has a r magnitude of @xmath1512.5 ( odell et al . 1978 ) and the 0.5 - 10 kev x - ray flux of 23@xmath2210@xmath34 erg/@xmath24/s ( reeves & turner 2000 ) . neglecting the k - correction and simply scaling the r magnitude to the x - ray flux we find that if a 3c273-like quasar is a source of the detected x - ray flux of @xmath154.7@xmath2210@xmath33 erg/@xmath24/s ( 0.5 - 10 kev ) then its optical counterpart would have a r magnitude of @xmath1521.7 . as in fact an optical counterpart for grb 001025 had not been found down to r=24.5 ( fynbo et al . 2000 ) we consider unlikely that the x - ray source is an agn . we have detected power - law spectra with index @xmath152 from two grb afterglows . in case of grb 011211 an identification of the x - ray source with the grb is supported by the observation of optical transient and also by the decline of x - ray flux during the _ xmm - newton _ observation . for grb 001025 we do not have such supporting evidence , but the spectral shape itself together with the position of x - ray source inside ipn / rxte error box allows us to suggest that the source is indeed an afterglow of grb 001025 . power - law fit with slope @xmath152 is very typical for detected x - ray afterglows of grbs ( see e.g. harrison et al . 2001 , int zand et al . 2001 , antonelli et al . we did not detect any significant changes in the x - ray spectrum during the long observation of grb 011211 . afterglow of grb 001025 was observed significantly later after the burst , and measured x - ray flux was much lower , but the spectrum was almost identical to grb 011211 . conspicuously the x - ray spectra in all or most of the observed afterglows are generated by common physical process and do not depend much on the differences in the burst environments . overall spectral shape can be fit to popular model of synchrotron emission with possible inclusion of inverse compton scattering ( piran 1999 ; granot & sari 2002 ; sari & esin 2001 ) . no significant absorption above the galactic values has been detected in the x - ray spectra . high absorption would be naturally expected if the burst occurs in a high - density star - forming region . ramirez - ruiz , trentham & blain ( 2002 ) suggested that high absorption in nearby ( relative to the burst birthplace ) interstellar media may be the reason for a lack of optical detections in a significant fraction of grb . the absorption should be detectable in soft x - rays ( 0.5 - 2 kev ) . contrary to such expectations , we did not detect any significant absorption above the galactic value in both a grb with optical afterglow ( grb 011211 ) and without it ( grb 001025 ) . the lack of x - ray absorption in grb 011211 is consistent with the lack of detected reddenning in the spectrum of optical afterglow ( simon et al . 2001 ) . reeves et al . ( 2002a , hereafter rwo ) reported on the discovery of a blue - shifted line complex in the spectrum of grb 011211 . the lines have been detected only with pn detector and only during first 5 ks of the observation , with the source located close to the ccd chip boundary ( fig . we have extracted source and background spectra from the same regions as rwo ( reeves et al . . the spectrum can be satisfactory fit to power - law with @xmath5=2.4 and n@xmath31 = 8 @xmath22 10@xmath32 @xmath2 ( @xmath35=1.03 for 44 d.o.f . joint fit of pn , mos1 and mos2 data gives @xmath35=0.92 ( 72 d.o.f . ) for @xmath5=2.15@xmath60.06 and an absorption fixed at the galactic value . hence we are able to get the perfect fit using a simple power - law model we do not see much value in adding extra lines to the model . addition of 5 lines at the energies specified by rwo gives @xmath35=0.93 ( 62 d.o.f . ) for joint fit of the pn and mos data . the analysis of pn data alone allowed us to reproduce line fluxes reported by rwo with somewhat lower significance ( fig . 4c and table 2 ) . we note that the pn data should be treated with special caution for the first 5 ks of the observation because of the unfortunate position of the source on the ccd . in their analysis rwo collected source data from two different pn chips and from the areas near the interchip boundary , which are the least suitable for fine spectroscopy . our extensive analysis showed that any alternative choice of extraction regions for the source and background leads to further reduction in the lines significance . we also found that background spectrum collected over the interchip edge is dominated by a strong feature at 0.7 kev , exactly at the energy of the most significant `` line '' reported by rwo ( see fig.4f ) . though the bulk of the events forming this line are eliminated by the event filtering we are still concerned that some of these bad events may be present in the spectrum of grb 011211 . our concern is amplified by the lack of any of the rwo `` lines '' in the pn spectrum after the reorientation ( fig.4d ) . spectral evolution of the source in sync with satellite revolutions looks quite suspicious unless one suggests that presense of the `` lines '' depends on the position of the source on the chip . such alternative hypothesis is further confirmed by a lack of `` lines '' in the mos data ( fig . it may therefore be concluded that there is no definitive evidence for the presence of the line complex at a redshift of 1.88 in the x - ray spectrum of grb 011211 . the existence of such complex is a possibility , but its statistical significance is greatly overestimated by rwo . our analysis suggests that the spectrum of grb 011211 is featureless and does not contain any significant line emission . we have used publicly available data obtained with _ xmm - newton _ satellite . we are grateful to the personnel of the _ xmm - newton _ science operations centre at vilspa , spain for satellite operations and expedited preparation of data products for scientific analysis . we are thankful to w.priedhorsky and l.titarchuk for encouraging advices , and to anonymous referee for her / his helpful comments . we acknowledge an interesting discussion of the draft version of our paper with j.osborne . altieri , b. , schartel , n. , santos , m. , tomas , l. , guainazzi , m. , piro , l. , parmar , a. 2000 , gcn circ . 869 antonelli , a. , piro , l. , vietry , m. , et al . 2000 , apj , 545 , l39 burud , i. , rhoads , j. , fruchter a. , & hjorth j. , on behalf of grace ( gamma - ray afterglow collaboration at eso ) , 2001 , gcn circ . 1213 costa , e. , frontera , f. , heise , j. , et al . 1997 , _ nature _ , 387 , 783 covino , s. , ghisellini , g. , saracco , p. , et al . , 2002 , gcn circ . 1214 dickey , j. m. , & lockman , f. j. 1990 , araa , 28 , 215 frontera , f. , amati , l. , guidorzi , c. , montanari , e. , costa , e. , feroci , m. , piro , l. , heise , j. , int zand , j.j.m . , 2002 , gcn circ . 1215 fruchter , a. , vreeswijk , p. , rhoads , j. , burud , i. , 2001 , gcn circ . 1200 fynbo , j. p. u. , moller , p. , milvang - jensen , b. , burud , i. , andersen , m. i. , pedersen , h. , jensen , b. l. , hjorth , j. , gorosabel , j. 2000 , gcn circ . 867 gandolfi , g. , on behalf of bepposax mission scientist , 2001 , gcn circ . 1189 gladders , m. , holland , s. , garnavich , p. m. , jha , s. , stanek , k. z. , bersier , d. , barrientos , l.f . , 2001 , 1209 granot , j. , sari , r. 2002 , apj , 568 , 820 grav , t. , hansen , m.w . , pedersen , h. , hjorth , j. , michelsen , r. , jensen , b.l . , andersen , m.i . , gorosabel , j. , fynbo , j.u . , 2001 , 1191 harrison , f.a . , yost , s.a . , sari , r. , berger , e. , galama , t. , et al . 2001 , apj , 559 , 123 int zand , j.j.m . , kuiper , l.m . , amati , l. , antonelli , l.a . , butler , r.c . , et al . 2001 , apj , 559 , 710 odell , s. l. , puschell , j. j. , stein , w. a. , owen , f. , porcas , r. w. , mufson , s. , moffett , t. j. , ulrich , m .- h . 1978 , apj , 224 , 22 piran , t. 1999 , phys.rep . , 314 , 575 ramirez - ruiz , e. , trentham , n. , & blain , a. w. 2002 , m.n.r.a.s . , 329 , 465 reeves , j. n. , & turner , m. j. l. 2000 , mnras , 316 , 234 reeves , j. n. , watson , d. , osborne , j. p. , pounds , k. a. , obrien , p. t. et al . , 2002a , _ nature _ , 416 , l512 ( rwo ) reeves , j. n. , watson , d. , osborne , j. p. , pounds , k. a. , obrien , p. t. 2002b , astro - ph/0206480 santos - lleo , m. , loiseau , n. , rodriguez , p. , altieri , b. , and schartel , n. , 2001 , gcn circ . 1192 sari , r. , esin , a.a . 2001 , apj , 548 , 787 simon , v. , hudec , r. , pizzichini , g. , masetti , n. , 2001 , gcn circ . 1211 smith , d. a. , levine , a. m. , remillard , r. , hurley , k. , cline , t. 2000 , gcn circ . 861 strueder , l. et al . , 2001 , a&a , l18 turner , m. et al . , 2001 , a&a , 365 , l27	we present the _ xmm - newton _ observations of x - ray afterglows of the @xmath0-ray bursts grb 011211 and grb 001025 . for grb 011211 _ xmm _ detected fading x - ray object with an average flux in 0.2 - 10 kev declining from @xmath1 erg @xmath2 s@xmath3 during the first 5 ks of 27-ks observation to @xmath4 erg @xmath2 s@xmath3 toward the end of the observation . the spectrum of the afterglow can be fit to a power law with @xmath5=2.16@xmath60.03 modified for the galactic absorption . no significant evolution of spectral parameters has been detected during the observation . similar x - ray spectrum with @xmath5=2.01@xmath60.09 has been observed by the xmm from the grb 001025 . the non - detection of any extra absorption in these spectra above the galactic value is an interesting fact and may impose restrictions to the favorable grb models involving burst origin in star - forming regions . finally we discuss soft x - ray lines from grb 011211 reported by reeves et al . and conclude that there is no definitive evidence for the presense of these lines in the spectrum . * # 1#1	5,589	358
the layered organic material 3 ( ai3 ) , which has been studied since the 1980s,@xcite has recently attracted renewed interest because it reveals low - energy massless dirac fermions under hyrdostatic pressure ( @xmath2 gpa).@xcite compared to graphene , certainly the most popular material with low - energy dirac fermions@xcite or electronic states at the surface of three - dimensional topological insulators,@xcite ai3 is strikingly different in several respects . apart from the tilt of the dirac cones and the anisotropy in the fermi surface,@xcite its average fermi velocity is roughly one order of magnitude smaller than that in graphene . this , together with an experimentally identified low - temperature charge - ordered phase at ambient pressure,@xcite indicates the relevance of electronic correlations . indeed , because the effective coupling constant for coulomb - type electron - electron interactions is inversely proportional to the fermi velocity , it is expected to be ten times larger in ai3 than in graphene . the material ai3 thus opens the exciting prospective to study strongly - correlated dirac fermions that are beyond the scope of graphene electrons.@xcite . another specificity of ai3 is the presence of additional massive carriers in the vicinity of the fermi level , as recently pointed out in ab - initio band - structure calculations.@xcite however , the interplay between massless dirac fermions and massive carriers has , to the best of our knowledge , not yet been proven experimentally . finally , one should mention a topological merging of dirac points that is expected for high but experimentally accessible pressure.@xcite here , we present magneto - transport measurements of ai3 crystals under hydrostatic pressure larger than @xmath3 gpa where dirac carriers are present . we show not only the existence of high - mobility dirac carriers as reported elsewhere,@xcite but we prove also experimentally the presence of low - mobility massive holes , in agreement with recent band - structure calculations.@xcite the interplay between both carrier types at low energy is the main result of our studies . furthermore , we show that the measured mobilities for the two carrier types hint at scattering mechanisms due to strongly screened interaction potentials or other short - range scatterers . the remainder of the paper is organized as follows . in sec . [ sec:1 ] , we present the experimental set - up and the results of the magneto - transport measurements ( sec . [ sec:1.1 ] ) under hydrostatic pressure . the subsection [ sec:1.2 ] is devoted to a discussion of the temperature dependence of the carrier densities , in comparison with the model of ( a ) massless dirac fermions and ( b ) massive carriers . furthermore thermopower measurements are presented to corroborate the two - carrier scenario . the measured temperature dependence of the extracted carrier mobilities is exposed in sec . [ sec:1.3 ] , and a theoretical discussion of the experimental results , in terms of short - range ( such as screened coulomb ) scatterers may be found in sec . [ sec:2 ] . we present our conclusions and future perspectives in sec . [ sec:3 ] . the single crystals of ai3 used in our study have been synthesized by electro - crystallization . their typical size is @xmath4 mm@xmath5 ( @xmath6 plane ) x @xmath7 m ( @xmath8 direction ) . six @xmath9 nm thick gold contacts were deposited by joule evaporation on both sides of the sample , allowing for simultaneous longitudinal and transverse resistivity measurements . a picture of one of the three samples studied is shown in the inset of figure [ magneto ] . the resistivities were measured using a low - frequency ac lock - in technique . the magnetic field @xmath10 , oriented along the @xmath8 axis , was swept between @xmath11 and @xmath12 t at constant temperature between @xmath13 and @xmath3 k. to account for alignment mismatch of patterned contacts , the longitudinal ( transverse ) resistivity has been symmetrized ( anti - symmetrized ) with respect to the orientation of @xmath10 to obtain even [ @xmath14 and odd [ @xmath15 functions respectively . hydrostatic pressure was applied at room temperature in a nicral clamp cell using daphne 7373 silicone oil as the pressure transmitting medium . the pressure was determined , at room temperature , using a manganine resistance gauge located in the pressure cell close to the sample . the values given below take into account the pressure decrease during cooling . the analysis of our data is based on the study of the magneto - conductivity and is similar to the one presented in ref . for multi - carrier semiconductor systems . the magneto - conductivity is obtained from the measured resistivity tensor by means of @xmath16 $ ] . for a single carrier system , its analytical expression reads@xcite @xmath17 where @xmath18 , @xmath19 is the electron charge , @xmath20 the mobility , and @xmath21 is the carrier density . figure [ magneto ] displays a typical magneto - conductivity curve of ai3 under pressure , where two ` plateaus ' can be clearly seen . as conductivity in ai3 has a strong 2d character , conductivity is shown both as 3d conductivity ( @xmath22 ) and as 2d conductivity ( @xmath23 of each bedt - ttf plane ) according to @xmath24 . as conductivity is additive , in a two - carrier system , the contributions of each carrier type a and b can be added , @xmath25 the two `` plateaus '' , observed in fig . [ magneto ] , indicate the existence of two different carrier types ( @xmath26 or @xmath27 ) with significantly different mobilities . from this curve , we can extract the mobilities , @xmath28 , of each carrier type , their zero - field conductivities , @xmath29 , and their carrier densities , @xmath30 , by @xmath31 . regime at high fields . the left axis shows the square ( 2d ) conductivity of each bedt - ttf plane while the right axis shows the `` bulk '' ( 3d ) longitudinal conductivity ( see text ) . inset : photograph of one sample . ] figure [ sigmaxxt ] shows magneto - conductivity curves of ai3 at a fixed pressure for several temperatures . the previous analysis has been repeated for each of these magneto - conductivity curves to obtain the densities ( fig . [ density ] ) and mobilities ( fig . [ mobility ] ) for each carrier type as a function of temperature and for three different pressures , @xmath32 , @xmath33 and @xmath34 gpa . the strong temperature dependence of the carrier density is a signature that temperature is higher than @xmath35 for both a and b carriers even at the lowest measured temperature , @xmath36 k. this low fermi temperature hints at the absence of charge inhomogeneities that prevent the approach of the dirac point in graphene on si0@xmath37 substrates.@xcite the carrier density can be calculated from @xmath38 , where @xmath39 is the fermi - dirac distribution and @xmath40 is the density of states for massive ( @xmath41 ) and dirac ( @xmath42 ) carriers:@xcite gpa for different temperatures , from bottom to top : @xmath3 , @xmath43 , @xmath34 , @xmath44 , @xmath45 , @xmath46 , @xmath47 , @xmath48 , @xmath49 , @xmath50 and @xmath51 k. the left axis shows the square ( 2d ) conductivity of each bedt - ttf plane while the right axis shows the `` bulk '' ( 3d ) longitudinal conductivity . ] gpa , triangles @xmath33 gpa and squares @xmath34 gpa ; blue thin symbols for a carriers and red thick symbols for b carriers ) . the left axis shows the density for each bedt - ttf plane ( @xmath52 ) while the right axis shows the bulk density ( @xmath53 ) . the lines represent power - law fits of to the a and b carrier densities that yield exponents 0.9 and 2.2 , respectively . inset : band structure calculations at @xmath54 gpa where both dirac cones ( a ) and parabolic bands ( b ) cross the fermi level ( adapted from ref . ) . ] gpa , triangles : @xmath33 gpa and squares : @xmath34 gpa ; blue thin symbols for a carriers and red thick symbols for b carriers ) . the continuous lines represent power laws fits for the mobilities dependences with temperature , which gives exponents @xmath55 ( a carriers ) and @xmath56 ( b carriers ) . the low temperature dispersion of a carriers mobility is due to a decrease of the saturating mobility ( dotted line ) by increasing pressure . ] @xmath57 @xmath58 where @xmath59 and @xmath60 are valley and spin degeneracies and @xmath61 is the effective mass of massive carriers described by a schrdinger equation . in ai3 under pressure , two dirac cones but only one massive band exist at the fermi level.@xcite for large temperatures , @xmath62 , the carrier density depends linearly on temperature for massive carriers and quadratically for dirac carriers : @xmath63 @xmath64 figure [ density ] represents the measured temperature dependence of the mobilities and reveals a power - law behavior , @xmath65 . indeed one obtains an exponent of @xmath66 for the low - mobility carriers ( b ) , in good agreement with what [ eq . ( [ nmassif ] ) ] expected for massive carriers , whereas one finds @xmath67 for the high - mobility carriers ( a ) , as roughly expected for massless dirac particles [ eq . ( [ ndirac ] ) ] . besides , the nature of the carriers can be extracted from hall measurements . furthermore , we have performed thermopower measurements under pressure on a second sample ( figure [ thermopower ] ) . these data show a sign change for the seebeck coefficient ( s ) around 5k . thermopower is the voltage per unit of temperature produced by a thermal gradient . the carrier type determines the sign while the density and mobility of the carriers establish the amplitude . thus , a sign change of the thermopower indicates that the relevant carriers at low temperature have a different charge than those at high temperature , requiring a two - carrier scenario . in agreement with ref . , a carriers which dominate the low - field conduction are electrons . on the contrary , at large fields the conduction is dominated by holes ( b carriers ) . notice that our results are consistent with ab - initio calculations of the band structure of ai3 under a pressure of @xmath68 gpa ( inset of figure [ density])@xcite and do not depend on pressure ( within the range @xmath69 gpa ) . this supports the idea that massless and massive particles coexist in a broad pressure range . however , since @xmath70 in the whole temperature range under study , both dirac electrons and dirac holes are excited . thus there are indeed not two but three carrier types : dirac holes , dirac electrons and massive electrons . for @xmath62 , the electron and hole densities are actually identical ( semimetal with symmetric band structure ) : @xmath71 . the absence of a third ` plateau ' in the magneto - conductivity data allows us to consider that dirac electrons and holes have roughly the same mobilities : @xmath72 . therefore , the results obtained in figure [ density ] and [ mobility ] still hold when we consider two types of dirac carriers ( electrons and holes ) in addition to the massive holes . this analysis allows us to avoid using hall effect measurements for the determination of carrier densities . indeed , hall effect interpretation becomes challenging as dirac electron and hole contributions partially compensate , leading to the determination of only an ` effective ' dirac carrier density , and they are both mixed with massive carriers contribution . this problem is solved here by analyzing the magneto - conductivity where all carriers contributions are additive . the effective mass of the massive carriers has been extracted from eq . ( [ nmassif ] ) . the obtained value is quite small @xmath73 ( @xmath74 is the free electron mass ) . meanwhile , from eq . ( [ ndirac ] ) , @xmath75 m / s can be extracted , in agreement with previous theoretica@xcite and experimental estimates.@xcite in fig . [ density ] , no significant variation of this argument is observed upon sweeping pressure ( which should appear as a vertical shift of the @xmath76 line ) . this indicates that @xmath77 does not change with pressure . in principle , pressure should enhance hopping while reducing the unit cell volume . thus , an enhancement of the @xmath77 with pressure could be expected according to the approximate expression @xmath78 , where @xmath79 is the hopping integral . this expression can be simplified by means of harrison s law ( @xmath80 ) into @xmath81 . as pressure slightly modifies the lattice constant ( @xmath82gpa @xcite ) , @xmath77 is expected to vary by the same order of magnitude which is smaller than our current experimental uncertainty . this accounts for the apparent absence of pressure effects on the carrier density in the range @xmath83 gpa . in fig . [ mobility ] , the mobility of the dirac carriers ( a ) reaches @xmath84 cm@xmath5/vs at low temperatures ( @xmath86 k ) , a value comparable to already published values.@xcite it is quite high compared to typical graphene on 2 values ( @xmath87 to @xmath88 cm@xmath5/vs ) but similar to suspended graphene and graphene on bn mobilities at very low carrier density.@xcite on the other hand , the mobility for massive carriers is @xmath89 cm@xmath5/vs at @xmath86 k , which is two orders of magnitude smaller than for dirac carriers . the temperature dependence of the mobility follows power laws for both massive ( exponent @xmath56 ) and dirac carriers ( exponent @xmath55 ) . moreover , the dirac carrier mobility seems to saturate at @xmath90 . a similar saturation has been reported in others dirac systems.@xcite table [ tabcomp ] summarizes the main parameters of massive and dirac carriers in ai3 , in comparison with graphene on 2 . .dirac and massive carriers parameters in ai3 at high pressure , in comparison with graphene electrons . [ cols="<,<,<,<",options="header " , ] in order to better understand the difference in the mobility , we investigate the ratio @xmath91 , in terms of the scattering times @xmath92 and @xmath93 for the massless dirac and massive carriers , respectively . furthermore , @xmath94 is the density - dependent cyclotron mass of the dirac carriers , in terms of the fermi energy @xmath95 . the scattering times may be obtained from fermi s golden rule ( for @xmath96 ) @xmath97 in terms of the impurity density @xmath98 , the matrix element @xmath99 , and the density of states ( [ ddirac ] ) for dirac and ( [ dmassif ] ) for massive carriers . we consider implicitly that both carrier types are affected by the same impurities , and the matrix element is independent of @xmath100 for short - range impurity scattering . apart from atomic defects , screened coulomb - type impurities approximately fulfill this condition , as it may be seen within the thomas - fermi ( tf ) approximation . indeed , the screening length of the coulomb interaction is dominated by the thomas - fermi wave vector @xmath101 m@xmath102 of the massive carriers , for an effective bohr radius @xmath103 , whereas the thomas - fermi wave vector for massless dirac carriers @xmath104 m@xmath102 , for a density @xmath105 cm@xmath102 and a fine - structure constant @xmath107 . the thomas - fermi wave vector is thus roughly one order of magnitude larger than the fermi wave vector of the massive carriers , which is itself much larger than that of the dirac carriers . the screened coulomb potential for @xmath100-type carriers may therefore be approximated by its @xmath108 value , @xmath109 , which is thus the same for both carrier types , as mentioned above . here , @xmath110 is the permittivity of the dielectric environment and @xmath111 is the dielectric function calculated within the thomas - fermi approximation . in view of the above considerations , we thus obtain , for the mobility ratio in the limit @xmath112 @xmath113 which does neither depend on the form of the matrix element nor on the impurity density . one expects a ratio in the @xmath114 range , whereas the measured ratio is @xmath115 at @xmath116 k. notice that for @xmath62 , that is in the experimentally relevant regime here , one may replace the energy dependence in the density of states of the massless dirac carriers by a linear dependence in temperature , @xmath117 , such that one expects a linear temperature dependence of the mobility ratio ( [ eq : mu ] ) , in agreement with our experimental findings ( @xmath118 for @xmath119 , see fig . [ mobility ] ) . to conclude , we present an interpretation of magneto - transport in ai3 that indicates that both massive and dirac carriers are present even at high pressures . thermopower measurements performed on one of the three studied samples are also in agreement with this two carrier scenario . so far in the literature , the conduction in this system has been attributed solely to dirac carriers.@xcite moreover , this coexistence holds with little perturbation in the whole range of pressure under study . as dirac carriers have high mobility , they dominate the conduction at low magnetic field and high temperatures . on the contrary , for high magnetic fields and low temperatures , the massive holes drive the conduction properties . this crossover can be clearly seen from our magneto - conductivity curves and is responsible for their peculiar ` plateau ' shape . it should also be noted that a proper separation of massive carriers has to be done prior to using any expression that concerns solely dirac carriers . in order to confirm the picture of coexisting dirac and massive carriers , complementary studies , such as spectroscopic measurements , are highly desirable but beyond the scope of the present paper . k. s. novoselov , a. k. geim , s. v. morozov , d. jiang , m. i. katsnelson , i. v. gregorieva , s. v. dubonos , and a. a. firsov , nature * 438 * , 197 ( 2005 ) ; y. zhang , y .- w . tan , h. l. stormer , and p. kim , nature * 438 * 201 , ( 2005 ) . castro neto , f. guinea , n.m.r . peres , k.s . novoselov , and a.k . geim , rev . phys * 81 * , 109 ( 2009 ) ; m. o. goerbig rev . phys * 83 * , 1193 ( 2011 ) ; v. n. kotov , b. uchoa , v. m. pereira , f. guinea , and a. h. castro neto , rev . * 84 * , 1067 .	transport measurements were performed on the organic layered compound 3 under hydrostatic pressure . the carrier types , densities and mobilities are determined from the magneto - conductance of 3 . while evidence of high - mobility massless dirac carriers has already been given , we report here , their coexistence with low - mobility massive holes . this coexistence seems robust as it has been found up to the highest studied pressure . our results are in agreement with recent dft calculations of the band structure of this system under hydrostatic pressure . a comparison with graphene dirac carriers has also been done . 3 @xmath0 2 @xmath1	5,228	149
the unwavering obsession to which the title refers applies only to the first author since the other co - authors are still too young to be obsessed by such a thing as the mass distribution of white dwarf stars . as early as 1976 , it was suggested that below @xmath0 k , convective mixing between the thin superficial hydrogen layer and the more massive underlying helium layer could turn a hydrogen - rich star into a helium - rich star , provided the mass of the hydrogen layer is small enough ( a modern value yields @xmath1 ) . furthermore , the effective temperature at which this mixing occurs is a function of the mass of the hydrogen layer : for thicker hydrogen layers , the mixing occurs at lower effective temperatures . since the process of convective mixing is still poorly understood , the exact ratio of helium to hydrogen after the mixing occurs remains unknown . in particular , it is possible that instead of turning a da star into a featureless helium - rich dc star , convective mixing may simply enrich the hydrogen - rich atmosphere with large quantities of helium , leading to a mixed hydrogen and helium atmospheric composition . such a hypothesis is difficult to test , however , since helium becomes spectroscopically invisible below @xmath2 k , and its presence can only be inferred through indirect methods . such a method has been proposed by @xcite who showed that the atmospheric helium abundance could be determined from a detailed examination of the high balmer lines , since the presence of helium increases the photospheric pressure , and thus produces a quenching of the upper levels of the hydrogen atom which , in turn , affects the line profiles . this method has been put forward on a more quantitative basis by @xcite who analyzed 37 cool da stars using the spectroscopic method of fitting high balmer line spectroscopy with the predictions of detailed model atmospheres with mixed hydrogen and helium compositions . their analysis first showed that the effects produced on the hydrogen lines at high @xmath3 could not be distinguished from those produced by the presence of large amounts of helium . hence , the problem could only be approached from a statistical point of view by assuming a mean value of @xmath4 for all stars , and then by determining individual helium abundances . under this assumption , the analysis of bergeron et al . revealed that the atmospheres of most objects below @xmath5 k were contaminated by significant amounts of helium , with abundances sometimes as high as @xmath6 . we show in figure 1 an update of this result using a sample of 232 da stars analyzed with our most recent grid of model atmospheres . on the left panel we show the surface gravity as a function of effective temperature for each object . clearly , the values determined here are significantly higher than the canonical value of @xmath4 for da stars ( shown by the dashed line ) ; the mean surface gravity of this sample is actually @xmath7 . if we assume instead that our sample is representative of other da stars and adopt @xmath4 for each object , we can determine individual helium abundances . this is shown on the right panel of figure 1 . as can be seen , non - negligible amounts of helium in the range @xmath8 at the surface of these da stars can easily account for the high @xmath3 values inferred under the assumption of pure hydrogen compositions . ( right panel).,height=245 ] the results discussed above rest heavily on the abililty of the models to describe accurately the physical conditions encountered in cool white dwarf atmospheres , but also on the reliability of the spectroscopic method to yield accurate measurements of the atmospheric parameters . it is with this idea in mind that ( * ? ? ? * bsl hereafter ) decided to test the spectroscopic method using da white dwarfs at higher effective temperatures ( @xmath9 k ) where the atmospheres are purely radiative and thus do not suffer from the uncertainties related to the treatment of convective energy transport , and where the assumption of a pure hydrogen composition is certainly justified . from the analysis of a sample of 129 da stars , bsl determined a mean surface gravity of @xmath10 , in much better agreement with the canonical value of @xmath4 for da stars . k ( solid line ; left axis ) compared with that of 54 db and dba stars above @xmath11 k ( hatched histogram ; right axis ) . the average masses are 0.585 and 0.598 @xmath12 , respectively . ] more recently , @xcite obtained high signal - to - noise spectroscopy of all 348 da stars from the palomar green survey and determined the atmospheric parameters for each object using nlte model atmospheres . if we restrict the range of effective temperature to @xmath13 k , the mean surface gravity of their sample is @xmath14 , in excellent agreement with the results of bsl . the corresponding mean mass for this sample is @xmath15 using evolutionary models with thick hydrogen layers . as part of our ongoing survey aimed at defining more accurately the empirical boundaries of the instability strip ( see gianninas , bergeron , & fontaine , these proceedings ) , we have been gathering for several years optical spectroscopy of da white dwarfs from the mccook & sion catalog using the steward observatory 2.3 m telescope facility . the mass distribution for the 667 da stars above 13,000 k is displayed in figure 2 , together with the mass distribution for 54 db and dba stars taken from @xcite ; for the latter , uncertainties with the line broadening theory of helium lines limits the accuracy of the spectroscopic method to @xmath16 k. both mass distributions are in excellent agreement . the results discussed in the last section indicate that the atmospheric parameters of hot ( @xmath13 k ) da stars are reasonable , and that the high @xmath3 values obtained for cool da stars are not related directly to the spectroscopic method itself . one of the largest uncertainties in cool white dwarf atmospheres is the treatment of convective energy transport . in an effort to parameterize the convective efficiency in da stars , @xcite studied a sample of 22 zz ceti stars and showed that the so - called ml2/@xmath17 version of the mixing - length theory yields the best internal consistency between optical and uv effective temperatures , trigonometric parallaxes , @xmath18 magnitudes , and gravitational redshift measurements . the mass vs. temperature distribution for our complete sample of da stars using this parameterization is displayed in figure 3 . the problem with the high masses or high @xmath3 values towards low effective temperatures is clearly apparent here . however , this increase in mass begins not only where convective mixing is believed to occur , but even in the temperature range where zz ceti stars are found . if the larger - than - average mass for zz ceti stars is explained in terms of helium enrichment from the deeper helium convection zone , the hydrogen layers in zz ceti stars need to be as thin as @xmath19 ( see fig . 4 of dufour & bergeron , these proceedings ) , in sharp contrast with values determined for zz ceti stars from asteroseismology . further insight into the problem with high masses at low effective temperatures may be gained by comparing the masses inferred from the spectroscopic method with those obtained from other methods , namely from trigonometric parallax and gravitational redshift measurements . we have thus secured high signal - to - noise spectroscopy for 129 da stars , 92 of which have parallaxes available , and 49 of which have gravitational redshifts . in figure 4 we reproduce the spectroscopic masses as a function of effective temperature for our complete sample of da stars discussed in the previous section , and we overplot in the top and bottom panels the masses derived from trigonometric parallax and gravitational redshift measurements , respectively . the parallax method relies on optical @xmath20 and infrared @xmath21 photometric measurements to constrain the effective temperature and the solid angle @xmath22 between the flux received at earth and that emitted at the surface of the star . given the distance @xmath23 obtained from the parallax , we obtain directly the radius @xmath24 and thus the photometric mass using evolutionary models . one of the most obvious features of the distribution of photometric masses in the top panel of figure 4 is the large number of low mass ( @xmath25 ) white dwarfs . most of these objects are probably unresolved double degenerates for which the flux received at earth corresponds to the contribution of both stars . hence the radii are overestimated and the masses are underestimated . if we ignore these low mass objects , the photometric masses do exhibit higher than average masses when compared with the spectrocopic masses at higher temperatures , which are closer to the canonical value of @xmath26 , although the dispersion of the photometric masses is also significantly larger . in some , but not all cases , the photometric mass exceeds the spectrocopic mass . it is then possible to adjust individually the atmospheric helium abundances until both methods yield consistent masses . since the parallax method uses broadband energy distributions that are not affected significantly by the presence of helium ( * ? ? ? * fig . 4 ) , the photometric masses are almost completely independent of the assumed atmospheric composition , in sharp contrast with the spectroscopic method . we finally note that in the case of the massive ( @xmath27 ) da star lhs 4033 @xcite , the parallax and spectrosocpic masses agree to within 0.01 @xmath12 . the gravitational redshift method used in the bottom panel of figure 4 is based on the relation between the measured redshift velocity , the mass @xmath28 , and the radius @xmath24 of the star , @xmath29 . this method provides mass measurements that are almost completely independent of anything else , although the velocity measurements are intrinsically more difficult to obtain than the other types of measurements used with other techniques . in particular , since the redshift mass measurements scale linearly with velocity , low mass white dwarfs are intrisically more difficult to measure , and the errors are correspondingly larger . nevertheless , if we take at face value the results shown in figure 4 , the redshift masses do not reveal any particular trend at low effective temperatures . more accurate trigonometric parallax and gravitational redshift measurements of _ individual _ stars may be required , together with some unwavering obsession , to help us further understand the high mass problem observed at low temperatures .	we discuss some of our current knowledge of the mass distribution of da and non - da stars using various methods for measuring white dwarf masses including spectroscopic , trigonometric parallax , and gravitational redshift measurements , with a particular emphasis on the problems encountered at the low end of the cooling sequence where energy transport by convection becomes important .	2,622	82
imagine a space craft travelling with constant speed along an unknown and possibly quite irregular closed path @xmath15 in an unexplored territory of the universe . after some time @xmath16 the loop is completed at least once , and the only data the astronauts can measure at time @xmath17 are the ratios of the squared distance from any previous position @xmath18 , to the distance of the current line of direction @xmath19 from that previous position @xmath18 , i.e. , the quotients @xmath20\quad{\,\,\,\text{for } \,\,}s < t.\ ] ] what can the astronauts say about their path of travel ? in other words , how much information about a closed curve of finite length in euclidean space is encoded in the relative tangent - point data ? the answer is : if the astronauts obtain a finite integral mean of some inverse power of all these data ( after time @xmath21 ) they can extract essential topological information as well as explicit smoothness properties of their path of travel ! to make this precise we assume from now on that the path @xmath22 is a rectifiable curve of finite length , parametrized by arclength on the circle @xmath23 of perimeter @xmath24 . hence , @xmath15 is a ( not necessarily injective ) lipschitz continuous mapping with @xmath25 a.e . on @xmath26 . geometrically , the tangent - point function @xmath27 involving the tangent line @xmath28 and defined for all @xmath29 and almost all @xmath30 , determines the radius of the unique circle that is tangent to @xmath15 at the position @xmath31 and passes through @xmath18 . ( this radius is set to be zero if @xmath32 , and is infinite if the vector @xmath33 is parallel to the tangent @xmath34 ) . the only assumption in the result indicated above is finiteness of the _ tangent - point potential _ @xmath35 [ thm:1.1 ] if @xmath36 for some @xmath5 then the image @xmath37 is a one - dimensional topological manifold ( possibly with boundary ) , embedded in @xmath38 in particular , the image curve has no self - intersections , although there is no chance to deduce injectivity of the arclength parametrization @xmath15 itself , since the integrand depends only on the image @xmath39 take , for example , a @xmath40-times covered circle of length @xmath41 , for which the integrand is constant , @xmath42 for all @xmath43 , so that the energy amounts to @xmath44 so the space craft s course can not be too wild , since it traces a one - dimensional manifold without any non - tangential self - crossings . but without further input the astronauts have no clue of how often they have completed that course . moreover , in case their path forms a manifold with boundary , say , a circular arc , there would be an abrupt ( and for the crew probably quite noticeable ) change of direction at the endpoints of that arc . mathematically , one can easily reparametrize the manifold to obtain a new injective arclength parametrization , which translates to the additional information that the spacecraft does not pass by any previous position at all , @xmath45 for all @xmath46 which we will assume from now on . in light of theorem [ thm:1.1 ] the tangent - point potential @xmath0 evaluated on closed curves in @xmath10 may serve as a valid _ knot energy _ as suggested by gonzalez and maddocks in ( * ? ? ? * section 6 ) , that is , as a functional separating different knot types by infinite energy barriers . it was shown by sullivan ( * ? ? ? 2.2 ) that for @xmath9 the energy @xmath0 blows up on a sequence of smooth knots converging smoothly to a smooth curve with self - crossings . ( his proof uses the taylor formula up to order two for the converging curves , and a uniform bound for the remainders . ) as a consequence of our analysis we generalize this result to continuous curves replacing smooth convergence by uniform convergence ( see proposition [ self - repulsive ] ) . thus @xmath0 for @xmath9 is indeed _ self - repulsive _ or _ charge _ , and hence a knot energy according to the definition given by ohara ( * ? ? ? 1.1 ) , which provides an affirmative answer to an open question posed in ( * ? ? ? * problem 8.1 ) . it also turns out that @xmath0 is _ strong _ for @xmath9 : among all continuous closed curves @xmath6 of fixed length @xmath24 and @xmath47 there are only finitely many knot types , see proposition [ prop : strong ] . this gives a partial answer to a conjecture expressed by sullivan in @xcite ( leaving open the case @xmath48 , and we do not consider links with more than one component ) . both these knot - theoretic results are based on a priori @xmath49-estimates for curves of finite @xmath0-energy , discussed in more detail later on . we will show in addition that two curves , whose hausdorff - distance is bounded above by an explicit small constant depending only on the energy values , are in fact in the same knot class . a qualitative version of such an isotopy result is well - known in the smooth category ; see e.g. ( * ? ? ? * chapter 8) , or @xcite . here , however , we have explicit quantitative bounds . notice also that hausdorff - distance alone , no matter how small , does not suffice to separate knot classes ; bounded @xmath0-energy is crucial here . [ thm:1.2 ] for any @xmath9 there is an explicit constant @xmath50 depending only on @xmath11 such that any two closed rectifiable curves with injective arclength parametrizations @xmath51 @xmath52 , with finite @xmath0-energy , are ambient isotopic if their hausdorff - distance is less than @xmath53 our proof of theorem [ thm:1.2 ] follows closely the arguments of marta szumaska who proved in her ph.d . thesis a similar result ( * ? ? ? * chapter 5 ) for a related three - point potential , the _ integral menger curvature _ @xmath54 where @xmath55 denotes the circumcircle radius of three points @xmath56 in euclidean space . essentially one reduces the isotopy question to that between polygons inscribed in @xmath57 and @xmath52 , whose edge lengths are solely controlled in terms of the energy . for polygonal knots a similar result is contained in the work of millet , piatek , and rawdon ( * ? ? ? * theorem 4.2 ) , where instead of , the polygonal thickness of the polygons together with their edge length determines the smallness condition on the hausdorff distance that guarantees isotopy of two polygonal knots . for general curves , thickness was defined by gonzalez and maddocks in @xcite as the smallest possible circumcircle radius @xmath58 when evaluated on all triples of distinct curve points . this concept of thickness was used as a tool in variational applications involving curves and elastic rods subject to various topological constraints ; see e.g. @xcite , @xcite , @xcite@xcite , @xcite , @xcite , and has been studied numerically , @xcite , @xcite , @xcite . the inverse of thickness of a curve @xmath15 can be obtained as limits @xmath59 for @xmath60 , or @xmath61 for @xmath62 in our papers @xcite we have studied regularizing , self - avoidance and compactness effects of several integral energies , including @xmath63 , which involve , vaguely speaking , various bounds for @xmath64 understood as a function of three variables , including bounds in @xmath65 , in @xmath66 where @xmath67 and @xmath68 ( or vice versa ) , and in spaces that resemble the classic morrey spaces @xmath69 . in each case we were able to detect similar phenomena : there is a certain limiting exponent for which an appropriate functional is scale invariant , and above this exponent three sorts of effects take place . first , curves with finite energy have no self - intersections . second , these energies serve well as knot energies allowing for valuable compactness results for equibounded families of loops in fixed isotopy classes , which is due to the third , the regularizing effect : curves with finite energy are more regular than initially assumed . for the present tangent - point potential @xmath0 we obtain the following regularity theorem , which shows that the astronauts would not experience any sudden change of direction during their travel . [ thm:1.3 ] if @xmath9 and the arclength parametrization @xmath70 is chosen to be injective , then @xmath15 is continuously differentiable with a hlder continuous tangent , i.e. , @xmath15 is of class @xmath71 more precisely , for each @xmath9 there exist two constants @xmath50 and @xmath72 depending only on @xmath11 such that each injective arclength parametrization @xmath15 with @xmath36 satisfies @xmath73 . the exponent @xmath48 is a limiting one here . it is relatively easy to use scaling arguments and check that @xmath74 for each @xmath5 when @xmath15 parametrizes a closed polygonal curve , but polygons have finite energy for all @xmath75 . the resulting hlder exponent @xmath76 is reminiscent of the classic sobolev imbedding theorem in the supercritical case : the domain of integration is two - dimensional , and the integrand is related to curvature . for @xmath77-curves the behaviour of @xmath78 close to the diagonal of @xmath79 ( where @xmath78 might blow up for curves with low regularity ) encodes some information about curvature , i.e. about second derivatives of the arclength parametrization @xmath15 . the point is that we need no information about the existence of @xmath80 in order to prove theorem [ thm:1.3 ] . a priori , we deal with curves that are rectifiable only , and even the existence of @xmath81 at all parameters can not be taken for granted . note that inequality is qualitatively optimal : for curves of class @xmath82 the integrand @xmath78 is bounded , and yields then @xmath83 . before describing the main ideas of the proof and the structure of the paper we would like to mention that while working on generalizations of self - avoidance energies to surfaces in @xmath84 , see @xcite , which involved a search for suitable integrands , we have realized that @xmath0 is a model energy that might be the easiest one to extend to the fully general case , i.e. to submanifolds of arbitrary dimension and co - dimension @xcite . this was one of the motivations to write the present note : to lay out in a simple , relatively easily tractable case all the arguments that should be applicable in much greater generality . theorem [ thm:1.1 ] is obtained as a corollary of a slightly more general result , see theorem [ beta - mfd ] below . we first prove a technical lemma ( see section [ beta - section ] ) which shows how @xmath0 can be used to control the behaviour of the so - called p. jones _ @xmath85-numbers _ , @xmath86 for small @xmath87 and closed balls @xmath88 of radius @xmath3 with center @xmath89 . it turns out that if @xmath90 then @xmath91 as @xmath92 uniformly with respect to @xmath89 , see lemma [ lem:2.3 ] and the remark at the end of section [ beta - section ] . and this is the key point to prove that @xmath93 is a topological manifold , as we have the following . [ beta - mfd ] if @xmath94 is arclength , and the image @xmath95 satisfies @xmath96 where @xmath97\to { { { \mathbb r}}}$ ] is a continuous nondecreasing function with @xmath98 , then @xmath6 is a one - dimensional submanifold of @xmath99 ( possibly with boundary ) . the main idea behind the proof of theorem [ beta - mfd ] is simple : if the result were false , then we could find a point @xmath89 in @xmath6 where a _ triple junction _ occurs ; in a small ball @xmath100 centered at @xmath89 we would have ( at least ) three disjoint arcs of @xmath6 in a long narrow tube . two of them would then be very close ( i.e. , would leave @xmath100 crossing @xmath101 in the same spherical cap at one end of the tube ) . observing points of those two arcs , and using on smaller and smaller scales , we are able to obtain a contradiction and eventually show that there could be no triple junction at @xmath89 . for details , see section [ image ] . by the preliminary results of section [ beta - section ] , if @xmath36 for some @xmath9 , then the control of @xmath85 numbers is much better than just . namely , @xmath102 for @xmath103 ; the constant in depends on @xmath104 . applying iteratively , we find in section [ diff ] suitably defined cones that contain short arcs of @xmath6 and obtain an estimate for their opening angles , proving that @xmath81 exists everywhere and is of class - dimensional set @xmath105 that is _ reifenberg flat with vanishing constant _ uniform estimates of @xmath85-numbers imply that @xmath106 is a @xmath107-manifold , see david , kenig and toro @xcite and preiss , tolsa and toro @xcite . here , we have no reifenberg flatness a priori in general rectifiable curves do not have to be reifenberg flat ; in fact we prove it by hand , using energy bounds leading to . ] @xmath108 . section [ sec:5 ] contains the proof of the isotopy result , theorem [ thm:1.2 ] . in the last section we show how to bootstrap the initial gain of @xmath107-regularity obtained in section [ diff ] , to the optimal regularity @xmath109 , and we will establish . we stress the fact that inequality in theorem [ thm:1.3 ] provides a uniform a priori estimate . this can be used in variational applications and to ensure compactness for infinite families of curves with uniformly bounded energy . some results of that type have been stated in @xcite ; we do not follow that thread here . finally , let us say that , at the moment , we have no clue how @xmath81 behaves in the limiting case @xmath48 ( we do not even know if it is defined everywhere for curves with finite @xmath110-energy ) but we are tempted to think that @xmath81 has vanishing mean oscillation for @xmath48 and that local oscillations of the tangent can be controlled by the local energy of the curve . we write @xmath111 to denote the straight line through two distinct points @xmath112 . if @xmath113 , then , abusing the notation slightly , we write sometimes @xmath114 instead of @xmath115 . for a closed set @xmath116 in @xmath99 we set @xmath117 in some places , it will be more convenient to work directly with the slabs @xmath118 around appropriately selected lines than to deal with the information expressed only in the language of @xmath85-numbers . finally , in section [ diff ] we work with cones . for @xmath119 and @xmath120 we denote by @xmath121 the double cone whose vertex is at the point @xmath89 , with cone axis passing through @xmath122 , and with opening angle @xmath123 . all balls @xmath88 with radius @xmath87 and center @xmath124 are closed balls throughout the paper . [ beta ] let @xmath104 be finite . there exists a constant @xmath125 such that if @xmath126 and @xmath127 satisfy @xmath128 then for every two points of the curve such that @xmath129 we have @xmath130 in particular , @xmath131 for @xmath9 we set @xmath132 . there exists a @xmath133 such that if @xmath134 , then @xmath135 * proof of lemma [ beta ] . * for @xmath136 , @xmath137 and @xmath138 small , we set @xmath139\}\ , , \\ n_d(s , t,{\epsilon } ) & : = & a_d(s,{\epsilon } ) \setminus x_d(s , t,{\epsilon})\ , .\end{aligned}\ ] ] note that @xmath140 . the proof has two steps : * we use the inequality @xmath141 to show that @xmath142 must be a small subset of @xmath143 , so that @xmath144 ; * we argue by contradiction , using energy estimates again , and show the desired inclusion . * step 1 . * fix @xmath145 and @xmath146 . we shall show that @xmath147 . since @xmath148 , the triangle inequality yields @xmath149 let @xmath150 then @xmath151 and @xmath152 by the triangle inequality . by definition of @xmath142 , the angle @xmath153 is contained between @xmath154 and @xmath155 . therefore @xmath156 and @xmath157 ( here we use @xmath126 ) . combining and , we obtain @xmath158 integration gives @xmath159 as @xmath160 . if @xmath161 , then @xmath162 which gives a contradiction for an appropriate choice of @xmath163 in the lemma . thus , we have @xmath164 . fix a @xmath165 . since then @xmath166 and the ( acute ) angle between the vectors @xmath167 and @xmath168 is very close to @xmath169 or @xmath170 ( the difference is at most @xmath154 ) , one can check that in fact @xmath171 therefore the distance from @xmath172 to @xmath173 is at least @xmath174 . if @xmath175 , then @xmath176 and @xmath177 integrating this inequality , we obtain @xmath178 again , for an appropriate choice of @xmath179 this gives a contradiction with . @xmath180 since the assumption @xmath9 was not used at all in the proof of the lemma , it is easy to check that the same reasoning that was used to obtain gives in fact the following [ lem:2.3 ] assume that @xmath48 and @xmath90 . then there exists a constant @xmath181 such that @xmath182 where @xmath183 the supremum being taken over all pairs of subsets @xmath184 with @xmath185 . * remark . * by the absolute continuity of the integral , this lemma implies that every curve with finite @xmath110 energy satisfies the assumptions of theorem [ beta - mfd ] . this section is devoted to the proof of theorem [ beta - mfd ] . we will argue by contradiction . the proof has two steps ; one of them has preparatory topological character and the second one shows how to use the assumption on the uniform decay of @xmath85 s . * proof of theorem [ beta - mfd ] . * we recall the assumption of the theorem that the arclength parametrization @xmath70 with image @xmath93 satisfies for some continuous nondecreasing function @xmath186\to{{\mathbb r}}$ ] with @xmath187 in addition , however , we assume that @xmath6 is neither homeomorphic to the unit circle @xmath7 nor to the unit interval @xmath188 $ ] . our goal is to show that this leads to a contradiction . * step 1 . triple junctions . * * claim : * _ there exists a triple junction @xmath189 , i.e. there are three closed sets @xmath190 , @xmath191 such that @xmath192 is a continuous image of the unit interval with @xmath193 for @xmath191 and such that @xmath194 _ * remark . * we allow the @xmath192 to have self - intersections , i.e. we do not require @xmath192 to be a homeomorphic image of the interval . moreover , more than three arcs of the curve may meet at @xmath89 ; we just need three of them to obtain the desired contradiction in step 2 in order to complete the proof of theorem [ beta - mfd ] . * proof of the claim . * we consider two distinct cases . * case 1 . * suppose that @xmath6 contains a proper closed subset @xmath195 that is homeomorphic to @xmath7 . take a point @xmath196 , @xmath197 . suppose w.l.o.g . that @xmath198 for some @xmath199 , @xmath200 $ ] ( otherwise just reverse the parametrization ) . let @xmath201 it is easy to see that @xmath202 is a triple junction ; two of the arcs @xmath192 of @xmath6 are contained in @xmath195 and the third one joins @xmath203 to @xmath89 . * case 2 . * suppose that case 1 fails and @xmath6 contains no proper closed subset homeomorphic to @xmath7 . consider the family of all proper subarcs of @xmath6 , @xmath204 which is partially ordered by inclusion . we will prove in detail below that every chain in @xmath205 has an upper bound in @xmath205 , so that by the kuratowski zorn lemma @xmath205 has a maximal element , @xmath206 . we have @xmath207 , as @xmath6 is not homeomorphic to @xmath8 by assumption . now , take a point @xmath208 , @xmath197 , and proceed like in case 1 joining @xmath122 with an arc to a point @xmath209 notice that @xmath89 can not be an endpoint of @xmath206 , since this would contradict the maximality of @xmath206 . it remains to be shown that every chain in @xmath205 indeed has an upper bound in @xmath205 , which is obvious for any finite chain . for an infinite chain @xmath210 where the index may be chosen to coincide with the length of the respective arc , @xmath211 for @xmath212 , i.e. where the index set @xmath106 is a ( in general uncountable ) subset of @xmath213 $ ] , we can choose a nondecreasing sequence of indices @xmath214 with @xmath215 , @xmath216 and @xmath217 $ ] . has positive diameter , otherwise the claim is trivially true . ] now continuously extend the corresponding nested injective arclength parametrizations @xmath218\to{{\mathbb r}}^n\quad\textnormal{with $ \gamma_i([-l_i/2,l_i/2 ] ) = \gamma_i$\ , and $ \gamma_{i+1}\|_{[-l_i/2,l_i/2]}=\gamma_i$ for all $ i\in{{\mathbb n}}$}\ ] ] by virtue of @xmath219 \end{cases}\ ] ] to all of @xmath220.$ ] since @xmath221 for all @xmath222 $ ] , @xmath223 , we obtain the uniform bound @xmath224,{{\mathbb r}}^n)}\le c$ ] for all @xmath225 which implies by the theorem of arzela - ascoli that some subsequence @xmath226 converges to some curve @xmath227,{{\mathbb r}}^n)$ ] uniformly on @xmath220.$ ] for distinct parameters @xmath228 one can find @xmath229 such that for all @xmath230 we have @xmath231 so that by @xmath232 which means that @xmath15 is injective , hence a homeomorphism on the open interval @xmath233 . but if @xmath234 were equal to @xmath172 for some @xmath235 then the arc @xmath236)$ ] would be homeomorphic to @xmath7 which would contradict our assumption that @xmath6 is neither homeomorphic to @xmath7 nor contains a proper closed subset homeomorphic to @xmath7 . the same contradiction would occur if @xmath237 for some @xmath238.$ ] hence @xmath239)$ ] is homeomorphic to the unit interval @xmath8 , that is @xmath240 . finally @xmath241 is maximal for the chain @xmath242 . indeed , if @xmath243 then @xmath241 is the desired upper bound because for @xmath244 it can not be that @xmath245 is contained in @xmath246 , so that total ordering in the chain implies that @xmath247 . if @xmath248 , on the other hand , we have @xmath244 for any @xmath249 which implies that the corresponding arc @xmath246 is contained in one of the @xmath250 for @xmath251 sufficiently large , and hence also @xmath252 the proof of our claim on the existence of ( at least one ) triple junction is complete now . * step 2 . tilting tubes . * we now fix a point @xmath253 that is a triple junction , and a small distance @xmath254 , @xmath255 where @xmath192 denote the closed , connected subsets of @xmath6 satisfying above . let @xmath256 for @xmath257 $ ] . shrinking @xmath254 if necessary , we can ensure the initial smallness condition @xmath258 rotating and translating the coordinate system in @xmath99 , we can assume without loss of generality that @xmath259 and select the three distinct points @xmath260 where @xmath261 . assumption implies now @xmath262 the intersection of the sphere @xmath263 with the tube @xmath264 consists of two symmetric spherical caps ; by dirichlet s pigeon - hole principle , one of these caps must contain two of the three distinct points @xmath265 . renumbering the @xmath192 and @xmath265 if necessary , we may assume that @xmath266 is as above and @xmath267 with @xmath268 and @xmath269 , @xmath270 . let @xmath271 and @xmath272 . fix a point @xmath273 . from now on , we will work only with @xmath274 and @xmath275 . proceeding inductively , we shall define a sequence of distances @xmath276 , unit vectors @xmath277 , linear @xmath278-dimensional subspaces @xmath279 and points @xmath280 such that @xmath281 as @xmath276 and @xmath282 as @xmath283 , this will yield @xmath284 , a contradiction . the distances @xmath285 , auxiliary vectors @xmath277 and hyperplanes @xmath279 will be defined in such a way that for all @xmath286 @xmath287 for @xmath288 we shall also show that @xmath289 for each @xmath286 . notice that in connection with the initial smallness condition will yield @xmath276 as @xmath290 we begin the construction for @xmath291 . select @xmath292 , @xmath293 . such a point exists since @xmath274 joins @xmath294 to @xmath295 and by continuity must intersect all planes @xmath296 , @xmath297 , while staying in the tube @xmath298 . let @xmath299 , @xmath300 , and @xmath301 . note that @xmath302 by construction . set @xmath303 . we already have for @xmath291 ; condition for @xmath291 follows directly from . to obtain for @xmath291 , we just use and continuity . assume now that @xmath285 , @xmath277 , @xmath304 , @xmath305 , and @xmath306 have already been defined for @xmath307 so that are satisfied for all @xmath308 . we use for @xmath309 to select a point @xmath310 , @xmath311 clearly , @xmath312 ( the second estimate is a simple application of the triangle inequality ) . thus , @xmath313 satisfies for @xmath314 , and choosing @xmath315 we also have for @xmath314 . again , for @xmath314 follows from the assumption on the decay of @xmath85 s . thus , the intersection @xmath316 , @xmath317 ; combining these inclusions with and with continuity , we obtain for @xmath314 . this completes the inductive construction . now , using for @xmath318 , we select for each @xmath319 a point @xmath320 by definition of @xmath321 , does hold . this completes the whole proof of theorem [ beta - mfd ] . throughout this section , we fix @xmath9 and consider a rectifiable curve @xmath93 whose arclength parametrization @xmath15 is injective on @xmath26 . the first step towards the proof of theorem [ thm:1.3 ] is to establish the following . [ prop:5.1 ] let @xmath9 . assume that @xmath322 is injective and @xmath323 . then @xmath81 is well defined everywhere and @xmath324 for @xmath325 . moreover there exist two positive constants @xmath326 , @xmath327 such that whenever @xmath328 and @xmath329 satisfy @xmath330 , then @xmath331 and we have @xmath332 * proof . * the argument is in fact similar to the proof of corollary 2.6 and theorem 2.10 in @xcite . we just sketch the main points , leaving ( relatively easy ) computational details as an exercise . fix @xmath333 with @xmath334 * step 1 . * for @xmath335 set @xmath336 , and select points @xmath337 so that @xmath338 . let @xmath339 so that condition of lemma [ beta ] is satisfied for @xmath340 and @xmath341 . the lemma yields @xmath342 so that the lines @xmath343 satisfy @xmath344 thus , @xmath345 . using and summing a geometric series ( here the assumption @xmath9 is crucial ! ) , we obtain @xmath346 where @xmath347 . now , to guarantee @xmath348 , one just assumes that @xmath349 is sufficiently small , i.e. @xmath350 with @xmath351 . by induction , @xmath352 passing to the limit @xmath353 , we obtain @xmath354 with @xmath355 defined by . * step 2 . * reversing the roles of @xmath89 and @xmath122 we obtain @xmath356 where @xmath357 is defined by ; this is the desired condition . * step 3 . * assume now that @xmath15 is differentiable at @xmath358 and @xmath17 and recall that @xmath15 was supposed to be injective . condition yields then @xmath359 ( note that the difference quotients of @xmath15 at @xmath358 and @xmath17 must belong to cones with vertices at @xmath169 , axis parallel to @xmath360 and opening angle given by ) . * step 4 . * since @xmath15 is differentiable everywhere , and @xmath25 a.e . , gives on a ( dense ) set of full measure . thus , @xmath81 has a continuous extension @xmath116 to all of @xmath26 ; one easily checks that in fact @xmath361 _ everywhere_. finally , assuming without loss of generality that @xmath362 , we estimate @xmath363 } ( to check the last inequality , let @xmath364 be the closed slab bounded by two planes passing through @xmath89 and @xmath122 , and perpendicular to @xmath365 , i.e. to the common axis of the two cones , and note that for each @xmath366 $ ] we have in fact @xmath367 . this follows from the bound @xmath368 , injectivity of @xmath15 and . thus , for each such @xmath369 we also have @xmath370 . ) the bi - lipschitz condition follows . the proof of proposition [ prop:5.1 ] is complete now . ( see also ( * ? ? ? * proof of thm . 2.10 ) where a similar scheme of reasoning is used . ) we start this section with the observation that @xmath0 is repulsive ( or charge ) , that is , @xmath0 blows up on a sequence of knots converging uniformly to a limit curve with self - crossings . [ self - repulsive ] if @xmath70 is a closed arclength parametrized curve of length @xmath371 with @xmath372 for different arclength parameters @xmath373 @xmath374 and if there is a sequence of rectifiable closed injective curves @xmath375 converging uniformly to @xmath15 , then @xmath376 as @xmath377 for any @xmath378 proof:1.1emassume to the contrary that ( for a suitable subsequence ) @xmath379 we set @xmath380),{\mathop{\rm diam}\nolimits}\gamma(s_l\setminus [ s , t ] ) , \delta_2(q)e^{\frac{-1}{q-2}}\big\}>0,\ ] ] where @xmath326 is the constant of proposition [ prop:5.1 ] , and choose @xmath381 and @xmath382 $ ] such that @xmath383)\quad{\,\,\,\text{and } \,\,}\quad for sufficiently large @xmath384 we find @xmath385 for all @xmath386 in particular , by , @xmath387)-\frac { \epsilon}5\overset{\eqref{eps}}{\ge}\frac 45 { \epsilon } , \notag\\ \label{lb } \\ \textnormal{and , analagously,\,\ , } & & \|\gamma_k(\sigma)-\gamma_k(s)\|\ge \frac 45 { \epsilon } , \notag\end{aligned}\ ] ] but @xmath388 hence , we can apply of proposition [ prop:5.1 ] to obtain the inclusion @xmath389 since there is an integer @xmath390 such that @xmath391 for all @xmath392 we know that the corresponding injective arclength parametrizations @xmath393 are continuously differentiable according to proposition [ prop:5.1 ] , so that the points @xmath394 and @xmath395 must be connected by a subarc of @xmath396 that is completely contained in the doubly conical region @xmath397 of diameter @xmath398 ( otherwise , the unit tangent vector of the arclength parametrization @xmath393 would jump at @xmath394 and @xmath395 contradicting @xmath399-smoothness for @xmath392 . ) since all @xmath396 are simple , either the point @xmath400 , or @xmath401 lies on that connecting arc within @xmath402 , thus contradicting the lower bound @xmath403 in . + [ prop : strong ] if @xmath9 , then the @xmath0-energy is _ strong _ in the following sense : for each @xmath404 and @xmath16 there are at most finitely many knot types which have a representative @xmath6 such that @xmath405 * remark . * the length constraint @xmath406 is necessary here , since by rescaling an arbitrary smooth simple curve we can make its @xmath0-energy as small as one wishes . an alternative would be to consider @xmath407 . this is a scale invariant energy . * we argue by contradiction . assume there are infinitely many knot types of length @xmath24 with the same energy bound , and by translational invariance we can assume moreover that all these knots contain the origin . take their arclength representatives @xmath226 , @xmath408 and use inequality of proposition [ prop:5.1 ] to conclude that the family @xmath409 is eqicontinuous . invoking the arzela ascoli compactness theorem and passing to a subsequence , we may assume that @xmath226 converges in the @xmath399-topology to some limit @xmath410 . let @xmath6 be the curve parametrized by @xmath15 . we next check that @xmath6 is simple , i.e. @xmath15 is injective on @xmath411 . to this end , we shall rely on proposition [ prop:5.1 ] to prove that there exists an @xmath412 such that all @xmath226 satisfy @xmath413 upon passing to the limit @xmath414 , this implies the injectivity of @xmath15 . all @xmath415 with @xmath416 sufficiently large are contained in a small @xmath399 neighbourhood of @xmath6 . thus , according to a known isotopy result , see e.g. ( * ? ? ? * chapter 8) or @xcite , they would all have to be of the same knot type , thereby contradicting the assumption that each @xmath415 is in a different isotopy class . to complete the proof , it is now enough to prove . consider @xmath417 given by @xmath418 by proposition [ prop:5.1 ] the @xmath226 are uniformly bounded in @xmath107 , where @xmath419 . thus , it is easy to show that there is a constant @xmath420 such that @xmath421 since @xmath422 such that @xmath423 now , we either have @xmath424 in which case implies @xmath425 or , by minimality , we have @xmath426 , which is equivalent to @xmath427 fix @xmath416 . let @xmath428 . if @xmath429 where @xmath326 stands for the constant from proposition [ prop:5.1 ] , then , by and of that proposition , we have @xmath430 and @xmath431 the last condition , however , clearly contradicts . hence , @xmath432 summarizing , and , we obtain with @xmath433 . @xmath180 now we present the proof of the isotopy result , theorem [ thm:1.2 ] . the proof consists of two steps . the first one , see proposition [ poly-1 ] below , is preparatory : we use proposition [ prop:5.1 ] to show that a curve @xmath6 of length @xmath24 and finite energy at most @xmath434 is ambient isotopic to a polygonal line that has roughly @xmath435 vertices , all of them belonging to @xmath6 . in the second step , we replace two curves that are close in hausdorff distance by polygonal curves ( staying in the same knot class ) and exhibit a series of @xmath436 and @xmath437-moves transforming one of them into the other one . ( the proof that we present gives a value of @xmath438 which is far from optimal ; we do not know how to obtain a sharp result of that type . ) before passing to the details , let us recall a definition , see e.g. ( * ? ? ? * chapter 1 ) . let @xmath439 be one of the segments of a polygonal knot @xmath6 in @xmath10 and let @xmath440 be a triangular surface bounded by the segments @xmath441 such that @xmath442 . we say that @xmath443 _ results from @xmath6 by a @xmath436-move_. the inverse operation is called a _ @xmath437-move_. let @xmath195 and @xmath444 be two polygonal knots in @xmath445 if @xmath195 can be obtained from @xmath444 by a finite sequence of @xmath436 and @xmath437-moves , then one says that @xmath195 and @xmath444 are _ combinatorially equivalent_. two polygonal knots @xmath195 and @xmath444 are ambient isotopic if and only if they are combinatorially equivalent , see ( * ? ? ? * chapter 1 ) . [ poly-1 ] let @xmath9 . assume that @xmath446 is injective and @xmath447 . let @xmath448 be the constant defined in proposition [ prop:5.1 ] . then @xmath93 is ambient isotopic to the polygonal curve @xmath449\ ] ] with @xmath450 vertices @xmath451 , whenever the parameters @xmath452 and @xmath453 are chosen on @xmath26 so that @xmath454 we denote the closed halfspace @xmath455 we shall work with ` double cones ' @xmath456 for sake of brevity , set @xmath457 and @xmath458 . we are going to use proposition [ prop:5.1 ] to verify two properties of @xmath459 . * claim 1 . * _ for each @xmath460 the intersection of @xmath6 and the two - dimensional disk @xmath461 contains precisely one point . if @xmath462 , then this point of @xmath6 is in the interior of @xmath459 . _ indeed , note first that @xmath463 is nonempty , as an arc of @xmath6 joining @xmath464 with @xmath465 must be contained in @xmath459 since if this were not the case , then of proposition [ prop:5.1 ] would be impossible for an injective and differentiable @xmath15 . if there were two distinct points @xmath466 , then could not hold both for the couple @xmath467 , and for the couple @xmath468 , simultaneously . finally , the second statement of claim 1 follows from the fact that inequality is strict . * claim 2 . * _ whenever @xmath469 we find that the sets @xmath470 and @xmath471 are disjoint . _ suppose to the contrary that @xmath472 and assume without loss of generality @xmath473 if @xmath474 were contained in @xmath470 then we would either find that the disk @xmath475 contains two distinct curve points contradicting claim 1 , or that there is a parameter @xmath476 such that @xmath477 although @xmath15 is injective , which is absurd . the same reasoning can be applied to @xmath478 so that we conclude from and assumptions and that the two tips @xmath479 @xmath480 of @xmath481 are contained in the set @xmath482 defined as @xmath483,\ ] ] which is just the intersection of the two cones within the balls centered in @xmath464 and @xmath465 but without the slab bounded by the two parallel planes @xmath484 and @xmath485 . we know that @xmath486 since @xmath487 and @xmath15 is injective . if @xmath488 then , , and enforce @xmath489 and @xmath490 which by leads to @xmath491 contradicting unless @xmath492 . if in the latter case @xmath493 is contained in @xmath494 then we obtain @xmath495 contradicting our assumption . if , on the other hand , @xmath493 is in @xmath496 , it is by actually contained in @xmath497 , but then can not hold . finally , @xmath498 in combination with also leading to is a contradictory statement , since @xmath499 by . we are now in the position to define the ambient isotopy between @xmath6 and @xmath500 . note that @xmath501 given by @xmath502 \cap d_i(\gamma(t ) ) \qquad\mbox{for \ , $ t\in [ t_i , t_{i+1})$ , $ i=1,\ldots , n$}\ ] ] is a well defined homeomorphism , parametrizing @xmath503 . the desired isotopy @xmath504 \to { { \mathbb r}}^3\ ] ] is equal to the identity on @xmath505 , and on each ` double cone ' @xmath459 it maps each two - dimensional slice @xmath506 , @xmath460 , homeomorphically to itself , keeping the boundary circle of @xmath506 fixed and moving the point @xmath18 along a straight segment on @xmath507 until it hits @xmath508 $ ] . @xmath180 * proof of theorem [ thm:1.2 ] . * abbreviate the maximal energy value @xmath509 of the two simple arclength parametrized curves @xmath510 of respective ( and a priori possibly quite different ) lengths @xmath511 , @xmath512 recall the assumption that the two curves are close in hausdorff - distance : @xmath513 fix @xmath514 so that @xmath515 , set @xmath516 and let @xmath517 for @xmath518 , and @xmath519 . by proposition [ poly-1 ] , @xmath195 is ambient isotopic to the polygonal line @xmath520\ ] ] where @xmath521 . now , for @xmath522 we set @xmath523 , @xmath524\bigr)\subset\gamma_1 $ ] , and introduce the half - spaces @xmath525 and @xmath526 , which are bounded by affine planes @xmath527 . consider the tubular regions @xmath528 their union contains @xmath529 ; we clearly have @xmath530 as @xmath531 . moreover , @xmath532 also when @xmath533 . to see this , we will use proposition [ prop:5.1 ] to prove @xmath534 before doing so , let us conclude from : if there existed a point @xmath535 with @xmath533 , we could find @xmath536 and @xmath537 such that @xmath538 by triangle inequality , a contradiction to . to verify , we repeat the trick that has already been used in the proof of proposition[prop : strong ] . notice that applied to @xmath57 implies @xmath539 so that the continuously differentiable function @xmath540 given by @xmath541 attains a positive minimum @xmath542 on the compact set @xmath543 , where we set @xmath544 , i.e. , there is a pair of parameters @xmath545 such that @xmath546 for all @xmath547 if @xmath548 we can apply to find @xmath549 if , on the other hand , @xmath550 then by minimality @xmath551 , which implies that both tangents @xmath552 and @xmath553 are perpendicular to the segment @xmath554 thus the intersection @xmath555 can not be contained in the intersection @xmath556 which according to means that @xmath557 establishing also in this case . assume now that @xmath558 . we shall prove that @xmath444 is ambient isotopic to @xmath195 ; by the choice of @xmath123 , this will mean that theorem [ thm:1.2 ] holds with @xmath559 . * _ for each @xmath522 there is a point @xmath560 _ without loss of generality we can assume that the curve @xmath57 is oriented in such a way that @xmath561 that is , each tangent @xmath562 points into the set @xmath563 which readily implies for the hyperplanes @xmath564 , @xmath522 , @xmath565 and similarly @xmath566 . indeed , according to @xmath567 \subset k_i,\ ] ] which implies that the tangent direction of the curve @xmath57 at @xmath464 can not deviate too much from the straight line through @xmath464 and @xmath465 . the inequalities in provide a quantified version of this fact . since @xmath568 we find three points @xmath569 if @xmath570 we set @xmath571 , and we are done . else we know that @xmath572 or that @xmath573 in the first case we will work with the two points @xmath574 and @xmath575 , in the second with @xmath574 and @xmath576 in the same way , so let us assume the second situation @xmath573 we know that @xmath577 since by @xmath578 on the other hand , @xmath574 and @xmath576 are not too far apart , @xmath579 so that we can infer from applied to the points @xmath580 and @xmath581 that @xmath582 we will show that @xmath583 \subset b(x_i,2{\epsilon}).\ ] ] notice that @xmath584 consists of two components , one containing @xmath585 , and the other one containing @xmath586 which implies that the intersection in is not empty . since @xmath444 connects @xmath574 and @xmath576 by within the set @xmath587 , the inclusion in yields the desired curve point @xmath588 thus proving the claim . to prove we first estimate the angle @xmath589 by the largest possible angle between a line tangent to both @xmath590 and @xmath591 and the line connecting the centers @xmath592 @xmath465 : @xmath593 so that @xmath594 now , let @xmath595 be the orthogonal projection of @xmath574 onto @xmath596 . since @xmath597 , it is easy to see that @xmath598 where @xmath599 ( see figure 1 below ) , which establishes @xmath600 and hence . l[0cm]6.5 cm the intersection of the doubly conical region @xmath587 with the plane @xmath596 is contained in the ball @xmath601 . since @xmath602 , the curve @xmath444 is ambient isotopic to the polygonal curve @xmath603 $ ] . to finish the proof of theorem [ thm:1.2 ] , it is now sufficient to check that @xmath604 and @xmath605 are combinatorially equivalent . since the sets @xmath606 are pairwise disjoint , we have @xmath607.\ ] ] this guarantees that all steps in the construction that follows involve legitimate @xmath436 and @xmath437-moves . the first step , taking place in @xmath608 , is to replace @xmath609 $ ] by the union of @xmath610 $ ] and @xmath611 $ ] , and then to replace @xmath611 $ ] by the union of @xmath612 $ ] and @xmath613 $ ] . next we perform one @xmath437 and one @xmath436-move in each of the @xmath614 for @xmath615 , replacing first @xmath616 $ ] and @xmath617 $ ] by @xmath618 $ ] , and next trading @xmath618 $ ] for the union of @xmath619 $ ] and @xmath620 $ ] . finally , for @xmath621 we perform two @xmath437-moves : first replace @xmath622 $ ] and @xmath623 $ ] by @xmath624 $ ] , and then replace @xmath625 $ ] and @xmath610 $ ] ( which has been added at the very beginning of the construction ) by @xmath626 $ ] . this concludes the whole proof . in this section , we show how to derive theorem [ thm:1.3 ] . the overall idea is similar to the one in ( * ? ? ? * section 6 ) but here the proof is a little bit less involved . assume that @xmath15 is 11 , @xmath324 , @xmath627 . restricting @xmath15 to a sufficiently short interval @xmath8 in @xmath628 $ ] , and rotating the coordinate system if necessary , we may assume that the first component @xmath629 of the tangent vector satisfies @xmath630 on @xmath8 and @xmath631 on @xmath8 for all @xmath632 . in fact , to achieve such control of @xmath81 on @xmath8 it is enough to assume that @xmath633 for some @xmath634 sufficiently small ; the desired control of @xmath81 follows then from proposition [ prop:5.1 ] . let @xmath635 ( here , @xmath636 denotes an arbitrary subinterval of @xmath8 ) . we shall show that for every @xmath637 , @xmath638 , @xmath639 where @xmath640 , @xmath641 is a large number such that @xmath642 , and @xmath643 once is established , we can iterate it to get rid of the first term on the right hand side of and prove that @xmath644 and set @xmath645 \ , \colon { \mathscr{h}}^1(y_1(s))\ge 2\|u - v\|/n \}\ , , \\ y_1(s ) & : = & \ { t\in [ u , v ] \ , \colon 1/r(\gamma(s),\gamma(t))\ge k_0 \|u - v\|^{-2/q } \}\ , .\end{aligned}\ ] ] the reader should think of the parameters in @xmath646 and @xmath647 as ` bad ' ones . here is a word of informal explanation . suppose that a curve is just @xmath648 for @xmath640 and not smoother , say like the graph of @xmath649 near zero . we would then expect that a typical point @xmath650 can be roughly at the distance @xmath651 from the tangent line at @xmath652 when @xmath653 or , equivalently for a flat graph over some interval , @xmath654 . but then @xmath78 at these two points would not exceed a constant multiple of @xmath655 by the explicit formula for the radius @xmath3 . as we know nothing about the existence of @xmath80 , there are no a priori upper bounds for @xmath78 that we might use . however , it is illustrative to look at the sets of points where the model bound @xmath656 we have @xmath657 so that @xmath658 now , select @xmath659\setminus y_0 $ ] and @xmath660\setminus y_0 $ ] such that @xmath661 by the triangle inequality , @xmath662 and @xmath19 are parallel , we have @xmath663 and there is nothing more to prove . thus , let us assume that @xmath664 and @xmath19 are not parallel and proceed to estimate @xmath665 . let @xmath666 \setminus \bigl(y_1(s)\cup y_1(t)\bigr ) $ ] . by definition of @xmath667 and choice of @xmath668 , we have @xmath669 if @xmath670 , then by definition of @xmath647 and of the tangent - point radius ( see ) we obtain @xmath671 a similar inequality is satisfied by the distance of @xmath672 to the other line , @xmath19 . now , let @xmath673 be the two - dimensional plane spanned by the two tangent vectors @xmath674 and @xmath34 . choose two points @xmath675 and @xmath676 such that @xmath677 and let @xmath678 . ( if @xmath664 and @xmath19 intersect , @xmath679 is their common point ; otherwise , the segment @xmath680 $ ] is perpendicular to each of these two lines and @xmath89 is its midpoint . ) let @xmath681 . then @xmath682 and let @xmath683 the lines @xmath684 intersect at @xmath685 . note that since @xmath686 is nonempty by , we have in fact by virtue of @xmath687 and @xmath688 thus , @xmath689 therefore , the projection @xmath690 of @xmath691 onto @xmath692 is contained in a rhombus @xmath693 in @xmath692 . the center of symmetry of @xmath693 is at @xmath89 ; the sides of @xmath693 are parallel to @xmath694 and @xmath695 ; its height equals @xmath696 and its acute angle @xmath697 ( since @xmath630 on @xmath8 , the angle @xmath698 _ is _ acute ) . the longer half - diagonal @xmath699 of @xmath693 is given by @xmath700 and @xmath701 since @xmath702 , invoking and the triangle inequality we conclude that @xmath703 now , recall that @xmath630 on @xmath8 . let @xmath704 and @xmath705 . we then have @xmath706 thus @xmath707 combining two estimates of @xmath708 , and , we obtain @xmath709 . the proof of the second part of theorem [ thm:1.3 ] is now complete . * to see that the exponent @xmath14 is indeed optimal and can not be replaced by any larger exponent , we follow the idea given by m. szumaska in her phd thesis @xcite . one has to fix an arbitrary @xmath710 $ ] and consider @xmath6 that is the graph of @xmath711 say on @xmath712 $ ] . it is possible to check that @xmath4 is finite ; however , the derivative of the arclength parametrization of @xmath6 is not hlder continuous with any exponent larger than @xmath713 . since @xmath85 can be an arbitrary number in @xmath714 $ ] , the exponent @xmath14 is indeed optimal . we do not give here the computational details which are somewhat tedious but routine ; one just has to pass from the graph description of @xmath6 to the arclength parametrization and use taylor s formula in estimates . the key point is that @xmath715 is not hlder continuous with any exponent larger than @xmath713 , due to its behaviour near to 0 . carlen , m. ; laurie , b. ; maddocks , j.h . ; smutny , j. biarcs , global radius of curvature , and the computation of ideal knot shapes . in : calvo , millett , rawdon , stasiak ( eds . ) _ physical and numerical models in knot theory _ , pp . series on knots and everything vol . 36 , world scientific , singapore 2005 . schuricht , f. ; von der mosel , h. euler - lagrange equations for nonlinearly elastic rods with self - contact . arch . rat . 168 * ( 2003 ) , 3582 . schuricht , f. ; von der mosel , h. characterization of ideal knots . calc . var . partial differential equations * 19 * ( 2004 ) , 281305 . sullivan , j.m . approximating ropelength by energy functions . in : calvo , millet , rawdon ( eds . ) _ physical knots : knotting , linking , and folding geometric objects in @xmath10 _ , pp- 181186 . 304 , ams , providence ri 2002 .	we study a two - point self - avoidance energy @xmath0 which is defined for all rectifiable curves in @xmath1 as the double integral along the curve of @xmath2 . here @xmath3 stands for the radius of the ( smallest ) circle that is tangent to the curve at one point and passes through another point on the curve , with obvious natural modifications of this definition in the exceptional , non - generic cases . it turns out that finiteness of @xmath4 for @xmath5 guarantees that @xmath6 has no self - intersections or triple junctions and therefore must be homeomorphic to the unit circle @xmath7 or to a closed interval @xmath8 . for @xmath9 the energy @xmath0 evaluated on curves in @xmath10 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value . we also establish an explicit upper bound on the hausdorff - distance of two curves in @xmath10 with finite @xmath0-energy that guarantees that these curves are ambient isotopic . this bound depends only on @xmath11 and the energy values of the curves . moreover , for all @xmath11 that are larger than the critical exponent @xmath12 , the arclength parametrization of @xmath6 is of class @xmath13 , with hlder norm of the unit tangent depending only on @xmath11 , the length of @xmath6 , and the local energy . the exponent @xmath14 is optimal .	16,737	399
diffusion in stationary states may be encountered either in equilibrium , where no macroscopic mass or energy fluxes are present in a system of many diffusing particles , or away from equilibrium , where diffusion is often driven by a density gradient between two open segments of the surface that encloses the space in which particles diffuse . in equilibrium states , one is interested in the _ self - diffusion _ coefficient @xmath3 , as given by the mean - square displacement ( msd ) of a tagged particle . this quantity , also called tracer diffusion coefficient , can be measured using e.g. neutron scattering , nmr or direct video imaging in the case of colloidal particles . in gradient - driven non - equilibrium steady states , there is a particle flux between the boundaries which is proportional to the density gradient . this factor of proportionality is the so - called transport or collective diffusion coefficient @xmath4 . often these two diffusion coefficients can not be measured simultaneously under concrete experimental conditions and the question arises whether one can infer knowledge about the other diffusion coefficient , given one of them . generally , in dense systems these diffusion coefficients depend in a complicated fashion on the interaction between the diffusing particles . in the case of diffusion in microporous media , e.g. in zeolites , however , the mean free path of the particles is of the order of the pore diameter or even larger . then diffusion is dominated by the interaction of particles with the pore walls rather than by direct interaction between particles . in this dilute so - called knudsen regime neither @xmath3 nor @xmath4 depend on the particle density anymore , but are just given by the low - density limits of these two quantities . one then expects @xmath3 and @xmath4 to be equal . this assumption is a fundamental input into the interpretation of many experimental data , see e.g. @xcite for an overview of diffusion in condensed matter systems . not long ago this basic tenet has been challenged by monte - carlo simulation of knudsen diffusion in pores with fractal pore walls @xcite . the authors of these ( and further ) studies concluded that self - diffusion depends on the surface roughness of a pore , while transport diffusion is independent of it . in other words , the authors of @xcite argue that even in the low density limit , where the gas particle are independent of each other and interact only with the pore walls , @xmath5 , with a dependence of @xmath3 on the details of the pore walls that @xmath4 does not exhibit . this counterintuitive numerical finding was quickly questioned on physical grounds and contradicted by further simulations @xcite which give approximate equality of the two diffusion coefficients . these controversial results gave rise to a prolonged debate which finally led to the consensus that indeed both diffusion coefficients should agree for the knudsen case @xcite . it has remained open though whether these diffusion coefficients are generally exactly equal or only approximately to a degree depending on the details of the specific setting . a physical argument put forward in @xcite suggests general equality . to see this one imagines the following _ gedankenexperiment_. imagine one colours in a equilibrium setting of many non - interacting particles some of these particles without changing their properties . at some distance from this colouring region the colour is removed . then these coloured particles experience a density gradient just as `` normal '' particles in an open system with the same pore walls would . since the walls are essentially the same and the properties of coloured and uncoloured particles are the same , the statistical properties of the ensemble of trajectories remain unchanged . hence one expects any pore roughness to have the same effect on diffusion , irrespective of whether one consider transport diffusion or self - diffusion . notice , however , that this microscopic argument , while intuitively appealing , is far from rigorous . first , the precise conditions under which the independence of the diffusion coefficients on the pore surface is supposed to be valid , is not specified . this is more than a technical issue since one may easily construct surface properties leading to non - diffusive behaviour ( cf . second , there is no obvious microscopic interpretation or unique microscopic definition of the transport diffusion coefficient for arbitrary surface structures . @xmath4 is a genuinely macroscopic quantity and a proof of equality between @xmath4 and @xmath3 ( which is naturally microscopically defined through the asymptotic long - time behaviour of the msd ) requires some further work and new ideas . one needs to establish that on large scales the knudsen process converges to brownian motion ( which then also gives @xmath3 ) . moreover , in order to compare @xmath4 and @xmath3 one needs a precise macroscopic definition of @xmath4 which is independent of microscopic properties of the system . the first part of this programme is carried out in @xcite . there we proved the quenched invariance principle for the horizontal projection of the particle s position using the method of considering the environment viewed from the particle . this method is useful in a number of models related to markov processes in a random environment , cf . e.g. @xcite . the aim of this paper is to solve the second problem of defining @xmath4 and proving equality with @xmath3 . as in @xcite we consider a random tube to model pore roughness . in contrast to @xcite , we now have to consider tubes of finite extension along the tube contour and introduce open segments at the ends of the tube . doing this rigorously then clarifies some of the salient assumptions underlying the equality of @xmath4 and @xmath3 . naturally , since we are in the dilute gas limit , there is no dependence on the particle density in either of the two diffusion constants . this obvious point has not been controversial and will not be stressed below . we note that we define @xmath4 through stationary transport in an open system since this is accessible experimentally as well as numerically in monte carlo simulation . indeed , in the literature that gave rise to the controversy that we address here , this way of defining @xmath4 is used , albeit in a non - rigorous fashion . sticking to this experimentally motivated setting we shall give below a precise definition that can be used to prove rigorously that under rather generic circumstances @xmath6 , which means that both diffusion constants depend on the pore surface in the same way . as pointed out above , this equality is expected from independence of the particles and the invariance principle for the process and its time - reversed . however , we could not find a general result applying here , and moreover , as it turns out , the proof is not entirely trivial . there are some technical difficulties to overcome because the quenched invariance principle of definition [ def_invariance_principle ] below is not very `` strong '' ( there is no uniformity assumption on the speed of convergence as a function of the initial conditions ) and the jumps of the embedded discrete - time billiard are not uniformly bounded . let us mention here that it is generally difficult to obtain stronger results in the above sense , since the corrector technique , generally used in the proof of quenched central limit theorems for reversible markov processes in random environment , is still not sufficiently well understood . to further illuminate the contents of our results we point out that in a bulk system the equality of the self - diffusion coefficient and the transport diffusion coefficient for the spread of _ equilibrium _ density fluctuations in an infinite system may be taken for granted in the case of particles that have no mutual interaction . hence another way of stating the main conclusion of our work is the assertion that the transport diffusion coefficient as defined here in a stationary _ far - from - equilibrium _ setting coincides with the usual equilibrium transport diffusion coefficient . we also address finite - size effects coming from the fact that we are dealing with diffusion in a finite , open geometry . this causes deviations from bulk results for first - passage - time properties if a tagged particle starts its motion close to one boundary . in particular , we compute the permeation time and the milne extrapolation length that characterizes the survival time of a particle injected at a boundary . as a final introductory remark , it is worth noting that the case of knudsen gas with the cosine reflection law ( which is the model considered in this paper ) is particularly easy to analyse because the stationary state can be written in an explicit form , cf.theorem [ t_stat_measure ] . as explained below , this is related to the following facts : ( i ) there is no interaction between particles , ( ii ) for random billiard ( i.e. , a motion of only one particle in a closed domain ) with the cosine reflection law the stationary measure is quite explicit , as shown in @xcite . similar questions are much more complicated when the explicit form of the stationary state is not known . this is the general situation for non - equilibrium steady states . we refer to e.g. the model of @xcite ( a chain of coupled oscillators ) where one resorts to a bound on the entropy production . this paper is organized in the following way . in section [ s_defin ] we define the infinite random tube , and then introduce the process we call random billiard . in section [ s_transport ] , we then consider a gas of independent particles with absorption / injection in a finite piece of the random tube , and we formulate our results on the stationary measure for that gas and on the transport diffusion coefficient . in section [ s_perm ] , we go on to formulate first passage time results that concern exit from and crossing of the finite tube by a tagged particle . the remaining part of the paper is devoted to the proof of our results . in section [ s_pr_prelim ] we mainly use the reversibility of the process to obtain several technical facts used later . in section [ s_pr_steady ] we prove the result on the stationary measure of the knudsen gas in the finite tube . section [ s_pr_trans ] contains the proofs of the results related to the transport diffusion coefficient , and in section [ s_pr_perm ] we prove the results related to the crossing of the finite tube . naively the transport diffusion coefficient in tube direction @xmath7 may be defined through the diffusion equation for the probability density @xmath8 , where a possible @xmath7-dependence may originate from a spatial inhomogeneity of the tube . denote by @xmath9 the particle current in the system ; assuming stationarity with a probability density @xmath10 one has @xmath11 . with fixed external densities @xmath12 at @xmath13 and @xmath14 at @xmath15 one finds by integration @xmath16 with density gradient @xmath17 and @xmath18 . by measuring the current and the boundary densities one can thus obtain the transport diffusion coefficient without having to determine the local quantity @xmath19 . this result , however , implies knowledge of the local coarse - grained boundary densities @xmath20 to be able to make any comparison with @xmath3 . in a real experimental setting as well as for a given microscopic model these boundary densities @xmath20 are difficult to obtain . in particular , there is no well - defined prescription where precisely on a microscopic scale these boundary quantities should be measured . we circumvent the problem of computing these quantities from microscopic considerations by considering the total number of particles in the tube rather than local properties of the boundary region of the tube . together with proving a large - scale linear density profile in a stationary open random tube , one may then infer the macroscopic density gradient , see the definition ( [ defdtrans ] ) below . thus one obtains a macroscopic definition of the transport diffusion coefficient which is independent of microscopic details of the model . in order to fix ideas in a mathematically rigorous form we first recall some notations from @xcite . let us formally define the random tube in @xmath21 , @xmath22 . in this paper , @xmath23 will always stand for the linear subspace of @xmath21 which is perpendicular to the first coordinate vector @xmath24 , we use the notation @xmath25 for the euclidean norm in @xmath21 or @xmath23 . for @xmath26 let @xmath27 be the open @xmath28-neighborhood of @xmath29 . define @xmath30 to be the unit sphere in @xmath21 . let @xmath31 be the half - sphere looking in the direction @xmath32 . for @xmath33 , sometimes it will be convenient to write @xmath34 , being @xmath35 the first coordinate of @xmath7 and @xmath36 ; then , @xmath37 , and we write @xmath38 , being @xmath39 the projector on @xmath23 . fix some positive constant @xmath40 , and define @xmath41 let @xmath42 be an open connected domain in @xmath23 or @xmath21 . we denote by @xmath43 the boundary of @xmath42 and by @xmath44 the closure of @xmath42 . the random tube is viewed as a stationary and ergodic process @xmath45 , where @xmath46 is a subset of @xmath47 ; cf . @xcite for a more detailed definition . we denote by @xmath48 the law of this process ; sometimes we will use the shorthand notation @xmath49 for the expectation with respect to @xmath48 . with a slight abuse of notation , we denote also by @xmath50 the random tube itself , where the billiard lives . intuitively , @xmath46 is the `` slice '' obtained by crossing @xmath51 with the hyperplane @xmath52 . we will assume that the domain @xmath51 is defined in such a way that it is an open subset of @xmath21 , and that it is connected . we write also @xmath53 for the closure of @xmath51 . in order to define the random billiard correctly , following @xcite , throughout this paper we suppose that @xmath48-almost surely @xmath54 is a @xmath55-dimensional surface satisfying the lipschitz condition . this means that for any @xmath56 there exist @xmath57 , an affine isometry @xmath58 , a function @xmath59 such that * @xmath60 satisfies lipschitz condition , i.e. , there exists a constant @xmath61 such that @xmath62 for all @xmath63 ; * @xmath64 , @xmath65 , and @xmath66 roughly speaking , lipschitz condition implies that any boundary point can be `` touched '' by a piece of a cone which lies fully inside the tube . this in its turn ensures that the ( discrete - time ) process can not remain in a small neighborhood of some boundary point for very long time ; in section 2.2 of @xcite one can find an example of a non - lipschitz domain where the random billiard behaves in an unusual way . we keep the usual notation @xmath67 for the @xmath55-dimensional lebesgue measure on @xmath47 ( usually restricted to @xmath46 for some @xmath35 ) or haar measure on @xmath68 . we write @xmath69 for the @xmath70-dimensional lebesgue measure in case @xmath71 , and haar measure in case @xmath72 . also , we denote by @xmath73 the @xmath55-dimensional hausdorff measure on @xmath54 ; since the boundary is lipschitz , one obtains that @xmath73 is locally finite ( cf . the proof of lemma 3.1 in @xcite ) . we assume additionally that the boundary of @xmath48-a.e . @xmath51 is @xmath73-a.e . _ continuously _ differentiable , and we denote by @xmath74 the set of boundary points where @xmath54 is continuously differentiable . to avoid complications when cutting a ( large ) finite piece of the infinite random tube , we assume that there exists a constant @xmath75 such that for @xmath48-almost all environments @xmath51 we have the following : for any @xmath76 with @xmath77 there exists a path connecting @xmath78 that lies fully inside @xmath51 and has length at most @xmath75 . for all @xmath79 , let us define the normal vector @xmath80 pointing inside the domain @xmath51 . we say that @xmath81 is _ seen from _ @xmath82 if there exists @xmath83 and @xmath84 such that @xmath85 for all @xmath86 and @xmath87 . clearly , if @xmath88 is seen from @xmath7 then @xmath7 is seen from @xmath88 , and we write `` @xmath89 '' when this occurs . next , we construct the knudsen random walk ( krw ) @xmath90 , which is a discrete time markov process on @xmath54 , cf . section 2.2 of @xcite . it is defined through its transition density @xmath91 : for @xmath92 @xmath93 where @xmath94 is the normalizing constant , and @xmath95 stands for the indicator function . this means that , being @xmath96 the quenched ( i.e. , with fixed @xmath51 ) probability and expectation , for any @xmath79 and any measurable @xmath97 we have @xmath98 = { \int\limits}_b k(x , y)\ , d{\nu^\omega}(y).\ ] ] we also refer to the knudsen random walk as the random walk with cosine reflection law , since it is elementary to obtain from that the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector . [ rem_not_to_infty ] in fact , in the general setting of @xcite , for unbounded domains , one has to consider the following possibility : at some moment the particle chooses the outgoing direction in such a way that , moving in this direction , it never hits the boundary of the domain again , thus going directly to the infinity . however , it is straightforward to see that , since @xmath99 , in our situation @xmath100-a.s . this can not happen . it is immediate to obtain from that @xmath101 is symmetric ( that is , @xmath102 for all @xmath92 ) ; for both the discrete- and continuous - time processes this leads to some nice reversibility properties , exploited in @xcite . clearly , @xmath91 depends on @xmath51 as well , but we usually do not indicate this in the notations in order to keep them simple . also , let us denote by @xmath103 the @xmath104-step transition density ; clearly , one obtains that @xmath105 is symmetric too for any @xmath106 . now , we define the knudsen stochastic billiard ( ksb ) @xmath107 , which is the main object of study in this paper . first , we do that for the process starting on the boundary @xmath54 from the point @xmath108 . let @xmath109 be the trajectory of the random walk , and define @xmath110 then , for @xmath111 , define @xmath112 in proposition 2.1 of @xcite it was shown that , provided that the boundary satisfies the lipschitz condition , we have @xmath113 @xmath100-a.s . , and so @xmath114 is well - defined for all @xmath115 . the quantity @xmath114 stands for the position of the particle at time @xmath116 ; since it is not a markov process by itself , we define also the cdlg version of the motion direction at time @xmath116 : @xmath117 observe that @xmath118 . recall also another notation from @xcite : for @xmath119 , @xmath120 , define ( with the convention @xmath121 ) @xmath122 so that @xmath123 is the next point where the particle hits the boundary when starting at the location @xmath7 with the direction @xmath124 . of course , we can define also the stochastic billiard starting from the interior of @xmath51 by specifying its initial position @xmath125 and initial direction @xmath126 : the particle starts at the position @xmath125 and moves in the direction @xmath126 with unit speed until hitting the boundary at the point @xmath127 ; then , the previous construction is applied , being @xmath127 the starting boundary point . we denote by @xmath128 the ( quenched ) law of ksb in the tube @xmath51 starting from @xmath7 with the initial direction @xmath124 . consider the rescaled projected trajectory @xmath129 of ksb . [ def_invariance_principle ] we say that the quenched invariance principle holds for the knudsen stochastic billiard in the infinite random tube if there exists a positive constant @xmath130 such that , for @xmath48-almost all @xmath51 , for any initial conditions @xmath131 such that @xmath132 , the rescaled trajectory @xmath133 weakly converges to the brownian motion as @xmath134 . also , for some of our results we will have to make more assumptions on the geometry of the random tube . consider the following * condition t. * * there exists a positive constant @xmath135 and a continuous function @xmath136 such that @xmath137 * in the case @xmath138 , we assume that there exist @xmath139 such that for all @xmath140 with @xmath141 there exists @xmath142 such that @xmath143 . * in the case @xmath144 , we assume that @xmath145 [ rem_almost_all ] from the fact that @xmath99 and @xmath73-almost all points of @xmath54 belong to @xmath146 , it is straightforward to obtain that for lebesgue@xmath147haar - almost all @xmath148 we have @xmath149 ( see lemma 3.2 ( i ) of @xcite ) . [ rem_clt ] in the paper @xcite we prove that , if the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle is finite ( which is true for @xmath138 , but not always for @xmath144 ) , then under certain additional conditions ( related to condition t of the present paper ) , the quenched invariance principle holds for the knudsen stochastic billiard in the infinite random tube , cf . theorem 2.2 , propositions 2.1 and 2.2 of @xcite . let us comment more on the above condition t : * in @xcite , instead of the `` uniform dblin condition '' ( ii ) , we assumed a more explicit ( although a bit more technical ) condition p , which implies that ( ii ) holds ( see lemma 3.6 of @xcite ) . in fact , in the proof of the quenched invariance principle the technical condition of @xcite is used only through the fact that it implies the uniform dblin condition . * the assumption we made for @xmath144 may seem to be too restrictive . however , is it only a bit more restrictive that the assumption that the random tube does not contain an infinite straight cylinder . as it was shown in proposition 2.2 of @xcite , if the random tube contains an infinite straight cylinder , then the averaged second moment of the projected jump length is infinite in dimension @xmath150 , and so the ( quenched ) invariance principle can not be valid . now , let us introduce the notations specific to this paper . consider a positive number @xmath151 ( which is typically supposed to be large ) ; denote by @xmath152 the part of the random tube @xmath51 which lies between @xmath153 and @xmath151 : @xmath154\}.\ ] ] denote also @xmath155 so that @xmath156 ( see figure [ f_piece_of_tube ] ) . ] observe that @xmath152 can , in fact , consist of several separate pieces , namely , one big piece between @xmath153 and @xmath151 , and possibly several small pieces near the left and the right ends ( we suppose that @xmath157 , so that there could not be two or more big pieces ) . it can be easily seen that those small pieces have no influence on the definition of the transport diffusion coefficient ; for notational convention , we still allow @xmath152 to be as described above . then , we consider a gas of independent particles in @xmath152 , described as follows . there is usual reflection on @xmath158 ; any particle which hits @xmath159 , disappears . in addition , for a given @xmath160 , new particles are injected in @xmath161 with intensity @xmath162 per unit surface area . every newly injected particle chooses the initial direction at random according to the cosine law . in other words , the injection in @xmath161 is given by an independent poisson process in @xmath163 with intensity @xmath164 . [ rem_cosine_chosen ] the choice of the cosine law for the injection of new particles is justified by theorem 2.9 of @xcite : for the ksb in a finite domain , the long - run empirical law of intersection with a @xmath55-dimensional manifold is cosine . one may think of the following situation : the random tube is connected from its left side @xmath161 to a very large reservoir containing the knudsen gas in the stationary regime ; then , the particles cross @xmath161 with approximately cosine law ( at least on the time scale when the density of the particles in the big reservoir remains unaffected by the outflow through the tube ) . in section [ s_pr_steady ] ( proof of theorem [ t_stat_measure ] ) we use this kind of argument to obtain a rigorous characterization of the steady state of this gas . we now consider this gas in the stationary regime . let @xmath165}:=[a , b]\times\xi$ ] , and let @xmath166 be the mean number of particles in @xmath167}$ ] , in a fixed environment @xmath51 . in theorem [ t_dens_grad ] below we shall see that there exists a constant @xmath168 such that @xmath169 which means that , after coarse - graining , the particle density profile is asymptotically linear . the above quantity @xmath168 is called the ( rescaled ) density gradient . we define also the current @xmath170 as the mean number of particles absorbed in @xmath171 per unit of time , and let the rescaled current be defined as @xmath172 then , consistently with the discussion in the beginning of this section , the _ transport diffusion coefficient _ @xmath4 is defined by @xmath173 now , suppose that the quenched invariance principle with constant @xmath174 holds for the stochastic billiard . our goal is to prove that @xmath4 is equal to the _ self - diffusion coefficient _ @xmath175 . to this end , we prove the following two results . first , we prove that the coarse - grained density profile is indeed linear : [ t_dens_grad ] suppose that the quenched invariance principle holds . then , for any @xmath176 there exists @xmath177 such that @xmath48-a.s . @xmath178 then , we calculate the limiting current : [ t_current ] suppose that the quenched invariance principle holds with constant @xmath174 , and assume also that condition t holds . then , we have @xmath48-a.s . @xmath179 some remarks are in place that illustrate the significance of the above theorems . theorem [ t_dens_grad ] means that @xmath180 , and using also theorem [ t_current ] , we obtain that @xmath6 . at the same time it becomes clear that such a statement can be true only asymptotically since in a finite open tube one has to expect finite size corrections of the mean particle number . these corrections may , in fact , depend strongly on the microscopic shape of the tube near the open boundaries . this implies that in experiments on real spatially inhomogeneous systems some care has to be taken as to what is measured as macroscopic density gradient . notice that with theorem [ t_current ] we also prove fick s law for diffusive transport of matter in the random knudsen stochastic billiard . since the velocity of the particles does not change at collisions with the tube walls , mass transport is proportional to energy transport . in this interpretation theorem [ t_current ] implies fourier s law for heat conduction , see e.g. @xcite for recent work on other processes . for a function @xmath181 and @xmath182 , denote @xmath183 as mentioned in the introduction , in the proof of theorems [ t_dens_grad ] and [ t_current ] we use the explicit form of the steady state for the knudsen gas in the random tube with injection from one side . let us formulate the following theorem : [ t_stat_measure ] * for the knudsen gas with absorption / injection in @xmath184 ( as before , with intensity @xmath162 per unit surface area ) the unique stationary state is poisson point process in @xmath185 with intensity @xmath186 . * for the gas with injection in @xmath161 only , the unique stationary distribution of the particle configuration is given by a poisson point process in @xmath187 with intensity measure @xmath188 \,d\alpha\ , du\ , dh.\ ] ] also , in both cases , for any initial configuration the process converges to the stationary state described above . of course , the above result is not quite unexpected . it is well known that independent systems have poisson invariant distributions ( with the single particle invariant measure for poisson intensity ) , let us mention e.g. @xcite ( section viii.5 ) and @xcite . still , we decided to include the proof of this theorem because ( as far as we know ) , it does not directly follow from any of the existing results available in the literature . let us introduce some more notations for the finite random tube . we denote by @xmath189 the set of points of @xmath190 , from where the particle can reach @xmath171 by a path which stays within @xmath152 and set @xmath191 ( see figure [ f_permeation ] ) , and let @xmath192 be the corresponding finite tube . since we are going to study now how long a tagged particle stays inside the tube and how it crosses ( i.e. , goes to the right boundary without going back to the left boundary ) , the idea is to inject it in a place from where it can actually do it . our interest is then in certain first - passage properties , in particular , the total life time of the particle inside @xmath193 ( i.e. , the time until the particle first exits @xmath193 ) and the permeation time which the particle needs to first exit @xmath193 at the end of the tube segment `` opposite '' to that where it was injected , i.e. , after crossing the tube . , and the event @xmath194 ( a trajectory crossing the tube is shown ) ] so , suppose that one particle is injected ( uniformly ) at random at @xmath195 into the tube @xmath193 ( that is , the starting location has the uniform distribution in @xmath195 , and the direction is chosen according to the cosine law ) , and let us denote by @xmath194 the event that it crosses the tube without going back to @xmath195 , i.e. , @xmath196 ( here , @xmath197 and @xmath198 are , respectively , entrance and hitting times for the discrete - time process , see and for the precise definitions ) . also , define @xmath199 to be the total lifetime of the particle , i.e. , if @xmath114 is the location of the particle at time @xmath116 , then @xmath200 . first , we calculate the asymptotic behaviour of the quenched and annealed ( averaged ) expectation of @xmath199 : [ t_exp_perm ] suppose that the quenched invariance principle holds with constant @xmath174 . we have @xmath201 observe that condition t ( i ) implies that @xmath202 is bounded away from @xmath153 , and so @xmath203 . at this point we remind the reader that here and in the next theorem the expected `` times '' are actually expected lengths of flight , related through the corresponding times through the trivial generic relation _ _ length__@xmath204__velocity__@xmath147_time_. in our knudsen gas we always assume unit velocity @xmath205 so that times can be identified with the appropriate lengths . to elucidate the physical significance of theorem [ t_exp_perm ] we observe that for usual brownian motion the expected lifetime @xmath206 of particle in an interval @xmath207 $ ] is given by @xmath208 , where @xmath209 is the starting position and @xmath210 is the diffusion coefficient . so , in particular , for a particle starting at the boundary @xmath211 ( or at @xmath212 ) the expected life time is @xmath153 . however , in a microscopic model of diffusion in a finite open system , this result can not be expected to be generally valid because of a positive probability that a particle which starts at @xmath213 would escape through the other boundary at @xmath214 . often it is found empirically that the expected life time can be approximated by @xmath215 with an effective shifted coordinate @xmath216 and effective interval length @xmath217 . the empirical shift length @xmath218 is known as milne extrapolation length @xcite , for a recent application to diffusion in carbon nanotubes see @xcite . from the definition ( [ milne ] ) one can see that the life time of a particle starting at the origin @xmath213 allows for the computation of the milne extrapolation length through the asymptotic relation @xmath219 provided the diffusion coefficient @xmath210 is known . in a physical system the milne extrapolation length depends on molecular details of the gas such as type of molecule or temperature , but in a knudsen gas also on the tube surface . in our model the properties of the gas are encoded in the unit velocity @xmath205 of the particles . observe now that the quantity @xmath220 corresponds to @xmath221 in our setting . hence , by identifying @xmath222 and using @xmath223 , theorem [ t_exp_perm ] furnishes us with the dependence of the milne extrapolation length on the tube properties through @xmath224 interestingly , @xmath218 depends only on very few generic properties of the random tube . the next result relies on theorem [ t_current ] , so we need to assume a stronger condition on the geometry of the tube . [ t_perm ] let us suppose that the quenched invariance principle is valid with @xmath174 , and assume that condition t holds . for the asymptotics of the probability of crossing , we have @xmath225 = \frac{\gamma_d\|{{\mathbb s}}^{d-1}\|{\hat\sigma}^2{\big\langle \|\omega_0\| \big\rangle_{\!{}_{{\mathbb p } } } } } { 2\|{\tilde\omega}_0\| } \qquad \text{${{\mathbb p}}$-a.s . , } \label{q_perm_prob}\\ \lim_{h\to\infty } h{\big\langle { { \mathtt p}_\omega}[{{\mathfrak c}}_h ] \big\rangle_{\!{}_{{\mathbb p } } } } = \frac{1}{2 } \gamma_d\|{{\mathbb s}}^{d-1}\|{\hat\sigma}^2{\big\langle \|\omega_0\| \big\rangle_{\!{}_{{\mathbb p } } } } { \big\langle \|{\tilde\omega}_0\|^{-1 } \big\rangle_{\!{}_{{\mathbb p}}}}. \label{a_perm_prob}\end{aligned}\ ] ] for the quenched behaviour of the conditional expectations , we have , @xmath48-a.s . @xmath226 and for the annealed ones @xmath227 as one sees from theorems [ t_exp_perm ] and [ t_perm ] , all our annealed results in fact say that one can interchange the limit as @xmath228 with integration with respect to @xmath48 . we still decided to include these results ( even though they are technically not difficult ) because , in models related to random environment , it is frequent that the annealed behaviour differs substantially from the quenched behaviour . one may find it interesting to observe that , by and @xmath229 to obtain another interesting consequence of our results , let us suppose now that @xmath48-a.s . the random tube is such that we have @xmath230 . observe that , by jensen s inequality , it holds that @xmath231 ( and the inequality is strict if the distribution of @xmath232 is nondegenerate ) , so `` roughness '' of the tube makes the quantities @xmath233 and @xmath234 increase . in other words , these quantities as well as the milne correlation length are minimized on the tubes with constant section ( which , by the way , do not have to be necessarily `` straight cylinders '' ! ) . the remaining part of the paper is devoted to the proofs of our results , and , as mentioned in the introduction , it is organized in the following way . in section [ s_pr_prelim ] we obtain several auxiliary results related to hitting of sets by the random billiard . in section [ s_pr_steady ] we obtain the explicit form of the stationary measure of the knudsen gas in the finite tube @xmath152 by using the corresponding result from @xcite about the stationary distribution of one particle in a finite domain . then , in section [ s_pr_trans ] , we apply the results of sections [ s_pr_prelim ] and [ s_pr_steady ] to obtain the explicit form of the transport diffusion coefficient . finally , in section [ s_pr_perm ] we use little s theorem to prove the results related to the crossing time of the random tube . we need first to prove several auxiliary facts for random billiard in arbitrary finite domains . as in @xcite , let @xmath235 be a bounded domain with lipschitz and a.e . continuously differentiable boundary . we keep the notation @xmath100 to denote the law of our processes , and we still use @xmath73 to denote the @xmath55-dimensional hausdorff measure on the boundary @xmath236 . consider a markov chain @xmath237 on @xmath236 , which has a transition density @xmath238 with the property @xmath239 for all @xmath240 . observe that the knudsen random walk @xmath241 has the above property , but we need to formulate the next results in a slightly more general framework , since we shall need to apply them to some other processes built upon @xmath241 . let us introduce the notations @xmath242 for the entrance and the hitting time of @xmath243 . also , for measurable @xmath244 such that @xmath245 we shall write @xmath246 = \frac{1}{{\nu^\omega}(b)}{\int\limits}_b{{\mathtt p}_\omega}^x[\cdot]\,d{\nu^\omega}(x),\ ] ] so that @xmath247 is the law for the process starting from the uniform distribution on @xmath248 . taking advantage of the reversibility of the process @xmath241 , we prove the following [ l_hitting_revers ] consider two arbitrary measurable sets @xmath249 such that @xmath250 . * suppose that @xmath251 . for any @xmath252 , we have @xmath253 } \nonumber\\ & = \frac{1}{{\nu^\omega}(b){{\mathtt p}_\omega}^b[\tau(f)<\tau^+(b ) ] } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[\tau(b)<\tau^+(f)]\ , d{\nu^\omega}(y ) \nonumber\\ & = \frac{1}{{\nu^\omega}(f){{\mathtt p}_\omega}^f[\tau(b)<\tau^+(f ) ] } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[\tau(b)<\tau^+(f)]\ , d{\nu^\omega}(y ) . \label{hitting_rev1}\end{aligned}\ ] ] * suppose that @xmath254 . for any @xmath255 , we have @xmath256 \,d{\nu^\omega}(x ) } \nonumber\\ & = { \int\limits}_{b '' } { { \mathtt p}_\omega}^x[\xi_{\tau^+(b)}\in b ' , \tau^+(b)<\tau(f ) ] \,d{\nu^\omega}(x ) . \label{hitting_rev2}\end{aligned}\ ] ] one immediately obtains the following consequence of lemma [ l_hitting_revers ] ( ii ) : [ c_trans_dens ] for any @xmath249 such that @xmath257 and @xmath254 , we have the following . * for @xmath258 , let us define the conditional ( on the event @xmath259 ) transition density @xmath260 : @xmath261 = { \int\limits}_{b '' } { \bar k}_{b , f}(x , y)\,d{\nu^\omega}(y).\ ] ] then , we have @xmath262 { \bar k}_{b , f}(x , y ) = { { \mathtt p}_\omega}^y[\tau^+(b)<\tau(f ) ] { \bar k}_{b , f}(y , x),\ ] ] that is , the random walk conditioned to return to @xmath248 without hitting @xmath263 is reversible with the reversible measure @xmath264 defined by @xmath265.\ ] ] * in particular ( take @xmath266 in the previous part ) the random walk observed at the moments of successive visits to @xmath248 is reversible with the reversible measure @xmath73 . _ proof of lemma [ l_hitting_revers ] . _ abbreviate for the moment @xmath267 . first , write using the fact that @xmath268 is symmetric @xmath269 & = \sum_{n=1}^\infty { { \mathtt p}_\omega}^b[\tau(f)=n,\tau^+(b)>n]\\ & = \sum_{n=1}^\infty { \int\limits}_b \frac{d{\nu^\omega}(x_0)}{{\nu^\omega}(b ) } { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_1 ) \ldots d{\nu^\omega}(x_{n-1 } ) \\ & \qquad\qquad\times { \int\limits}_f d{\nu^\omega}(x_n ) { \bar k}(x_0,x_1)\ldots { \bar k}(x_{n-1},x_n)\\ & = \frac{{\nu^\omega}(f)}{{\nu^\omega}(b)}\sum_{n=1}^\infty { \int\limits}_f \frac{d{\nu^\omega}(x_n)}{{\nu^\omega}(f ) } { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_{n-1 } ) \ldots d{\nu^\omega}(x_1 ) \\ & \qquad\qquad\times { \int\limits}_b d{\nu^\omega}(x_0 ) { \bar k}(x_n , x_{n-1})\ldots { \bar k}(x_1,x_0)\\ & = \frac{{\nu^\omega}(f)}{{\nu^\omega}(b ) } \sum_{n=1}^\infty { { \mathtt p}_\omega}^f[\tau(b)=n,\tau^+(f)>n]\\ & = \frac{{\nu^\omega}(f)}{{\nu^\omega}(b ) } { { \mathtt p}_\omega}^f[\tau(b)<\tau^+(f)].\end{aligned}\ ] ] then , similarly @xmath253 } \\ & = \frac{1}{{{\mathtt p}_\omega}^b[\tau(f)<\tau^+(b ) ] } \sum_{n=1}^\infty { { \mathtt p}_\omega}^b[\tau(f)=n,\xi_{\tau(f)}\in f ' , \tau^+(b)>n]\\ & = \frac{{\nu^\omega}(b)}{{\nu^\omega}(f){{\mathtt p}_\omega}^f[\tau(b)<\tau^+(f ) ] } \sum_{n=1}^\infty { \int\limits}_b \frac{d{\nu^\omega}(x_0)}{{\nu^\omega}(b ) } { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_1 ) \ldots d{\nu^\omega}(x_{n-1 } ) \\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\times { \int\limits}_{f ' } d{\nu^\omega}(x_n ) { \bar k}(x_0,x_1)\ldots { \bar k}(x_{n-1},x_n)\\ & = \frac{1}{{\nu^\omega}(f){{\mathtt p}_\omega}^f[\tau(b)<\tau^+(f ) ] } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[\tau(b)<\tau^+(f)]\ , d{\nu^\omega}(y),\end{aligned}\ ] ] so is proved . let us prove . analogously to the previous computation , we write @xmath270 \,d{\nu^\omega}(x)}\\ & = { \int\limits}_{b'}d{\nu^\omega}(x ) \sum_{n=1}^\infty { { \mathtt p}_\omega}^x[\xi_{\tau^+(b)}\in b '' , \tau^+(b)=n,\tau(f)>n]\\ & = \sum_{n=1}^\infty { \int\limits}_{b ' } d{\nu^\omega}(x_0 ) { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_1 ) \ldots d{\nu^\omega}(x_{n-1 } ) \\ & \qquad\qquad\times { \int\limits}_{b '' } d{\nu^\omega}(x_n ) { \bar k}(x_0,x_1)\ldots { \bar k}(x_{n-1},x_n)\\ & = \sum_{n=1}^\infty { \int\limits}_{b '' } d{\nu^\omega}(x_n ) { \int\limits}_{u^{n-1 } } d{\nu^\omega}(x_{n-1 } ) \ldots d{\nu^\omega}(x_1 ) \\ & \qquad\qquad\times { \int\limits}_{b ' } d{\nu^\omega}(x_0 ) { \bar k}(x_n , x_{n-1})\ldots { \bar k}(x_1,x_0)\\ & = { \int\limits}_{b '' } { { \mathtt p}_\omega}^x[\xi_{\tau^+(b)}\in b ' , \tau^+(b)<\tau(f ) ] \,d{\nu^\omega}(x),\end{aligned}\ ] ] and is proved . this concludes the proof of lemma [ l_hitting_revers ] . next , we recall the dirichlet s principle : [ p_dirichlet ] consider @xmath271 with @xmath272 and @xmath254 , and denote @xmath273 $ ] ( so that , in particular , @xmath274 for all @xmath275 and @xmath276 for all @xmath277 ) . define @xmath278 , h(x)=0 \text { for all } x\in b , h(x)=1 \text { for all } x\in f\}.\ ] ] then @xmath279 = { { \mathcal e}}({\hat h},{\hat h } ) = \min_{h\in { { \mathcal h}}}{{\mathcal e}}(h , h),\ ] ] where @xmath280 _ proof . _ for the proof , we refer to the discrete case , e.g. proposition 3.8 in @xcite , and observe that the proof applies to the space - continuous case , using that , on general spaces , harmonicity in the analytic sense and in the probabilistic sense are equivalent notions by @xcite . indeed , minimizers @xmath32 of the dirichlet form are harmonic in the analytic sense , i.e. , there are in the kernel of the form ( see ( 2.10 ) in @xcite ) , though the left - hand side of ( 24 ) is the value of @xmath281 when @xmath32 is harmonic in the probabilistic sense , i.e. , the expectation of the process at some exit time ( see theorem 2.7 in @xcite ) with the appropriate boundary conditions . now , we go back to the knudsen random walk in the random tube @xmath51 . recall that @xmath105 stands for the the @xmath104-step transition density of krw , and that we have @xmath282 for all @xmath78 . let us define for an arbitrary @xmath283 @xmath284 in case @xmath42 is an interval , say , @xmath285 , we write @xmath286 instead of @xmath287 . there is the following apriory bound on the size of the jump of the random billiard : there exists a constant @xmath288 , depending only on @xmath289 and the dimension , such that for @xmath48-almost all @xmath51 @xmath290 \leq { \tilde\gamma}_1 u^{-(d-1)},\ ] ] for all @xmath56 , @xmath291 , see formula ( 54 ) of @xcite . moreover , using , for any @xmath292 it is straightforward to obtain that , for some @xmath293 @xmath294 \leq { \tilde\gamma}^{(n)}_1 u^{-(d-1)},\ ] ] for all @xmath56 , @xmath291 ( also , without restriction of generality , we can assume that @xmath295 is nondecreasing in @xmath104 ) . now , with the help of the above formula we prove the following result : [ l_separated ] for any @xmath106 there exists @xmath296 such that for all @xmath291 and @xmath182 we have @xmath297 _ proof . _ abbreviate @xmath298 . the main idea is the following : if at some step the knudsen random walk jumped from some point of @xmath299 to @xmath300 , it must cross @xmath301 , so the probability of such a jump is the same as the probability of the jump to @xmath301 in the semi - infinite tube with the boundary @xmath302 . so , we obtain @xmath303}d{\nu^\omega}(x ) { \int\limits}_{{\tilde f}^\omega[a+u,\infty)}d{\nu^\omega}(y ) k^n(x , y)}\qquad\\ & = { \int\limits}_{{\tilde f}^\omega(-\infty , a ] } { { \mathtt p}_\omega}^x[\xi_n\in { \tilde f}^\omega[a+u,\infty)]\,d{\nu^\omega}(x)\\ & \leq { \int\limits}_{{\tilde f}^\omega(-\infty , a ] } { { \mathtt p}_\omega}^x\big[\bigcup_{k=1}^n\{\xi_k\cdot{\mathbf{e}}\geq a+u , \xi_j\cdot{\mathbf{e } } < a+u \text { for all } j < k\}\big]\ , d{\nu^\omega}(x)\\ & \leq { \int\limits}_{{\tilde f}^\omega(-\infty , a]}d{\nu^\omega}(x_0 ) \sum_{k=1}^n { \int\limits}_{({\tilde f}^\omega(-\infty , a+u))^{k-1 } } d{\nu^\omega}(x_0)\ldots d{\nu^\omega}(x_{k-1})\\ & \qquad\qquad\qquad\qquad\qquad\qquad { \int\limits}_{{\tilde f}^\omega[a+u,\infty)}d{\nu^\omega}(x_k ) k(x_0,x_1)\ldots k(x_{k-1},x_k)\\ & \leq { \int\limits}_{{\tilde f}^\omega(-\infty , a]}d{\nu^\omega}(x ) { \int\limits}_v d{\nu^\omega}(y ) \big(k(x , y)+k^2(x , y)+\cdots+k^n(x , y)\big).\end{aligned}\ ] ] by symmetry of @xmath91 , we have for any @xmath177 @xmath304\ , d{\nu^\omega}(y),\ ] ] so lemma [ l_separated ] now follows from . let us consider a sequence of i.i.d . random variables @xmath305 with uniform distribution on @xmath306 ( where @xmath307 is from condition t ( ii ) ) , independent of everything . also , let us define @xmath308 . then , it is straightforward to obtain that , for any @xmath56 and @xmath309 , we have @xmath310 \geq n^{-1}r_1{\nu^\omega}(b)\ ] ] for some @xmath311 . let @xmath312 be the transition density of the process @xmath313 . observe that this process is still reversible with the reversible measure @xmath73 , so that @xmath314 for all @xmath92 . similarly to @xcite , let us define @xmath315 and @xmath316 we suppose that @xmath317 and @xmath318 are defined as in but with @xmath241 instead of @xmath237 , and let @xmath319 and @xmath320 be the corresponding quantities for the process @xmath321 . [ l_escape_upper ] suppose that @xmath322 with @xmath323 . moreover , assume that @xmath324 for all @xmath275 , and @xmath325 for all @xmath326 ( of course , the same result is valid if we assume that @xmath324 for all @xmath277 , and @xmath325 for all @xmath327 ) . then , there exist positive constants @xmath328 , @xmath329 , such that @xmath330 \leq { \tilde\gamma}_3u^{-(d-1 ) } + \frac{1}{u^2}{\int\limits}_{{\tilde f}^\omega[a , a+u]}b(x)\ , d{\nu^\omega}(x),\ ] ] and @xmath331 \leq { \tilde\gamma}_4u^{-(d-1 ) } + \frac{1}{u^2}{\int\limits}_{{\tilde f}^\omega[a , a+u]}{\hat b}(x)\ , d{\nu^\omega}(x).\ ] ] moreover , and are valid also in the finite tube @xmath152 ( in this case we assume that @xmath332 and @xmath333 ) . we now work in finite tube @xmath152 . let us use the abbreviations @xmath339 , and @xmath340 . observe that , by condition t ( i ) , we have that for some @xmath341 @xmath342 for all @xmath104 and for @xmath48-a.a . @xmath51 . to distinguish between the seconds moments of the projected jump length in finite and infinite tubes , we modify our notations in the following way . for @xmath343 , let @xmath344 and @xmath345 be the quantities defined as in and , but in the finite tube @xmath152 . let us use the notations @xmath346 and @xmath347 for the corresponding quantities in the infinite tube . now , we need an estimate on the integrals appearing in the right - hand sides of and , for the case of the finite tube : [ l_int_b ] suppose that @xmath348 and assume that @xmath138 and condition t holds . then , we have @xmath349}b_h(x)\ , d{\nu^\omega}(x ) < \infty \qquad { { \mathbb p}}\text{-a.s.},\ ] ] and the same is valid with @xmath350 on the place of @xmath351 . _ let us recall some notations from @xcite . define @xmath352 define the probability measure @xmath353 on @xmath354 by @xmath355 where @xmath356 is the @xmath357-dimensional hausdorff measure on the boundary of @xmath190 , @xmath358 is the scalar product of the normal vectors pointing inside the section and inside the tube ( see section 2 of @xcite for details ) , and @xmath359 is the normalizing constant . in lemma 3.1 of @xcite it is shown that @xmath353 is the invariant law of the environment seen from the walker , that is @xmath360 \big\rangle_{\!{}_{{\mathbb q}}}}={\big\langle f \big\rangle_{\!{}_{{\mathbb q}}}}.\ ] ] using also that @xmath361 and , it is straightforward to obtain that @xmath362 implies @xmath363 . so , using the notations of @xcite , by the ergodic theorem we obtain @xmath364}b_\infty(x)\ , d{\nu^\omega}(x ) & = \frac{1}{h } { \int\limits}_0^h d\alpha { \int\limits}_{\xi}d{\mu^\omega}_\alpha(v)\kappa^{-1}_{\alpha , v } b_\infty(\theta_\alpha\omega , v)\nonumber\\ & \to { \big\langle b_\infty \big\rangle_{\!{}_{{\mathbb q } } } } \qquad \text{as $ h\to\infty$ } , \label{conv_b}\end{aligned}\ ] ] a.s . and in @xmath365 , and the same with @xmath366 on the place of @xmath367 . then , follows from the fact that , for all @xmath151 , @xmath368 for all @xmath369 . now , with @xmath366 instead of @xmath367 , the previous inequality is not necessarily valid . so , to prove for @xmath350 instead of @xmath351 , consider @xmath56 such that @xmath370 $ ] , and write ( note that for all @xmath343 we have @xmath371 ) @xmath372\\ & \leq c_1 h^{-(d-3)}\end{aligned}\ ] ] ( recall that @xmath138 ) , and then we obtain for @xmath350 as well . next , we obtain a lower bound for certain escape probabilities : [ l_dirichlet_lower ] suppose that @xmath373 , and @xmath374 . also , assume that @xmath138 and condition t holds . then , there exist positive constants @xmath375 , @xmath376 , such that @xmath377 \geq \frac{{\tilde\gamma}_7}{n - m},\ ] ] and @xmath378 \geq \frac{{\tilde\gamma}_8}{h}.\ ] ] _ proof . _ let @xmath379 be the dirichlet form corresponding to @xmath380 ( cf . ) . first , let us prove . as in proposition [ p_dirichlet ] , we use the notation @xmath381 $ ] ; observe that @xmath274 for all @xmath382 and @xmath383 for all @xmath384 ( and hence for all @xmath385 ) . using this fact together with and cauchy - schwarz inequality , we write ( abbreviating @xmath386 ) @xmath387}\\ & = { \hat{{\mathcal e}}}({\hat h},{\hat h})\\ & \geq \sum_{j=0}^u { \int\limits}_{u_{m+j}}d{\nu^\omega}(x_j ) { \int\limits}_{u_{m+j+1}}d{\nu^\omega}(x_{j+1 } ) { \hat k}(x_j , x_{j+1 } ) ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & = \big(\prod_{j=0}^{u+1}{\nu^\omega}(u_{m+j})\big)^{-1 } { \int\limits}_{u_m}d{\nu^\omega}(x_0)\ldots { \int\limits}_{u_{m+u+1}}d{\nu^\omega}(x_{u+1})\\ & \qquad\qquad\qquad \sum_{j=0}^{u}{\nu^\omega}(u_{m+j}){\nu^\omega}(u_{m+j+1 } ) { \hat k}(x_j , x_{j+1 } ) ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & \geq n^{-1}r_1{\tilde\gamma}_5 ^ 2 \big(\prod_{j=0}^{u+1}{\nu^\omega}(u_{m+j})\big)^{-1 } { \int\limits}_{u_m}d{\nu^\omega}(x_0)\ldots\\ & \qquad\qquad\qquad \ldots { \int\limits}_{u_{m+u+1}}d{\nu^\omega}(x_{u+1})\sum_{j=0}^{u } ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & \geq \frac{n^{-1}r_1{\tilde\gamma}_5 ^ 2}{u+1 } \big(\prod_{j=0}^{u+1}{\nu^\omega}(u_{m+j})\big)^{-1 } { \int\limits}_{u_m}d{\nu^\omega}(x_0)\ldots{\int\limits}_{u_{m+u+1}}d{\nu^\omega}(x_{u+1})\\ & = \frac{n^{-1}r_1{\tilde\gamma}_5 ^ 2}{n - m+1},\end{aligned}\ ] ] and this proves . by denoting @xmath388 $ ] and writing @xmath389}\\ & \geq \sum_{j=1}^{n+1 } { \int\limits}_{u_j}d{\nu^\omega}(x_j ) { \int\limits}_{u_{j+1}}d{\nu^\omega}(x_{j+1 } ) { \hat k}(x_j , x_{j+1 } ) ( { \hat h}(x_j)-{\hat h}(x_{j+1}))^2\\ & \quad + { \int\limits}_{{{\hat d}_\ell}}d{\nu^\omega}(x_0 ) { \int\limits}_{u_1}d{\nu^\omega}(x_1 ) { \hat k}(x_0,x_1 ) ( { \hat h}(x_0)-{\hat h}(x_1))^2\end{aligned}\ ] ] in exactly the same way one can show . this concludes the proof of lemma [ l_dirichlet_lower ] . next , we need ( pointwise ) estimates on the probabilities of exiting the tube at the left boundary : [ l_gamblers_ruin ] assume condition t and @xmath138 . suppose also that @xmath390 , and @xmath391 $ ] . then , there exists @xmath392 such that for all @xmath382 we have @xmath393 \leq \frac{{\tilde\gamma}_9(n - m+1)}{h}.\ ] ] _ proof . _ from now on , we assume for technical reasons that @xmath394 ( in any case , otherwise the upper bound @xmath395 is good enough for us ) . first , by lemmas [ l_escape_upper ] and [ l_int_b ] , we obtain that @xmath396 \leq \frac{c_1}{h}.\ ] ] next , lemma [ l_dirichlet_lower ] implies that @xmath397 \geq \frac{c_2}{n - m+1}.\ ] ] also , from it is clear that for any @xmath382 we have @xmath398 \geq { { \mathtt p}_\omega}^x[{\hat\xi}_1\in u_m ] \geq c_3\ ] ] for some @xmath399 . now , denote @xmath400 , @xmath401 to be the successive times when the set @xmath402 is visited . by corollary [ c_trans_dens ] ( i ) and , we obtain that , conditional on not hitting @xmath403 , the process of successive returns to @xmath402 is reversible with the reversible density @xmath404 , such that for all @xmath382 @xmath405 for some positive constants @xmath406 . using also and , we obtain that there are constants @xmath407 such that for any @xmath70 @xmath408 & \leq \frac{c_6}{h},\\ { { \mathtt p}_\omega}^{u_m}[{\hat\tau}(v_n)<{\hat\tau}^+(u_m ) \mid { \hat\tau}({{\hat d}_\ell}\cup v_n)>\sigma_k ] & \geq \frac{c_7}{n - m+1}.\end{aligned}\ ] ] so , we can write @xmath409 & = \sum_{k=1}^\infty { { \mathtt p}_\omega}^{u_m}\big[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n ) \mid { \hat\tau}({{\hat d}_\ell}\cup v_n)\in(\sigma_{k-1},\sigma_k]\big ] \nonumber\\ & \qquad\qquad \times { { \mathtt p}_\omega}^{u_m}\big[{\hat\tau}({{\hat d}_\ell}\cup v_n ) \in(\sigma_{k-1},\sigma_k]\big ] \nonumber\\ & \leq \sum_{k=1}^\infty \frac{c_6/h}{c_7/(n - m+1 ) } { { \mathtt p}_\omega}^{u_m}\big[{\hat\tau}({{\hat d}_\ell}\cup v_n ) \in(\sigma_{k-1},\sigma_k]\big ] \nonumber\\ & = \frac{c_6c_7^{-1}(n - m+1)}{h}. \label{bound_set}\end{aligned}\ ] ] now , the `` pointwise '' version of is substantially more difficult to prove . consider a sequence of i.i.d . random variables @xmath410 with @xmath411 = n^{-1}r_1{\tilde\gamma}_5\ ] ] ( recall and ) . then , one can couple the random sequences @xmath412 with @xmath413 in such a way that when the event @xmath414 occurs , @xmath415 has the stationary distribution on @xmath416}$ ] . we denote by @xmath417 and @xmath418 the probability and expectation with fixed @xmath51 and @xmath419 , and let @xmath420 be the expectation with respect to @xmath419 . one can formally define @xmath417 in the following way . for any @xmath421 , define the transition density @xmath422 by @xmath423 let @xmath424 be the distribution on @xmath54 with the density @xmath422 , and let @xmath425 be the uniform distribution on @xmath426 . then , given @xmath427 , the law of @xmath415 under @xmath417 is given by @xmath428 also , let us define @xmath429 . now , observe that @xmath430 = [ { \hat\xi}_{j-1}\cdot{\mathbf{e } } ] \text { on } \{j={\hat\kappa}\}\ ] ] and , for @xmath431 such that @xmath432 , @xmath433 \mid { \hat\kappa}=j \big ) } \nonumber\\ & = e^{\zeta } \big({{\mathtt p}_{\omega,\zeta}}^x\big[\|({\hat\xi}_i- { \hat\xi}_{i-1})\cdot{\mathbf{e}}\| \geq u \big ] \mid \zeta_i=0 \big ) \nonumber\\ & \leq \frac{1}{p^{\zeta}[\zeta_i=0 ] } { { \mathtt p}_\omega}^x\big[\|({\hat\xi}_i- { \hat\xi}_{i-1})\cdot{\mathbf{e}}\| \geq u \big ] \nonumber\\ & \leq c_8h^{-(d-1 ) } , \label{<j}\end{aligned}\ ] ] recall . then , write using and @xmath434 e^{\zeta}\big({{\mathtt p}_{\omega,\zeta}}^{u_0}\big [ \max_{\ell\leq{\hat\kappa } } \|({\hat\xi}_\ell-{\hat\xi}_0)\cdot{\mathbf{e}}\| \geq s\big]\mid { \hat\kappa}=j\big ) \nonumber\\ & \leq \sum_{j=1}^\infty p^{\zeta}[{\hat\kappa}=j ] e^{\zeta}\big({{\mathtt p}_{\omega,\zeta}}^x\big[\text{there exists } i\leq j \text { such that } \nonumber\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad \geq s / j\big]\mid { \hat\kappa}=j\big ) \nonumber\\ & \leq \sum_{j=1}^\infty p^{\zeta}[{\hat\kappa}=j ] jc_9\big(\frac{s}{j}\big)^{-(d-1 ) } \nonumber\\ & = c_9 s^{-(d-1 ) } \sum_{j=1}^\infty j^d p^{\zeta}[{\hat\kappa}=j ] \nonumber\\ & = c_{10 } s^{-(d-1)}. \label{oc_kappa_1}\end{aligned}\ ] ] now , using , we have for an arbitrary @xmath382 @xmath435 } \nonumber\\ & = e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n ) ] \nonumber\\ & \leq e^{\zeta } { { \mathtt p}_{\omega,\zeta}}^x\big[\max_{j\leq { \hat\kappa } } { \hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n)\big ] \nonumber\\ & \qquad + e^{\zeta } { { \mathtt p}_{\omega,\zeta}}^x\big[\max_{j\leq { \hat\kappa } } \nonumber\\ & \leq e^{\zeta } { { \mathtt p}_{\omega,\zeta}}^x\big[\max_{j\leq { \hat\kappa } } { \hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n)\big ] + c_{11}h^{-(d-1)}. \label{i+ii}\end{aligned}\ ] ] let us deal with the first term in . we have , taking advantage of and ( recall that @xmath138 ) @xmath436}\\ & \leq \sum_{\ell\geq h/16 } e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x\big[[{\hat\xi}_{{\hat\kappa}}]=\ell\big ] { { \mathtt p}_\omega}^{u_\ell}[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_n ) ] \\ & \leq \frac{c_{12}(n - m+1)}{h}\sum_{\ell\geq m } e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x\big[[{\hat\xi}_{{\hat\kappa}}]=\ell\big]\\ & \quad + \sum_{\frac{h}{16}\leq\ell < m } \frac{c_{12}\big((n - m+1)+(m-\ell)\big)}{h } e^{\zeta}{{\mathtt p}_{\omega,\zeta}}^x\big[[{\hat\xi}_{{\hat\kappa}}]=\ell\big]\\ & \leq \frac{c_{13}(n - m+1)}{h},\end{aligned}\ ] ] and this concludes the proof of lemma [ l_gamblers_ruin ] . next , we prove a result which shows that it is unlikely that a particle crosses the tube @xmath152 `` too quickly '' . suppose that one particle is injected ( uniformly ) at random at @xmath161 into the tube @xmath152 , and we still denote by @xmath194 the event that it crosses the tube without going back to @xmath161 , i.e. , @xmath437 ( one can see that there is no conflict with the notation of section [ s_perm ] ) . also , recall that @xmath199 stands for the total lifetime of the particle as defined in section [ s_perm ] , i.e. , if @xmath114 is the location of the particle at time @xmath116 , then @xmath438 . [ l_cross ] for any @xmath439 there exists ( large enough ) @xmath177 with the following property : there exists large enough @xmath440 such that for all @xmath441 @xmath442 \leq \frac{{\varepsilon}}{h}.\ ] ] _ proof . _ for @xmath443 , we say that @xmath56 is @xmath444-good if @xmath445 \geq 1-{\varepsilon}_1.\ ] ] let @xmath446 be a large positive parameter to be specified later ; for @xmath447 denote @xmath448 ; denote also @xmath449 now , consider first the case @xmath138 . from now on we suppose that @xmath177 is sufficiently large to assure the following : @xmath450 \geq 1-\frac{{\varepsilon}_1}{2},\ ] ] where @xmath451 is the standard brownian motion and @xmath452 is the corresponding probability measure . in this case , if the invariance principle holds , then for any fixed @xmath453 every @xmath7 is @xmath444-good for all large enough @xmath151 . using the monotone convergence theorem , it is straightforward to obtain that for fixed @xmath454 there exists large enough @xmath455 such that for all @xmath441 @xmath456 > \frac{3}{4}.\ ] ] then , by the ergodic theorem , there exists large enough @xmath455 such that for all @xmath441 there exists @xmath457 such that @xmath458 , and @xmath459 . now , let us consider also the event @xmath460 ( that is , with respect to the process @xmath461 , the particle enters @xmath462 before coming back to @xmath161 ) . then , write @xmath463 & \leq { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h , { { \mathcal t}}_h\leq m^{-1}h^2 ] + { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h^c,{{\mathfrak c}}_h ] \nonumber\\ & = { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h]{{\mathtt p}_\omega}^{{{\hat d}_\ell}}[{{\mathcal t}}_h\leq m^{-1}h^2\mid { \hat{{\mathfrak c}}}_h ] \nonumber\\ & \quad + { { \mathtt p}_\omega}^{{{\hat d}_\ell}}[{{\mathfrak c}}_h]{{\mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h^c\mid{{\mathfrak c}}_h ] . \label{cross_decompose}\end{aligned}\ ] ] now , by lemmas [ l_escape_upper ] and [ l_int_b ] , we can write for some @xmath464 @xmath465,{{\mathtt p}_\omega}^{{{\hat d}_\ell}}[{\hat{{\mathfrak c}}}_h]\ } \leq \frac{c_1}{h}.\ ] ] then , from we obtain that @xmath466 \leq \sup_{x\in{{\hat d}_r } } { { \mathtt p}_\omega}^x\big[\max_{j\leq n}\|\xi_j\cdot{\mathbf{e}}-h\|\geq h/4\big ] \leq c_2 h^{-(d-1)}\ ] ] for some @xmath467 . so , to complete the proof of , it remains to prove that the term @xmath468 $ ] in is small . to do this , let us recall that , by lemma [ l_hitting_revers ] ( i ) , for any @xmath469 , we have @xmath470 = \big({\nu^\omega}({{\hat d}_\ell}){{\mathtt p}_\omega}^{{{\hat d}_\ell}}[\hat{{\mathfrak c}}_h]\big)^{-1 } { \int\limits}_{f'}{{\mathtt p}_\omega}^y[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}^+(v_{ln_0})]\,d{\nu^\omega}(y).\ ] ] by lemma [ l_dirichlet_lower ] , we have that for some @xmath399 @xmath471\big)^{-1 } \leq c_3 h.\ ] ] for @xmath472 denote @xmath473 . using lemma [ l_gamblers_ruin ] , we can write for any @xmath474 @xmath475 & = { \int\limits}_{\partial\omega}{\hat k}(y , z ) { { \mathtt p}_\omega}^z[{\hat\tau}({{\hat d}_\ell})<{\hat\tau}(v_{ln_0})]\,d{\nu^\omega}(z ) \nonumber\\ & \leq { \int\limits}_{{{\hat d}_\ell}}{\hat k}(y , z)\,d{\nu^\omega}(z ) \nonumber\\ & \qquad + \sum_{j=1}^{ln_0 } \frac{{\tilde\gamma}_9(ln_0-j+1)}{h } { \int\limits}_{u_j}{\hat k}(y , z)\,d{\nu^\omega}(z)\nonumber\\ & \leq \frac{{\tilde\gamma}_9}{h}\sum_{j=1}^{ln_0 } { \int\limits}_{s_j}{\hat k}(y , z)\,d{\nu^\omega}(z ) . \label{upper_py}\end{aligned}\ ] ] so , by , in the case @xmath138 , we obtain from that for some positive constant @xmath476 @xmath477 \leq \frac{c_4}{h}\ ] ] and , by , , and the construction of @xmath478 we obtain that @xmath479 \leq c_3c_4{\varepsilon}_2.\ ] ] next , integrating over @xmath480 , we obtain from lemma [ l_separated ] that @xmath481\,d{\nu^\omega}(y ) & \leq \frac{{\tilde\gamma}_9}{h}\sum_{j=1}^{ln_0 } { \int\limits}_{v_{n_0}\setminus i_{n_0}}d{\nu^\omega}(y ) { \int\limits}_{s_j}d{\nu^\omega}(z){\hat k}(y , z)\\ & \leq \frac{c_5{\tilde\gamma}_9}{h } \sum_{j=1}^{ln_0 } ( ln_0+l - j)^{-(d-1)}\\ & \leq \frac{c_6}{h } l^{-(d-2)}.\end{aligned}\ ] ] again using , , we obtain that @xmath482 \leq c_3c_6 l^{-(d-2)}.\ ] ] so , and imply that for any @xmath483 there exists large enough @xmath214 such that for all large enough @xmath151 we have @xmath484 \geq 1-{\varepsilon}_3.\ ] ] but then , since all @xmath485 are @xmath444-good , from we obtain that @xmath486 \leq 1-(1-{\varepsilon}_1)(1-{\varepsilon}_3).\ ] ] using , , and in , we conclude the proof of in the case @xmath138 . let us prove the lemma in the case @xmath144 . take @xmath487 note that @xmath488 , so lemma [ l_escape_upper ] implies that @xmath489\leq c_7h^{-1}$ ] for some @xmath490 . by condition t ( iii ) we obtain that @xmath491 = 0,\ ] ] and , since for any @xmath492 , @xmath493 we have @xmath494 , we then obtain @xmath495 \geq 1-{\varepsilon}_4\ ] ] for a small @xmath496 . the proof of in the case @xmath144 then follows in the same way . in this section we prove the theorem that characterizes the stationary regime for the knudsen gas in a finite tube . _ proof of theorem [ t_stat_measure ] . _ in order to prove item ( i ) , we consider the process with absorbing / injection boundaries in _ both _ @xmath161 and @xmath171 ( that is , the injection is given by two independent poisson processes in @xmath497 and @xmath498 with intensities @xmath499 in both cases ) . fix a sequence of positive numbers @xmath500 such that @xmath501 for all @xmath70 . for each @xmath70 , consider a domain @xmath502 with the following properties * @xmath503 , @xmath504 , @xmath505 ; * @xmath506 ; * any segment @xmath507 with @xmath508 , @xmath509 has length at least @xmath510 ] ( one may construct such a domain e.g. as shown on figure [ f_reservoir ] ) . now , let us consider @xmath511 independent particles in @xmath502 . by theorem 2.4 of @xcite , the unique invariant measure of this system is product of uniform measures in location and direction . we are going to compare this process ( observed only on @xmath152 ) with the process with absorbing / injection boundaries in both @xmath161 and @xmath171 ( naturally , we assume that the injection is with the cosine law and with the same intensity mentioned in theorem [ t_stat_measure ] . let @xmath512 be the expectation for the above process in @xmath502 with @xmath511 particles , with respect to the invariant measure . also , we denote by @xmath513 the expectation with respect to the process with absorbing / injection boundaries in @xmath514 at time @xmath116 , with the initial configuration chosen from the poisson point process in @xmath185 with intensity @xmath186 . let @xmath515 be a function on @xmath187 , taking values on the interval @xmath516 $ ] . for a configuration @xmath517 in @xmath187 ( which means that we have @xmath518 particles with positions @xmath519 and vector speeds @xmath520 ) , write @xmath521 denote also by @xmath522 the mean value of @xmath515 on @xmath187 . clearly , we have @xmath523 also , it is straightforward to obtain that @xmath524 since , as @xmath525 , the binomial distribution with parameters @xmath511 and @xmath526 converges to the poisson distribution with parameter @xmath527 , for any @xmath515 we have @xmath528 now , let us fix @xmath529 and prove that for any @xmath439 @xmath530 for all large enough @xmath70 . for this , denote by @xmath531 the total number of particles which entered @xmath152 through the right boundary @xmath171 up to time @xmath529 . for the process with absorption / injection , an elementary calculation shows that @xmath531 has poisson distribution with parameter @xmath532 . let us suppose without restriction of generality that @xmath533 and denote @xmath534 \text { such that } x+vt\in{{\hat d}_r}\};\ ] ] observe that @xmath535 . now , a particle starting in @xmath536 with the direction @xmath124 will cross @xmath171 by time @xmath529 iff @xmath537 . so , it is straightforward to obtain that , for the process in @xmath502 , the random variable @xmath531 has the binomial distribution with parameters @xmath511 and @xmath538 as @xmath525 . then , conditioned on @xmath539 , for both processes the @xmath104 entering particles to @xmath171 ( seen as a point process on @xmath540 $ ] ) are independent , each having density @xmath541 . observe that the same considerations apply also to the particles which enter through @xmath161 . to obtain , we use now the following coupling argument . first of all , as we already know , the initial configurations restricted to @xmath152 for both processes can be successfully coupled with probability that converges to @xmath395 as @xmath525 . then , by the argument we just presented , the same applies for the process of particles entering through @xmath514 . this shows that , with large probability , both processes can be successfully coupled . now , combining with and using the fact that a point process is uniquely determined by its characteristic functional ( cf . e.g. section 5.5 of @xcite ) , we obtain that the poisson point process in @xmath185 with intensity @xmath186 is invariant for the knudsen gas with absorption / injection in @xmath184 . as for the convergence to the stationary state and the uniqueness , this follows from an easy coupling argument . indeed , consider one process starting from the invariant measure defined above , and another process starting from an arbitrary ( fixed ) configuration . the initial particles are independent , but the newly injected particles are the same for both processes . then , since any fixed particle will eventually disappear , the coupling time is a.s.finite , and so the system converges to the unique stationary state . ( using theorem 2.1 of @xcite , with some more work one can show that , for _ fixed _ tube , this convergence is exponentially fast ; however , we do not need this kind of result in the present paper . ) this concludes the proof of the part ( i ) . let us prove the part ( ii ) . still considering the process with absorption and injection in @xmath184 , suppose that the particles entering through @xmath171 are coloured red , and the particles entering through @xmath161 are coloured green . so , we need to compute the stationary measure for green particles . using the ( quasi ) reversibility of knudsen stochastic billiard ( see theorem 2.5 of @xcite ) , we obtain that , given that there is a particle in @xmath34 with the vector speed @xmath32 , the probability that it is green equals @xmath542.\ ] ] using also the part ( i ) , we obtain that , for the gas with injection only in @xmath161 , the stationary measure is that of poisson point process with intensity @xmath543 \,d\alpha\ , du\ , dh.\ ] ] note also that convergence and uniqueness follow from the same coupling argument as in part ( i ) . this concludes the proof of theorem [ t_stat_measure ] . let us observe also that theorem [ t_stat_measure ] allows us to characterize the stationary measure for knudsen gas where the injection takes place from both sides , but with different intensities ( which are constant on @xmath161 and @xmath171 ) . we have [ cor_different ] consider now knudsen gas with injection from both sides , with respective intensities @xmath162 and @xmath544 on @xmath161 and @xmath171 ( without restriction of generality , let us suppose that @xmath545 ) . then , a poisson point process with intensity measure @xmath546\big ) \,d\alpha\ , du\ , dh\ ] ] is the steady state of the knudsen gas . indeed , one may imagine that particles of type @xmath395 are injected from both sides with intensity @xmath544 and particles of type @xmath150 are injected only from the left with intensity @xmath547 , and use theorem [ t_stat_measure ] . for integers @xmath548 define @xmath549 let @xmath550 be a brownian motion with diffusion constant @xmath174 , starting from the origin ; we define ( being @xmath551 the expectation with respect to the probability measure on the space where the brownian motion is defined ) @xmath552 to be the probabilities of the corresponding events for this brownian motion . fix an integer @xmath177 . for @xmath553 and @xmath453 define @xmath554 intuitively , @xmath555 is the scaling factor one needs to use in order to assure that the rescaled ( and projected on @xmath556 ) trajectory of the knudsen stochastic billiard stays sufficiently close to the brownian motion . by the portmanteau theorem , observe that , if the knudsen stochastic billiard starting from @xmath557 satisfies the quenched invariance principle , this means that for any @xmath453 it holds that @xmath558 . since , for @xmath48-almost every @xmath51 , the invariance principle holds for a.a . starting points @xmath557 , we have @xmath559 by the monotone convergence theorem , we obtain that for all @xmath560 there exists @xmath561 such that @xmath562 so , using the ergodic theorem , we obtain for almost all @xmath51 and all @xmath151 large enough @xmath563 then , by theorem [ t_stat_measure ] , we can write @xmath564.\ ] ] now , let us prove that the rescaled density gradient is given by @xmath565 . _ proof of theorem [ t_dens_grad ] . _ fix an arbitrary @xmath176 and suppose that @xmath177 is a ( large ) integer . consider the quantity @xmath566 defined by , and suppose that @xmath567 is large enough to assure that ( recall ) @xmath568 abbreviate @xmath569 and consider any integer @xmath570 $ ] . suppose that @xmath571 , @xmath572 are such that @xmath573 $ ] , and @xmath574 . then , since @xmath575 , from we obtain that @xmath576 & \leq { { \mathtt p}_\omega}^{z , h}[\wp_{-(m - j)}({\hat z}^{(\phi)})<\wp_j({\hat z}^{(\phi ) } ) ] \nonumber\\ & = { { \mathtt e}_\omega}^{z , h } \big(1-r_{m - j , j}({\hat z}^{(\phi)})\big)\nonumber\\ & \leq \frac{j}{m } + m^{-2}\nonumber\\ & \leq \frac{j+1}{m},\label{upper_b_gambler}\end{aligned}\ ] ] and @xmath576 & \geq { { \mathtt p}_\omega}^{z , h}[\wp_{-(m - j+1)}({\hat z}^{(\phi)})<\wp_{j-1}({\hat z}^{(\phi ) } ) ] \nonumber\\ & = { { \mathtt e}_\omega}^{z , h } \big(1-r_{m - j+1,j-1}({\hat z}^{(\phi)})\big ) \nonumber\\ & \geq \frac{j-1}{m } - m^{-2}\nonumber\\ & \geq \frac{j-2}{m}. \label{lower_b_gambler}\end{aligned}\ ] ] also , by the ergodic theorem , we can choose @xmath151 large enough so that for all @xmath577 @xmath578 } \big\| -{\big\langle \|\omega_0\| \big\rangle_{\!{}_{{\mathbb p}}}}\big\| = \bigg\|\frac{m}{h}{\int\limits}_{\frac{(j-1)h}{m}}^{\frac{jh}{m } } - { \big\langle \|\omega_0\| \big\rangle_{\!{}_{{\mathbb p}}}}\bigg\| \leq m^{-1}.\ ] ] so , by , , , , @xmath579 analogously , using instead of , we obtain @xmath580 then , we obtain from and , and so the proof of theorem [ t_dens_grad ] is concluded . at this point , let us formulate an additional result which will be used in section [ s_pr_perm ] . [ p_1/6 ] define @xmath581\nonumber\\ & \qquad\qquad\qquad\qquad\qquad \times { { \mathtt p}_\omega}^{(\alpha , u),h}[\wp_{h-\alpha}(x\cdot{\mathbf{e } } ) < \wp_{-\alpha}(x\cdot{\mathbf{e } } ) ] , \label{def_m}\end{aligned}\ ] ] and suppose that the quenched invariance principle holds . then , for any @xmath176 there exists @xmath177 such that @xmath48-a.s . @xmath582 _ proof . _ the proof is quite analogous to the proof of theorem [ t_dens_grad ] . now , we calculate the limiting rescaled current . _ proof of theorem [ t_current ] . _ first , we obtain an upper and a lower bounds for @xmath583 , where @xmath584 . by e.g. the formula 1.2.0.2 of @xcite , we have @xmath585 = { \int\limits}_0^t\frac{\|a\|}{\sqrt{2\pi}{\hat\sigma}s^{3/2 } } \exp\big(-\frac{a^2}{2{\hat\sigma}^2s}\big)\ , ds.\ ] ] so , for @xmath586 @xmath587 - p[\wp_{-j}(b^{({\hat\sigma } ) } ) < \wp_i(b^{({\hat\sigma})})]\nonumber\\ & \geq -m^{-2/5 } + { \int\limits}_0^m\frac{i}{\sqrt{2\pi}{\hat\sigma}s^{3/2 } } \exp\big(-\frac{i^2}{2{\hat\sigma}^2s}\big)\ , ds . \label{gtil_lower}\end{aligned}\ ] ] also , for any @xmath588 , @xmath589\nonumber\\ & = { \int\limits}_0^m\frac{i}{\sqrt{2\pi}{\hat\sigma}s^{3/2 } } \exp\big(-\frac{i^2}{2{\hat\sigma}^2s}\big)\ , ds . \label{gtil_upper}\end{aligned}\ ] ] in particular , for @xmath590 , we obtain after some elementary computations that there exists a positive constant @xmath591 such that @xmath592 next , we employ the same strategy as in the proof of theorem [ t_dens_grad ] . fix a large @xmath177 , and suppose that @xmath593 is such that holds . now , let @xmath594 be the expected number of particles that were absorbed in @xmath161 up to time @xmath595 , in the stationary regime . clearly , we have then @xmath596 . so , one can write @xmath597\nonumber\\ & \qquad { } \times { { \mathtt p}_\omega}^{(\alpha , u),h}\big[\wp_{h-\alpha } ( x\cdot{\mathbf{e}})\leq \frac{h^2}{m } , \wp_{h-\alpha}(x\cdot{\mathbf{e } } ) < \wp_{-\alpha } ( x\cdot{\mathbf{e}})\big]\nonumber\\ & + { { \mathtt e}_\omega}{\widetilde w}_{h , m},\label{calc_current}\end{aligned}\ ] ] where @xmath598 is the mean number of particles that were injected in @xmath161 , successfully crossed the tube , and then hit @xmath171 before time @xmath595 . to obtain the corresponding upper bound , fix an arbitrary @xmath439 and suppose that @xmath177 is large enough so that of lemma [ l_cross ] holds for those @xmath610 . the term @xmath611 of can be estimated in the following way : @xmath612 \leq c_4 \frac{h^2}{m}\times\frac{{\varepsilon}}{h},\ ] ] so @xmath613 . then , analogously to , using also , we have for some @xmath614 @xmath615 now , observe that @xmath616 with this observation , theorem [ t_current ] follows from and . observe that , since the particles are independent , the knudsen gas in the finite tube @xmath193 can be regarded as a @xmath617 queueing system ; moreover , using e.g.theorem 2.1 of @xcite it is straightforward to obtain that the service time ( which is the lifetime of a newly injected particle ) is a random variable with exponential tail . then , let us recall the following basic identity of queuing theory ( known as little s theorem ) : [ prop_little ] suppose that @xmath618 is the arrival rate , @xmath619 is the mean number of customers in the system , and @xmath620 is the mean time a customer spends in the system , then @xmath621 . _ _ see e.g. section 5.2 of @xcite . to understand intuitively why this fact holds true , one may reason in the following way : by large time @xmath116 , the total time of all the customers in the system would be ( approximately ) @xmath622 on one hand , and @xmath623 on the other hand . _ proof of theorem [ t_exp_perm ] . _ this result almost immediately follows from theorem [ t_dens_grad ] by using proposition [ prop_little ] . first , for the gas of independent particles the arrival rate is @xmath624 recall that the particles are injected in @xmath195 only . then , from theorem [ t_dens_grad ] it is straightforward to obtain that for the mean number of particles @xmath625 in the system , we have @xmath626 then , proposition [ prop_little ] implies . to prove the corresponding annealed result , note that @xmath627 by theorem [ t_stat_measure ] ( ii ) . so , applying the bounded convergence theorem , we obtain . _ proof of theorem [ t_perm ] . _ first , observe that in the stationary regime the particles leave the system at the right boundary with rate @xmath628 , and this should be equal to the entrance rate @xmath629 $ ] of the particles which cross the tube , with @xmath618 from . so , follows from theorem [ t_current ] . to prove , observe that , by using lemma [ l_escape_upper ] with @xmath630 and @xmath631 , we obtain that for some positive constants @xmath632 which do not depend on @xmath51 @xmath633 \leq c_1 + \frac{c_2}{h } { \int\limits}_{{\tilde f}^\omega(0,h)}b(x)\ , d{\nu^\omega}(x).\ ] ] by , the collection of random variables @xmath634,h>1)$ ] is uniformly integrable , and this implies . in order to prove , denote by @xmath635 the mean number of particles in the stationary regime that _ will exit _ at @xmath171 . observe that , by theorem [ t_stat_measure ] ( ii ) and proposition [ p_1/6 ] , @xmath636 so , using and proposition [ prop_little ] , we obtain . the relations and follow from and . now , observe that and immediately follow from , , and , so now it remains only to prove . let @xmath637 , @xmath638 be the moments of successive visits to @xmath195 for the process in the finite tube . by corollary [ c_trans_dens ] , @xmath639 is uniformly distributed in @xmath195 for all @xmath70 , and so we can write @xmath640 \leq k{{\mathtt p}_\omega}[{{\mathfrak c}}_h].\ ] ] then , using , lemma [ l_dirichlet_lower ] , and the fact that the random variables @xmath641 are independent of everything , we obtain @xmath642\\ & \leq \sum_{k=1}^\infty { { \mathtt p}_\omega}[{\hat\tau}({{\hat d}_r})<{\hat\tau}({{\tilde d}_\ell } ) , \sigma_{k-1}<z_1+\cdots+z_{{\hat\tau}({{\hat d}_r})}<\sigma_k]\\ & \leq { { \mathtt p}_\omega}[z_1+\cdots+z_j\neq \sigma_\ell \text { for all $ \ell < k$ and all~$j$ } \mid \tau({{\hat d}_r})<\sigma_k]\\ & \qquad \times { { \mathtt p}_\omega}[\tau({{\hat d}_r})<\sigma_k]\\ & \leq { { \mathtt p}_\omega}[{{\mathfrak c}}_h ] \sum_{k=1}^\infty k(1-n^{-1})^{\lceil\frac{k-1}{n}\rceil},\end{aligned}\ ] ] and this implies that @xmath643\geq c_4/h$ ] for some @xmath644 not depending on @xmath51 . since @xmath645 , one obtains from the bounded convergence theorem . we thank takashi kumagai for pointing us reference @xcite . the work of f.c . was partially supported by cnrs ( umr 7599 `` probabilits et modles alatoires '' ) and anr polintbio . s.p.was partially supported by cnpq ( 300886/20080 ) . thanks dfg ( priority programme spp 1155 ) for financial support . the work of m.v . was partially supported by cnpq ( 304561/20061 ) . s.p . and m.v . also thank fapesp ( 2009/523798 ) , cnpq ( 471925/20063 , 472431/20099 ) , and capes / daad ( probral ) for financial support . f. comets , s. popov , g.m . schtz , m. vachkovskaia ( 2010 ) quenched invariance principle for knudsen stochastic billiard in random tube . to appear in : _ ann . cooper ( 1981 ) _ introduction to queueing theory _ ( 2nd ed . ) . north holland . s. zschiegner , s. russ , r. valiullin , m .- o . coppens , a .- dammers , a. bunde , j. krger ( 2008 ) normal and anomalous diffusion of non - interacting particles in linear nanopores . . phys . j. _ 161 * ( 109 ) .	we consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate ( the tube axis ) . inside the random tube , which is stationary and ergodic , non - interacting particles move straight with constant speed . upon hitting the tube walls , they are reflected randomly , according to the cosine law : the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector . steady state transport is studied by introducing an open tube segment as follows : we cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis . particles which leave this piece through the segment boundaries disappear from the system . through stationary injection of particles at one boundary of the segment a steady state with non - vanishing stationary particle current is maintained . we prove ( i ) that in the thermodynamic limit of an infinite open piece the coarse - grained density profile inside the segment is linear , and ( ii ) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube . thus we prove fick s law and equality of transport diffusion and self - diffusion coefficients for quite generic rough ( random ) tubes . we also study some properties of the crossing time and compute the milne extrapolation length in dependence on the shape of the random tube . + * keywords : * cosine law , knudsen random walk , random medium , self - diffusion coefficient , transport diffusion coefficient , random walk in random environment + * ams 2000 subject classifications : * 60k37 . secondary : 37d50 , 60j25 @xmath0universit paris 7 , ufr de mathmatiques , case 7012 , 2 , place jussieu , f75251 paris cedex 05 , france + e - mail : ` comets@math.jussieu.fr ` , url : ` http://www.proba.jussieu.fr/\simcomets ` @xmath1department of statistics , institute of mathematics , statistics and scientific computation , university of campinas unicamp , rua srgio buarque de holanda 651 , cep 13083859 , campinas sp , brazil + e - mails : ` popov@ime.unicamp.br ` , ` marinav@ime.unicamp.br ` + urls : ` http://www.ime.unicamp.br/\simpopov ` , ` http://www.ime.unicamp.br/\simmarinav ` @xmath2forschungszentrum jlich gmbh , institut fr festkrperforschung , d52425 jlich , deutschland + e - mail : ` g.schuetz@fz-juelich.de ` , + url : ` http://www.fz-juelich.de/iff/staff/schuetz_g/ `	31,320	797
microscopic _ ab - initio _ many - body approaches have significantly progressed in recent years @xcite . nowdays , due to increased computing power and novel techniques , _ ab - initio _ calculations are able to reproduce a large number of observables for atomic nuclei with mass up to a=14 . the light nuclei have also served as a crucial site to recognize the important role of three - body forces and three - body correlations . approaches like the no - core shell model ( ncsm ) @xcite , the green s function monte carlo ( gfmc ) @xcite and the coupled - cluster theory with single and double excitations ( ccsd ) @xcite can be formally extended for heavier nuclei . however , the explosive growth in computational power , required to achieve convergent results , severely hinders the detailed _ ab - initio _ studies of heavier , a@xmath3 , nuclei . in the case of the ncsm , the slow convergence of the calculated energies is caused by the adoption of a two - body cluster approximation , which does not take many - body correlations into account . straightforward employment of the three - body and higher - body interactions dramatically complicates the problem , even for light nuclei . an alternative approach is to construct a small - space effective two - body interaction , which would account for the many - body correlations for the a - body system in a large space . attempts to include many - body correlations approximately modifying the one - body part of the effective two - body hamiltonian and employing a unitary transformation have been reported recently @xcite . in this paper we derive a valence space ( @xmath1 ) effective two - body interaction that accounts for all the core - polarization effects available in the _ ab - initio _ ncsm wavefunctions . first , in the framework of the ncsm , we construct the effective hamiltonians on the two - body cluster level for a=6 systems in the @xmath4 space . @xmath5 represents the limit on the total oscillator quanta ( n ) above the minimum configuration . we take @xmath5 values from 2 to 12 . second , following the original idea of ref . @xcite , we employ an unitary many - body transformation and obtain the effective two - body hamiltonian in the @xmath6 space ( p - shell space ) , which exactly reproduces the lowest , @xmath6 space dominated , eigenstates of the 6-body hamiltonian in the large @xmath4 space . third , we perform ncsm calculations for a=4 and a=5 systems with the effective hamiltonian constructed on the two - body cluster level for the a=6 system and determine the core and one - body parts of the effective two - body hamiltonian for a=6 in the p - shell space . finally , the procedure is generalized for arbitrary mass number a. we analyze the properties of the constructed two - body hamiltonians , investigate their efficiency to reproduce the observables of different a - body systems calculated in large @xmath4 spaces and study the role of the effective p - shell space three - body interaction . the starting point of the no core shell model ( ncsm ) approach is the bare , exact a - body hamiltonian constrained by the harmonic oscillator ( ho ) potential @xcite : @xmath7 where @xmath8 is the one - body ho hamiltonian @xmath9 and @xmath10 is a bare nn interaction @xmath11 modified by the term introducing a- and @xmath12-dependent corrections to offset the ho potential present in @xmath8 : @xmath13 the eigenvalue problem for the exact a - body hamiltonian ( [ homa ] ) for @xmath14 is very complicated technically , since an extremely large a - body ho basis is required to obtain converged results . however , the @xmath15 problem is considerably simpler . for many realistic nn interactions its solution in the relative ho basis with @xmath16 accounts well for the short range correlations and is a precise approximation for the infinite space ( @xmath17 ) result . this allows one to adopt the two - body cluster approximation to construct the ncsm effective two - body hamiltonian @xmath18 for an a - body system in an @xmath19 space of tractable dimension , where the lower index @xmath20 stands for the number of particles in the cluster . this approximation consists of solving eq.([homa ] ) for the @xmath21 body subsystem of a leading to @xmath22 the information about the total number of interacting particles a enters the bare @xmath23 hamiltonian through the second term in the right hand side of ( [ voma ] ) . next , we find the unitary transformation @xmath24 which reduces the bare @xmath23 hamiltonian in the `` infinite space '' ( @xmath25 ) to the diagonal form : @xmath26 where , for the sake of simplicity , we omit the index @xmath27 for @xmath24 and keep only the index @xmath20 indicating the order of cluster approximation . the matrix @xmath24 can be split into 4 blocks : @xmath28 where the square @xmath29 @xmath30 matrix corresponds to the p - space ( or model space ) of dimension @xmath31 , characterized by the chosen @xmath5 value . taking into account that the @xmath32 matrix has a diagonal form @xmath33 one can calculate the effective @xmath34 hamiltonian using the following formula : @xmath35 it is easy to show by inserting eq.([homa2e ] ) into the eq.([heff1 ] ) , and taking into account eq.([udec ] ) that the unitary transformation ( [ heff1 ] ) is equivalent to the commonly used unitary transformation @xcite and that eq.([heff1 ] ) is identical to the eqs.(15,16 ) from @xcite . we note , that , by using eq.([heff1 ] ) one does not need to calculate and store a large number of matrix elements of the @xmath36-operator ( i.e. , @xmath37 ) . furthermore , the decoupling condition @xmath38 is automatically satisfied , which is obvious from the diagonal form of the @xmath32 matrix . we note that our treatment of center - of - mass motion remains the same as in the ncsm ( ref . we initiate all effective interaction developments at the a - body level , and , through a series of steps , arrive at a smaller space effective interaction appropriate for the a - body system . for this reason , our derived effective hamiltonians have their first subscript as `` a '' . the next step of the traditional ncsm prescription is to construct the full a - body hamiltonian using the effective two - body hamiltonian ( [ heff1 ] ) and to diagonalize it in the a - body @xmath5 model space . as we increase the number of nucleons , the dimension of the corresponding @xmath5 model space increases very rapidly . for instance , up - to - date computing resources allow us to go as high as @xmath39 for the lower part of the p - shell ( a=5,6 ) @xcite , while already for the upper part of the p - shell ( a@xmath4015 ) , we are limited to @xmath41 . the computational eigenvalue problem for many - body systems is complicated because of the very large matrix dimensions involved . the largest dimension of the model space that we encountered in this study for @xmath42li with @xmath43 exceeds @xmath44 . to solve this problem we have used the specialized version of the shell - model code antoine @xcite , recently adapted for the ncsm @xcite . in fact , the ncsm calculation for the a=6 system in the @xmath43 space yields nearly converged energies for the lowest states dominated by the @xmath45 components , while there is incomplete convergence for @xmath46 in @xmath41 space . therefore , considering the @xmath43 ncsm results as exact solutions for the lowest @xmath45 dominated 6-body states , we may construct the @xmath47 space hamiltonian for the a=6 system , which exactly reproduces those @xmath43 eigenvalues @xcite . moreover , if it is possible to solve the 6-body problem for a=6 , then it is possible to solve the 6-body problem for arbitrary a , using the corresponding effective hamiltonian @xmath34 obtained in the two - body cluster approximation . this means that we can determine for any a - body system the effective hamiltonian in the @xmath47 space , which accounts for 6-body cluster dynamics in the large @xmath43 space . to generalize , we start by defining the procedure for determining the effective hamiltonian matrix elements for the @xmath48-body cluster in the a - nucleon system . we do this by constructing the full @xmath48-body hamiltonian using the effective 2-body hamiltonian ( [ heff1 ] ) and diagonalizing it in the @xmath5 model space . in the spirit of eq.([homa2 ] ) , this yields the eigenenergies @xmath49 of the @xmath48-body system and their corresponding @xmath48 eigenvectors which make up the unitary transformation matrix @xmath50 . these @xmath48-body results can then be projected into a smaller , secondary @xmath51-space , given by @xmath52 with @xmath53 , where , similar to eqs.([udec ] ) and ( [ edec ] ) , @xmath49 and @xmath50 can be split into parts related to the two spaces , @xmath51 and @xmath54 , where @xmath55 . the new secondary effective hamiltonian then takes the following general form : @xmath56 where the @xmath12 superscript on the left - hand side is omitted for the sake of simplicity . as stated earlier , the new index @xmath48 determines the order of the cluster approximation in the smaller @xmath51 space , _ i.e. _ , @xmath53 . because the @xmath51 space has @xmath53 , the projection into this space `` freezes '' some number of the @xmath48 nucleons into fixed single particle configurations , which can be thought of as the `` inert core '' states in the standard shell model ( ssm ) approach . consequently , it is possible to write @xmath48 as @xmath57 , where @xmath58 is the number of nucleons making up the core configuration , while @xmath59 refers to the size of valence cluster . for instance , in the case of p - shell nuclei , @xmath60 , so , if @xmath61 ( _ i.e. _ the 5-body cluster approximation ) , then the effective hamiltonian @xmath62 is simply a one - body hamiltonian ( @xmath63 ) appropriate for the a - nucleon system . similarly , for the 6-body cluster approximation , _ i.e. _ , @xmath64 , we obtain the effective hamiltonian @xmath65 , which is a two - body hamiltonian ( @xmath66 ) for the a - body system , and , so on for larger values of @xmath48 . whatever the value of @xmath59 is , the effective hamiltonian @xmath67 contains the information about the @xmath48-body dynamics in the original large @xmath19 space , since it reproduces exactly the lowest @xmath68 eigenvalues @xmath69 of the @xmath48-body hamiltonian in the @xmath19 space , where @xmath68 is a dimension of the @xmath51 space . in the case of a doubly magic closed shell with two extra nucleons i.e. , @xmath70 , _ etc . _ , the dimension of the effective hamiltonian @xmath71 is a 2-body ( @xmath66 ) hamiltonian in the p- , sd- , pf - spaces , _ etc . _ , respectively . this means that the secondary effective hamiltonian ( [ heff2 ] ) contains only 1-body and 2-body terms , even after the _ exact _ a - body cluster transformation . this effective hamiltonian ( [ heff2 ] ) , which now contains the correlation energy of a nucleons , is the correct one - body plus two - body hamiltonian to use in a ssm calculation with inert core . the @xmath72 nucleon - spectators fully occupy the shells below the valence shell and the total a - body wave - function can be exactly factorized as the @xmath58-body `` core '' and the valence 2-body wave functions . this considerably simplifies the calculation of the effective hamiltonian , because only the @xmath6 part ( p@xmath73-space part ) of the complete @xmath4 wave function needs to be specified . utilizing the approach outlined above , we have calculated effective p - shell hamiltonians for @xmath74li , using the 6-body hamiltonians with @xmath75 and @xmath76 mev , constructed from the inoy ( inside nonlocal outside yukawa ) interaction @xcite . this is a new type of interaction , which has local behavior appropriate for traditional nn interactions at longer ranges , but exhibits a nonlocality at shorter distances . the nonlocality of the nn interaction has been introduced in order to account effectively for three - nucleon ( nnn ) interactions which correctly describe the nnn bound states @xmath77h and @xmath77he , whereas still reproducing nn scattering data with high precision . the corresponding excitation energies of p - shell dominated states and the binding energy of @xmath74li are shown in fig.[spectra6li14hw ] as a function of @xmath5 . the dimension of the configurational space for the @xmath43 case considered is 48 million ( m - scheme ) . a two orders of magnitude increase in the size of the model space , as compared to the previous @xmath78 study @xcite , allows us to determine a converged value of 31.681 mev for the @xmath42li binding energy . furthermore , the excitation energy of the highest lying p - space state , @xmath79 , is lowered by an amount of 2.1 mev in comparison to the @xmath78 case , indicating improved convergence for both the excited states and ground state for @xmath43 . in the ssm an effective two - body hamiltonian for a nucleus with mass number a is represented in terms of three components : @xmath80 where @xmath81 is the inert core part associated with the interaction of the nucleons occupying closed shells , @xmath82 is the one - body part corresponding to the interaction of valence nucleons with core nucleons , and @xmath83 is the two - body part referring to the interaction between valence particles . it is usually assumed that the core and one - body parts are constant for an arbitrary number of valence particles and that only the two - body part @xmath83 may contain mass dependence that includes effects of three - body and higher - body interactions . to represent the @xmath84 hamiltonian in the ssm format , we develop a valence cluster expansion ( vce ) , @xmath85 where the lower index , k , stands for the k - body interaction in the @xmath59-body valence cluster ( @xmath86 ) ; the first upper index a for the mass dependence ; and the second upper index , @xmath87 for the number of particles contributing to the corresponding k - body part . thus , we consider the more general case of allowing the core ( k=0 ) , one - body ( k=1 ) and other k - body parts to vary with the mass number a. this appears necessary to include the a - dependence of the excitations of the core ( @xmath58 ) nucleons treated in the original n@xmath88 basis space . for the a=6 case the two - body valence cluster ( 2bvc ) approximation is exact : @xmath89 where the core part , @xmath90 , is defined as the ground state @xmath91 energy of @xmath92he calculated in the @xmath19 space with the tbmes of the primary effective hamiltonian , @xmath93 for a=6 . then the one - body part , @xmath94,is determined as @xmath95 the tbmes of the one - body part , @xmath94 , @xmath96 may be represented in terms of single particle energies ( spe ) , @xmath97 : @xmath98 where the index a ( as well as b , c , and d ) denotes the set of single particle ho quantum numbers @xmath99 , upper index stands for proton ( p ) and neutron ( n ) , and the e(@xmath100li , j ) , e(@xmath100he , j ) are ncsm energies of the lowest @xmath101 and @xmath102 states calculated in the @xmath4 space for the @xmath103-body system using the tbmes of the @xmath104 effective hamiltonian , @xmath105 , which includes coulomb energy . finally , the two - body part @xmath106 is obtained by subtracting of two hamiltonians : @xmath107 it is worth noting that since the coulomb energy is included in the original hamiltonian , the proton - proton ( pp ) , neutron - neutron ( nn ) and proton - neutron ( pn ) @xmath108 tbmes of the two - body part , @xmath109 , have different values . the pn tbmes of the core , one - body and two - body parts of the expanded hamiltonian for @xmath42li are listed in the table i. [ cols="^,^,^,^,^,^ , > , > , > , > , > , > , > , > , > , > , > , > " , ] obtained results indicate that accounting for the effective 3-body interactions considerably improves the agreement with the exact ncsm for the @xmath110he , does not bring much change for @xmath111he and yields worse results for @xmath112he ( see fig.[he8 - 9 ] ) . performing a similar calculation with the effective interaction obtained in the 3bvc approximation starting from the cd - bonn interaction @xcite , we obtained results which are shown in fig.[he8 - 9cd ] . note , that the effective cd - bonn interaction constructed in the 2bvc approximation considerably underbinds the he isotopes in comparison to the exact ncsm results . the subsequent employment of the 3bvc approximation compensates these large differences and yields much better results for @xmath112he . however , to draw more quantitative conclusion about the 3-body and higher - body effective interactions , one needs to perform exact diagonalization using the 3bmes . we will evaluate this effect in future studies . within the ncsm approach we can calculate , by exact projection , full a - nucleon dependent tbmes ( and 3bmes ) . these a - dependent tbmes ( and 3bmes ) can be separated into core , one - body and two - body ( and three - body ) parts , all of which are also a - dependent , contrary to the ssm approach . when these a - dependent effective one- and two - body ( and three - body ) interactions are employed in ssm calculations , they exactly reproduce full ncsm calculations for a=6 ( a=7 ) isobars and yield results in good agreement with full ncsm calculations for @xmath113 performed in large basis spaces . our results for @xmath113 , which include the 3-body effective interaction , indicate that 3- and higher - body effective interactions may play an important role in determining their binding energies and spectra . future investigations will be extended to include effective 3-body interactions exactly and to explore other physical operators , such as transition operators and em moments . we thank the institute for nuclear theory at the university of washington for its hospitality and the department of energy for partial support during the development of this work . b.r.b . and a.f.l . acknowledge partial support of this work from nsf grants phy0244389 and phy0555396 ; p.n . acknowledges support in part by the u.s . doe / sc / np ( work proposal n. scw0498 ) and u.s department of energy grant de - fg02 - 87er40371 ; j.p.v . acknowledges support from u.s . department of energy grants de - fg02 - 87er40371 and de - fc02 - 07er41457 ; and the work of i.s . was performed under the auspices of the u.s . doe . prepared by llnl under contract de - ac52 - 07na27344 . thanks the gesellschaft fr schwerionenforschung mbh darmstadt , germany , for its hospitality during the preparation of this manuscript and the alexander von humboldt stiftung for its support . 100 p. navratil , v. g. gueorgiev , j.p . vary , w. e. ormand , and a. nogga , phys . . lett . * 99 * , 042501 ( 2007 ) . a. nogga , p.navratil , b.r.barrett , j.p.vary , phys . rev . c. * 73 * , 064002 ( 2006 ) . i. stetcu , b.r.barrett , p.navratil , j.p.vary , phys . rev . c. * 71 * , 044325 ( 2005 ) . p. navratil and w.e.ormand , phys . lett . * 88 * , 152502 ( 2002 ) ; phys . rev . c. * 68 * , 034305 ( 2003 ) . p. navratil , j.p.vary , b.r.barrett , phys . * 84 * , 5728 ( 2000 ) ; 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we will start by recalling from @xcite how to define intersection numbers in the algebraic setting . we will connect this with the natural topological idea of intersection number already discussed in the introduction . consider two simple closed curves @xmath1 and @xmath2 on a closed orientable surface @xmath0 . as in @xcite , it will be convenient to assume that @xmath1 and @xmath2 are shortest geodesics in some riemannian metric on @xmath0 so that they automatically intersect minimally . we will interpret the intersection number of @xmath1 and @xmath2 in suitable covers of @xmath28 , exactly as in @xcite and @xcite . let @xmath12 denote @xmath29 , let @xmath13 denote the infinite cyclic subgroup of @xmath12 carried by @xmath1 , and let @xmath30 denote the cover of @xmath0 with fundamental group equal to @xmath13 . then @xmath1 lifts to @xmath30 and we denote its lift by @xmath1 again . let @xmath31 denote the pre - image of this lift in the universal cover @xmath32 of @xmath0 . the full pre - image of @xmath1 in @xmath33 consists of disjoint lines which we call @xmath1lines , which are all translates of @xmath31 by the action of @xmath12 . ( note that in this paper groups act on the left on covering spaces . ) similarly , we define @xmath11 , @xmath34 , the line @xmath35 and @xmath2lines in @xmath32 . now we consider the images of the @xmath1lines in @xmath34 . each @xmath1line has image in @xmath34 which is a line or circle . then we define @xmath36 to be the number of images of @xmath1lines in @xmath37 which meet @xmath2 . similarly , we define @xmath38 to be the number of images of @xmath2lines in @xmath30 which meet @xmath1 . it is shown in @xcite , using the assumption that @xmath1 and @xmath2 are shortest closed geodesics , that each @xmath1line in @xmath34 crosses @xmath2 at most once , and similarly for @xmath2lines in @xmath30 . it follows that @xmath36 and @xmath38 are each equal to the number of points of @xmath39 , and so they are equal to each other . we need to take one further step in abstracting the idea of intersection number . as the stabiliser of @xmath31 is @xmath13 , the @xmath1lines naturally correspond to the cosets @xmath40 of @xmath13 in @xmath12 . hence the images of the @xmath1lines in @xmath41 naturally correspond to the double cosets @xmath42 . thus we can think of @xmath36 as the number of double cosets @xmath42 such that @xmath43 crosses @xmath35 . this is the idea which we generalise to define intersection numbers in a purely algebraic setting . first we need some terminology . two sets @xmath44 and @xmath45 are _ almost equal _ if their symmetric difference @xmath46 is finite . we write @xmath47 if a group @xmath12 acts on the right on a set @xmath48 , a subset @xmath44 of @xmath48 is _ almost invariant _ if @xmath49 for all @xmath50 in @xmath12 . an almost invariant subset @xmath44 of @xmath48 is _ non - trivial _ if @xmath44 and its complement @xmath51 are both infinite . the complement @xmath51 will be denoted simply by @xmath52 , when @xmath48 is clear from the context for finitely generated groups , these ideas are closely connected with the theory of ends of groups via the cayley graph @xmath53 of @xmath12 with respect to some finite generating set of @xmath12 . ( note that @xmath12 acts on its cayley graph on the left . ) using @xmath54 as coefficients , we can identify @xmath55cochains and @xmath3cochains on @xmath53 with sets of vertices or edges . a subset @xmath44 of @xmath12 represents a set of vertices of @xmath53 which we also denote by @xmath44 , and it is a beautiful fact , due to cohen @xcite , that @xmath56 is an almost invariant subset of @xmath12 if and only if @xmath57 is finite , where @xmath58 is the coboundary operator . now @xmath53 has more than one end if and only if there is an infinite subset @xmath44 of @xmath12 such that @xmath59 is finite and @xmath52 is also infinite . thus @xmath53 has more than one end if and only if @xmath12 contains a non - trivial almost invariant subset . if @xmath13 is a subgroup of @xmath12 , we let @xmath26 denote the set of cosets @xmath60 of @xmath13 in @xmath12 , ie , the quotient of @xmath12 by the left action of @xmath13 of course , @xmath12 will no longer act on the left on this quotient , but it will still act on the right . thus we also have the idea of an almost invariant subset of @xmath26 , and the graph @xmath61 has more than one end if and only if @xmath26 contains a non - trivial almost invariant subset . now the number of ends @xmath62 of @xmath12 is equal to the number of ends of @xmath53 , so it follows that @xmath63 if and only if @xmath64 contains a non - trivial almost invariant subset . similarly , the number of ends @xmath65 of the pair @xmath66 equals the number of ends of @xmath67 , so that @xmath68 if and only if @xmath26 contains a non - trivial almost invariant subset . now we return to the simple closed curves @xmath1 and @xmath2 on the surface @xmath0 . pick a generating set for @xmath12 which can be represented by a bouquet of circles embedded in @xmath0 . we will assume that the wedge point of the bouquet does not lie on @xmath1 or @xmath2 . the pre - image of this bouquet in @xmath32 will be a copy of the cayley graph @xmath53 of @xmath12 with respect to the chosen generating set . the pre - image in @xmath30 of the bouquet will be a copy of the graph @xmath61 , the quotient of @xmath53 by the action of @xmath13 on the left . consider the closed curve @xmath1 on @xmath30 . let @xmath44 denote the set of all vertices of @xmath61 which lie on one side of @xmath1 . then @xmath44 has finite coboundary , as @xmath57 equals exactly the edges of @xmath61 which cross @xmath1 . hence @xmath44 is an almost invariant subset of @xmath26 . let @xmath69 denote the pre - image of @xmath44 in @xmath70 , so that @xmath69 equals the set of vertices of @xmath53 which lie on one side of the line @xmath31 . now finally the connection between the earlier arguments and almost invariant sets can be given . for we can decide whether the lines @xmath31 and @xmath35 cross by considering instead the sets @xmath69 and @xmath71 . the lines @xmath31 and @xmath35 together divide @xmath12 into the four sets @xmath72 , @xmath73 , @xmath74 and @xmath75 , where @xmath76 denotes @xmath77 , and @xmath31 crosses @xmath35 if and only if each of these four sets projects to an infinite subset of @xmath78 now let @xmath12 be a group with subgroups @xmath13 and @xmath11 , let @xmath44 be a non - trivial almost invariant subset of @xmath26 and let @xmath45 be a non - trivial almost invariant subset of @xmath79 . we will define the intersection number @xmath80 of @xmath44 and @xmath45 . first we need to consider the analogues of the sets @xmath81and @xmath71 in the preceding paragraph , and to say what it means for them to cross . if @xmath12 is a group and @xmath13 is a subgroup , then a subset @xmath69 of @xmath12 is @xmath13__-almost invariant _ _ if @xmath69 is invariant under the left action of @xmath13 , and simultaneously @xmath82 is an almost invariant subset of @xmath83 . in addition , @xmath69 is a _ non - trivial _ @xmath13almost invariant subset of @xmath12 , if the quotient sets @xmath82 and @xmath84 are both infinite . note that if @xmath13 is trivial , then a @xmath13almost invariant subset of @xmath64 is the same as an almost invariant subset of @xmath12 . let @xmath69 be a @xmath13almost invariant subset of @xmath12 and let @xmath71 be a @xmath11almost invariant subset of @xmath12 . we will say that @xmath69 _ crosses _ @xmath71 if each of the four sets @xmath72 , @xmath85 , @xmath74 and @xmath86 projects to an infinite subset of @xmath78 we will often write @xmath87 instead of listing the four sets @xmath72 , @xmath85 , @xmath74 and @xmath88 if @xmath12 is a group and @xmath13 is a subgroup , then we will say that a subset @xmath89 of @xmath12 is @xmath13_finite _ if it is contained in the union of finitely many left cosets @xmath60 of @xmath13 in @xmath12 , and we will say that two subsets @xmath90 and @xmath89 of @xmath12 are @xmath13_almost equal _ if their symmetric difference is @xmath13finite . in this language , @xmath69 crosses @xmath71 if each of the four sets @xmath91 is not @xmath11finite . this definition of crossing is not symmetric , but it is shown in @xcite that if @xmath12 is a finitely generated group with subgroups @xmath13 and @xmath11 , and @xmath69 is a non - trivial @xmath13almost invariant subset of @xmath12 and @xmath71 is a non - trivial @xmath11almost invariant subset of @xmath12 , then @xmath69 crosses @xmath71 if and only if @xmath71 crosses @xmath69 . if @xmath69 and @xmath71 are both trivial , then neither can cross the other , so the above symmetry result is clear . however , this symmetry result fails if only one of @xmath69 or @xmath71 is trivial . this lack of symmetry will not concern us as we will only be interested in non - trivial almost invariant sets . now we come to the definition of the intersection number of two almost invariant sets . [ defnofintersectionnumber]let @xmath13 and @xmath11 be subgroups of a finitely generated group @xmath12 . let @xmath44 denote a non - trivial almost invariant subset of @xmath26 , let @xmath45 denote a non - trivial almost invariant subset of @xmath92 and let @xmath69 and @xmath71 denote the pre - images of @xmath44 and @xmath45 respectively in @xmath12 . then the intersection number @xmath80 of @xmath44 and @xmath45 equals the number of double cosets @xmath42 such that @xmath93 crosses @xmath94 [ almostequalsetshavesameintersectionnumber]the following facts about the intersection number are proved in @xcite . 1 . intersection numbers are symmetric , ie @xmath95 . @xmath80 is finite when @xmath12 , @xmath13 , and @xmath11 are all finitely generated . if @xmath96 is an almost invariant subset of @xmath26 which is almost equal to @xmath44 or to @xmath52 and if @xmath97 is an almost invariant subset of @xmath79 which is almost equal to @xmath45 or to @xmath98 , then @xmath99 we will often be interested in situations where @xmath69 and @xmath71 do not cross each other and neither do many of their translates . this means that one of the four sets @xmath87 is @xmath11finite , and similar statements hold for many translates of @xmath81and @xmath71 . if @xmath100 and @xmath101 do not cross , then one of the four sets @xmath102 is @xmath103finite , but probably not @xmath11finite . thus one needs to keep track of which translates of @xmath69 and @xmath71 are being considered in order to have the correct conjugate of @xmath11 , when formulating the condition that @xmath104 and @xmath90 do not cross . the following definition will be extremely convenient because it avoids this problem , thus greatly simplifying the discussion at certain points . let @xmath104 be a @xmath13almost invariant subset of @xmath12 and let @xmath90 be a @xmath11almost invariant subset of @xmath12 . we will say that @xmath105 is _ small _ if it is @xmath13finite . as the terminology is not symmetric in @xmath104 and @xmath90 and makes no reference to @xmath13 or @xmath11 , some justification is required . if @xmath104 is also @xmath106almost invariant for a subgroup @xmath106 of @xmath12 , then @xmath106 must be commensurable with @xmath13 . thus @xmath105 is @xmath13finite if and only if it is @xmath106finite . in addition , the fact that crossing is symmetric tells us that @xmath105 is @xmath13finite if and only if it is @xmath11finite . this provides the needed justification of our terminology . finally , the reader should be warned that this use of the word small has nothing to do with the term small group which means a group with no subgroups which are free of rank @xmath107 . at this point we have the machinery needed to define the intersection number of two splittings . this definition depends on the fact , which we recall from @xcite , that if a group @xmath12 has a splitting over a subgroup @xmath13 , there is a @xmath13almost invariant subset @xmath69 of @xmath12 associated to the splitting in a natural way . this is entirely clear from the topological point of view as follows . if @xmath108 , let @xmath109 denote a space with fundamental group @xmath12 constructed in the usual way as the union of @xmath110 , @xmath111 and @xmath112 . if @xmath113 , then @xmath109 is constructed from @xmath110 and @xmath112 only . now let @xmath114 denote the based cover of @xmath109 with fundamental group @xmath13 , and denote the based lift of @xmath115 into @xmath114 by @xmath112 . then @xmath69 corresponds to choosing one side of @xmath112 in @xmath114 . we now give a purely algebraic description of this choice of @xmath69 ( see @xcite for example ) . if @xmath108 , choose right transversals @xmath116 , @xmath117 of @xmath118 in @xmath119 , @xmath120 , both of which contain the identity element . ( a right transversal for a subgroup @xmath13 of a group @xmath12 consists of one representative element for each right coset @xmath40 of @xmath13 in @xmath121 each element of @xmath12 can be expressed uniquely in the form @xmath122 with @xmath123 , @xmath124 , @xmath125 , where only @xmath126 , @xmath127 and @xmath128 are allowed to be trivial . then @xmath69 consists of elements for which @xmath127 is non - trivial . in the case of a hnn extension @xmath18 , let @xmath129 , @xmath130 , @xmath107 , denote the two inclusions of @xmath13 in @xmath119 so that @xmath131 , and choose right transversals @xmath132 of @xmath133 in @xmath119 , both of which contain the identity element . each element of @xmath12 can be expressed uniquely in the form @xmath134 where @xmath135 lies in @xmath119 and , for @xmath136 , @xmath137 or @xmath138 , @xmath139 if @xmath137 , @xmath140 if @xmath141 and moreover @xmath142 if @xmath143 . in this case , @xmath69 consists of elements for which @xmath127 is trivial and @xmath144 . in both cases , the stabiliser of @xmath69 under the left action of @xmath12 is exactly @xmath13 and , for every @xmath145 , at least one of the four sets @xmath146 is empty . note that this is equivalent to asserting that one of the four inclusions @xmath147 , @xmath148 , @xmath149 , @xmath150 holds . the following terminology will be useful . a collection @xmath151 of subsets of @xmath12 which are closed under complementation is called _ nested _ if for any pair @xmath104 and @xmath90 of sets in the collection , one of the four sets @xmath102 is empty . if each element @xmath104 of @xmath151 is a @xmath152almost invariant subset of @xmath12 for some subgroup @xmath152 of @xmath12 , we will say that @xmath151 is _ _ almost nested _ _ if for any pair @xmath104 and @xmath90 of sets in the collection , one of the four sets @xmath102 is small . the above discussion shows that the translates of @xmath69 and @xmath153 under the left action of @xmath12 are nested . note that @xmath69 is not uniquely determined by the splitting . in both cases , we made choices of transversals , but it is easy to see that @xmath69 is independent of the choice of transversal . however , in the case when @xmath108 , we chose @xmath69 to consist of elements for which @xmath127 is non - trivial whereas we could equally well have reversed the roles of @xmath119 and @xmath120 . this would simply replace @xmath69 by @xmath154 . also either of these sets could be replaced by its complement . we will use the term _ standard almost invariant set _ for the images in @xmath26 of any one of @xmath69 , @xmath155 , @xmath153 , @xmath154 . in the case when @xmath113 , reversing the roles of the two inclusion maps of @xmath13 into @xmath119 also replaces @xmath69 by @xmath154 . again we have four standard almost invariant sets which are the images in @xmath83 of any one of @xmath69 , @xmath155 , @xmath153 , @xmath154 . there is a subtle point here . in the amalgamated free product case , we use the obvious isomorphism between @xmath17 and @xmath156 . in the hnn case , let us write @xmath157 to denote the group @xmath158 . then the correct isomorphism to use between @xmath159 and @xmath160 is not the identity on @xmath119 . instead it sends @xmath161 to @xmath162 and @xmath119 to @xmath163 . in all cases , we have four standard almost invariant subsets of @xmath164 [ defnofintersectionnumberofsplittings]if a group @xmath12 has splittings over subgroups @xmath13 and @xmath11 , and if @xmath44 and @xmath45 are standard almost invariant subsets of @xmath26 and @xmath79 respectively associated to these splittings , then the _ intersection number _ of this pair of splittings of @xmath12 is the intersection number of @xmath44 and @xmath165 [ xgisequivalenttox]as any two of the four standard almost invariant subsets of @xmath26 associated to a splitting of @xmath12 over @xmath13 are almost equal or almost complementary , remark [ almostequalsetshavesameintersectionnumber ] tells us that this definition does not depend on the choice of standard almost invariant subsets @xmath44 and @xmath166 . if @xmath69 and @xmath71 denote the pre - images in @xmath12 of @xmath167and @xmath45 respectively , and if we conjugate the first splitting by @xmath168 and the second by @xmath169 , then @xmath170 is replaced by @xmath171 and @xmath172is replaced by @xmath173 . now @xmath174 is @xmath13almost equal to @xmath69 and @xmath175 is @xmath11almost equal to @xmath71 , because of the general fact that for any subset @xmath89 of @xmath12 and any element @xmath50 of @xmath12 , the set @xmath176 lies in a @xmath31neighbourhood of @xmath89 , where @xmath31 equals the length of @xmath50 . this follows from the equations @xmath177 . it follows that the intersection number of a pair of splittings is unchanged if we replace them by conjugate splittings . now we can state two easy results about the case of zero intersection number . recall that if @xmath69 is one of the standard @xmath13almost invariant subsets of _ _ _ _ @xmath12 determined by a splitting of @xmath12 over @xmath13 , then the set of translates of @xmath81and @xmath153 is nested . it follows at once that the self - intersection number of @xmath82 is zero . also if two splittings of @xmath12 over subgroups @xmath13 and @xmath11 are compatible , and if @xmath170 and @xmath71 denote corresponding standard @xmath13almost and @xmath11almost invariant subsets of @xmath12 , then the set of all translates of @xmath69 , @xmath153 , @xmath71 , @xmath178 is also nested , so that the intersection number of the two splittings is zero . the next section is devoted to proving converses to each of these statements . before going further , we need to say a little more about splittings . recall from the introduction that a group @xmath12 is said to split over a subgroup @xmath13 if @xmath12 is isomorphic to @xmath18 or to @xmath17 , with @xmath179 . we will need a precise definition of a splitting . we will say that a _ splitting _ of @xmath12 consists either of proper subgroups @xmath119 and @xmath120 of @xmath12 and a subgroup @xmath13 of @xmath180 such that the natural map @xmath181 is an isomorphism , or it consists of a subgroup @xmath119 of @xmath182 and subgroups @xmath183 and @xmath184 of @xmath119 such that there is an element @xmath185 of @xmath12 which conjugates @xmath183 to @xmath184 and the natural map @xmath186 is an isomorphism . recall also that a collection of @xmath23 splittings of a group @xmath12 is _ compatible _ if @xmath12 can be expressed as the fundamental group of a graph of groups with @xmath23 edges , such that , for each @xmath24 , collapsing all edges but the @xmath24-th yields the @xmath24-th splitting of @xmath25 we note that if a splitting of a group @xmath12 over a subgroup @xmath13 is compatible with a conjugate of itself by some element @xmath50 of @xmath12 , then @xmath50 must lie in @xmath13 . this follows from a simple analysis of the possibilities . for example , if the splitting @xmath187 is compatible with its conjugate by some @xmath145 , then @xmath12 is the fundamental group of a graph of groups with two edges , which must be a tree , such that collapsing one edge yields the first splitting and collapsing the other yields its conjugate by @xmath50 . this means that each of the two extreme vertex groups of the tree must be one of @xmath119 , @xmath188 , @xmath120 or @xmath189 , and the same holds for the subgroup of @xmath12 generated by the two vertex groups of an edge . now it is easy to see that @xmath190 and @xmath191 , or the same inclusions hold with the roles of @xmath119 and @xmath120 reversed . in either case it follows that @xmath50 lies in @xmath13 as claimed . the case when @xmath192 is slightly different , but the conclusion is the same . this leads us to the following idea of equivalence of two splittings . we will say that two amalgamated free product splittings of @xmath12 are equivalent , if they are obtained from the same choice of subgroups @xmath119 , @xmath120 and @xmath13 of @xmath12 . this means that the splittings @xmath17 and @xmath156 of @xmath12 are equivalent . similarly , a splitting @xmath18 of @xmath12 is equivalent to the splitting obtained by interchanging the two subgroups @xmath183 and @xmath184 of @xmath119 . also we will say that any splitting of a group @xmath12 over a subgroup @xmath13 is equivalent to any conjugate by some element of @xmath13 . then the equivalence relation on all splittings of @xmath12 which this generates is the idea of equivalence which we will need . stated in this language , we see that if two splittings are compatible and conjugate , then they must be equivalent . note that two splittings of a group @xmath12 are equivalent if and only if they are over the same subgroup @xmath13 , and they have exactly the same four standard almost invariant sets . next we need to recall the connection between splittings of groups and actions on trees . serre theory , @xcite or @xcite , tells us that if a group @xmath12 splits over a subgroup @xmath13 , then @xmath12 acts without inversions on a tree @xmath193 , so that the quotient is a graph with a single edge and the vertex stabilisers are conjugate to @xmath119 or @xmath120 and the edge stabilisers are conjugate to @xmath13 . in his important paper @xcite , dunwoody gave a method for constructing such a @xmath12tree starting from the subset @xmath69 of @xmath12 defined above . the crucial property of @xmath170 which is needed for the construction is the nestedness of the set of translates of @xmath69 under the left action of @xmath12 . we recall dunwoody s result : [ dunwoodytreeconstruction]let @xmath151 be a partially ordered set equipped with an involution @xmath194 , where @xmath195 , such that the following conditions hold : 1 . if @xmath196 , @xmath197 and @xmath198 , then @xmath199 . 2 . if @xmath196 , @xmath197 , there are only finitely many @xmath200 such that @xmath201 . 3 . if @xmath196 , @xmath197 , at least one of the four relations @xmath198 , @xmath202 , @xmath203 , @xmath204 holds . if @xmath196 , @xmath197 , one can not have @xmath198 and @xmath205 then there is an abstract tree @xmath193 with edge set equal to @xmath151 such that the order relation which @xmath151 induces on the edge set of @xmath193 is equal to the order relation in which @xmath198 if and only if there is an oriented path in @xmath193 which begins with @xmath196 and ends with @xmath206 one applies this result to the set @xmath207 with the partial order given by inclusion and the involution by complementation . there is a natural action of @xmath12 on @xmath151 and hence on the tree @xmath193 . in most cases , @xmath12 acts on @xmath193 without inversions and we can recover the original decomposition from this action as follows . let @xmath196 denote the edge of @xmath193 determined by @xmath69 . then @xmath69 can be described as the set @xmath208 or @xmath209 . if the action of @xmath12 on @xmath193 has inversions , then the original splitting must have been an amalgamated free product decomposition @xmath210 , with @xmath13 of index @xmath107 in @xmath119 . in this case , subdividing the edges of @xmath193 yields a tree @xmath211 on which @xmath12 acts without inversions . if @xmath212 denotes the edge of @xmath211 contained in @xmath196 and containing the terminal vertex of @xmath196 , then @xmath69 can be described as the set @xmath213 or @xmath214 . now we will prove the following result . this implies part 2 ) of remark [ almostequalsetshavesameintersectionnumber ] . we give the proof here because the proof in @xcite is not complete , and we will need to apply the methods of proof later in this paper . [ finitenumberofdoublecosets]let @xmath12 be a finitely generated group with finitely generated subgroups @xmath13 and @xmath11 , a non - trivial @xmath13almost invariant subset @xmath69 and a non - trivial @xmath11almost invariant subset @xmath71 . then @xmath215 and @xmath71 are not nested } consists of a finite number of double cosets @xmath216 let @xmath53 denote the cayley graph of @xmath12 with respect to some finite generating set for @xmath12 . let @xmath44 denote the almost invariant subset @xmath217 of @xmath26 and let @xmath45 denote the almost invariant subset @xmath218 of @xmath79 . recall from the start of this section , that if we identify @xmath44 with the @xmath219cochain on @xmath220 whose support is @xmath44 , then @xmath44 is an almost invariant subset of @xmath83 if and only if @xmath57 is finite . thus @xmath57 is a finite collection of edges in @xmath61 and similarly @xmath221 is a finite collection of edges in @xmath222 . now let @xmath223 denote a finite connected subgraph of @xmath61 such that @xmath223 contains @xmath224 and the natural map @xmath225 is onto , and let @xmath151 denote a finite connected subgraph of @xmath222 such that @xmath151 contains @xmath221 and the natural map @xmath226 is onto . thus the pre - image @xmath227 of @xmath223 in @xmath53 is connected and contains @xmath228 , and the pre - image @xmath0 of @xmath151 in @xmath53 is connected and contains @xmath229 let @xmath230 denote a finite subgraph of @xmath227 which projects onto @xmath223 , and let @xmath231 denote a finite subgraph of @xmath0 which projects onto @xmath151 . if @xmath232 meets @xmath0 , there must be elements @xmath126 and @xmath4 in @xmath13 and @xmath11 such that @xmath233 meets @xmath234 . now @xmath235 meets @xmath236 is finite , as @xmath12 acts freely on @xmath53 . it follows that @xmath237 meets @xmath238 consists of a finite number of double cosets @xmath239 the result would now be trivial if @xmath69 and @xmath71 were each the vertex set of a connected subgraph of @xmath53 . as this need not be the case , we need to make a careful argument as in the proof of lemma 5.10 of @xcite . consider @xmath50 in @xmath12 such that @xmath232 and @xmath0 are disjoint . we will show that @xmath93 and @xmath71 are nested . as @xmath227 is connected , the vertex set of @xmath232 must lie entirely in @xmath71 or entirely in @xmath240 suppose that the vertex set of @xmath232 lies in @xmath71 . for a set @xmath2 of vertices of @xmath53 , let @xmath241 denote the maximal subgraph of @xmath53 with vertex set equal to @xmath2 . each component @xmath89 of @xmath242 and @xmath243 contains a vertex of @xmath227 . hence @xmath244 contains a vertex of @xmath232 and so must meet @xmath71 . if @xmath244 also meets @xmath178 , then it must meet @xmath0 . but as @xmath0 is connected and disjoint from @xmath232 , it lies in a single component @xmath244 . it follows that there is exactly one component @xmath244 of @xmath245 and @xmath246 which meets @xmath178 , so that we must have @xmath247 or @xmath248 . similarly , if @xmath232 lies in @xmath178 , we will find that @xmath249 or @xmath250 . it follows that in either case @xmath93 and @xmath71 are nested as required . in theorem 2.2 of @xcite , scott used dunwoody s theorem to prove a general splitting result in the context of surfaces in @xmath9manifolds . we will use the ideas in his proof a great deal . the following theorem is the natural generalisation of his result to our more general context and will be needed in the proofs of theorems [ disjointsplittings ] and [ splittingsexist ] . the first part of the theorem directly corresponds to the result proved in @xcite , and the second part is a simple generalisation which will be needed later . @xmath251 [ algebrafromtorustheorem ] 1 . let @xmath13 be a finitely generated subgroup of a finitely generated group @xmath12 . let @xmath69 be a non - trivial @xmath13almost invariant set in @xmath12 such that @xmath207 is almost nested and if two of the four sets @xmath252 are small , then at least one of them is empty . then @xmath12 splits over the stabilizer @xmath106 of @xmath69 and @xmath27 contains @xmath13 as a subgroup of finite index . further , one of the @xmath106almost invariant sets @xmath71 determined by the splitting is @xmath13almost equal to @xmath253 2 . let @xmath254 be finitely generated subgroups of a finitely generated group @xmath12 . let @xmath255 , @xmath256 , be a non - trivial @xmath257almost invariant set in @xmath12 such that @xmath258 is almost nested . suppose further that , for any pair of elements @xmath104 and @xmath90 of @xmath151 , if two of the four sets @xmath259 are small , then at least one of them is empty . then @xmath12 can be expressed as the fundamental group of a graph of groups whose @xmath24-th edge corresponds to a conjugate of a splitting of @xmath12 over the stabilizer @xmath260 of @xmath255 , and @xmath261 contains @xmath257 as a subgroup of finite index . further , for each @xmath24 , one of the @xmath261almost invariant sets determined by the @xmath24-th splitting is @xmath257almost equal to @xmath255 . most of the arguments needed to prove this theorem are contained in the proof of theorem 2.2 of @xcite , but in the context of @xmath9manifolds . we will present the proof of the first part of this theorem , and then briefly discuss the proof of the second part . the idea in the first part is to define a partial order on @xmath207 , which coincides with inclusion whenever possible . let @xmath104 and @xmath90 denote elements of @xmath151 . if @xmath262 is small , we want to define @xmath263 . there is a difficulty , which is what to do if @xmath104 and @xmath90 are distinct but @xmath264 and @xmath265 are both small . however , the assumption in the statement of theorem [ algebrafromtorustheorem ] is that if two of the four sets @xmath102 are small , then one of them is empty . thus , as in @xcite , we define @xmath263 if and only if @xmath262 is empty or the only small set of the four . note that if @xmath266 then @xmath263 . we will show that this definition yields a partial order on @xmath267 as usual , we let @xmath53 denote the cayley graph of @xmath12 with respect to some finite generating set . the distance between two points of @xmath12 is the usual one of minimal edge path length . our first step is the analogue of lemma 2.3 of @xcite . [ smallimpliesinboundednbhd]@xmath262 is small if and only if it lies in a bounded neighbourhood of each of @xmath268 , @xmath90 , @xmath269 as @xmath104 and @xmath90 are translates of @xmath69 or @xmath153 , it suffices to prove that @xmath270 is small if and only if it lies in a bounded neighbourhood of each of @xmath69 , @xmath153 , @xmath93 , @xmath271 . if @xmath272 is small , it projects to a finite subset of @xmath26 which therefore lies within a bounded neighbourhood of the image of @xmath273 . by lifting paths , we see that each point of @xmath270 lies in a bounded neighbourhood of @xmath273 , and hence lies in a bounded neighbourhood of @xmath69 and @xmath153 . by reversing the roles of @xmath93 and @xmath274 , we also see that @xmath270 lies in a bounded neighbourhood of each of @xmath93 and @xmath275 for the converse , suppose that @xmath270 lies in a bounded neighbourhood of each of @xmath69 and @xmath153 . then it must lie in a bounded neighbourhood of @xmath273 , so that its image in @xmath26 must lie in a bounded neighbourhood of the image of @xmath273 . as this image is finite , it follows that @xmath270 must be small , as required . now we can prove that our definition of @xmath276 yields a partial order on @xmath277 our proof is essentially the same as in lemma 2.4 of @xcite . [ partialorder]if a relation @xmath276 is defined on @xmath151 by the condition that @xmath263 if and only if @xmath262 is empty or the only small set of the four sets @xmath102 , then @xmath276 is a partial order . we need to show that @xmath276 is transitive and that if @xmath263 and @xmath278 then @xmath279 suppose first that @xmath263 and @xmath280 . the first inequality implies that @xmath262 is small and the second implies that @xmath265 is small , so that two of the four sets @xmath102 are small . the assumption of theorem [ algebrafromtorustheorem ] implies that one of these two sets must be empty . as @xmath263 , our definition of @xmath276 implies that @xmath262 is empty . similarly , the fact that @xmath280 tells us that @xmath265 is empty . this implies that @xmath281 as required . to prove transitivity , let @xmath104 , @xmath90 and @xmath89 be elements of @xmath151 such that @xmath282 . we must show that @xmath283 our first step is to show that @xmath284 is small . as @xmath285 and @xmath286 are small , we let @xmath287 be an upper bound for the distance of points of @xmath262 from @xmath90 and let @xmath288 be an upper bound for the distance of points of @xmath286 from @xmath89 . let @xmath289 be a point of @xmath284 . if @xmath289 lies in @xmath90 , then it lies in @xmath290 and so has distance at most @xmath288 from @xmath89 . otherwise , it must lie in @xmath262 and so have distance at most @xmath287 from some point @xmath291 of @xmath90 . if @xmath291 lies in @xmath89 , then @xmath289 has distance at most @xmath287 from @xmath89 . otherwise , @xmath291 lies in @xmath290 and so has distance at most @xmath288 from @xmath89 . in this case , @xmath289 has distance at most @xmath292 from @xmath89 . it follows that in all cases , @xmath293 has distance at most @xmath292 from @xmath89 , so that @xmath284 lies in a bounded neighbourhood of @xmath89 as required . as @xmath284 is contained in @xmath294 , it follows that it lies in bounded neighbourhoods of @xmath89 and @xmath294 , so that @xmath284 is small as required . the definition of @xmath276 now shows that @xmath295 , except possibly when two of the four sets @xmath296 are small . the only possibility is that @xmath297 and @xmath284 are both small . as one must be empty , either @xmath298 or @xmath299 . we conclude that if @xmath300 , then either @xmath295 or @xmath299 . now we consider two cases . first suppose that @xmath301 , so that either @xmath295 or @xmath302 . if @xmath299 , then @xmath303 , so that @xmath304 . as @xmath305 and @xmath306 , it follows from the first paragraph of the proof of this lemma that @xmath307 . hence , in either case , @xmath283 now consider the general situation when @xmath300 . again either @xmath308 or @xmath299 . if @xmath299 , then we have @xmath309 . now the preceding paragraph implies that @xmath304 . hence we again have @xmath305 and @xmath304 so that @xmath307 . hence @xmath295 still holds . this completes the proof of the lemma . next we need to verify that the set @xmath151 with the partial order which we have defined satisfies all the hypotheses of dunwoody s theorem [ dunwoodytreeconstruction ] . [ posatisfiesdunwoody]@xmath151 together with @xmath276 satisfies the following conditions . 1 . if @xmath104 , @xmath310 and @xmath263 , then @xmath311 . 2 . if @xmath104 , @xmath310 , there are only finitely many @xmath312 such that @xmath313 . 3 . if @xmath104 , @xmath310 , at least one of the four relations @xmath263 , @xmath314 , @xmath315 , @xmath316 holds . if @xmath104 , @xmath310 , one can not have @xmath263 and @xmath317 conditions ( 1 ) and ( 3 ) are obvious from the definition of @xmath276 and the hypotheses of theorem [ algebrafromtorustheorem ] . to prove ( 4 ) , we observe that if @xmath263 and @xmath318 , then @xmath319 and @xmath105 must both be small . this implies that @xmath104 itself is small , so that @xmath69 or @xmath153 must be small . but this contradicts the hypothesis that @xmath69 is a non - trivial @xmath13almost invariant subset of @xmath25 finally we prove condition ( 2 ) . let @xmath320 be an element of @xmath151 such that @xmath321 . recall that , as @xmath322 projects to a finite subset of @xmath83 , we know that @xmath322 lies in a @xmath323neighbourhood of @xmath69 , for some @xmath324 . if @xmath325 but @xmath48 is not contained in @xmath69 , then @xmath326 and @xmath69 are not nested . now lemma [ finitenumberofdoublecosets ] tells us that if @xmath48 is such a set , then @xmath50 belongs to one of only finitely many double cosets @xmath327 . it follows that if we consider all elements @xmath48 of @xmath151 such that @xmath325 , we will find either @xmath328 , or @xmath322 lies in a @xmath323neighbourhood of @xmath69 , for finitely many different values of @xmath329 hence there is @xmath330 such that if @xmath325 then @xmath48 lies in the @xmath331neighbourhood of @xmath69 . similarly , there is @xmath332 such that if @xmath333 , then @xmath48 lies in the @xmath288neighbourhood of @xmath334 let @xmath335 denote the larger of @xmath287 and @xmath288 . then for any elements @xmath104 and @xmath90 of @xmath151 with @xmath263 , the set @xmath262 lies in the @xmath323neighbourhood of each of @xmath104 , @xmath336 , @xmath90 and @xmath337 now suppose we are given @xmath263 and wish to prove condition ( 2 ) . choose a point @xmath338 in @xmath104 whose distance from @xmath336 is greater than @xmath323 , choose a point @xmath339 in @xmath340 whose distance from @xmath90 is greater than @xmath323 and choose a path @xmath1 in @xmath53 joining @xmath338 to @xmath339 . if @xmath341 , then @xmath342 must lie in @xmath48 and @xmath339 must lie in @xmath343 so that @xmath1 must meet @xmath344 . as @xmath1 is compact , the proof of lemma [ finitenumberofdoublecosets ] shows that the number of such @xmath48 is finite . this completes the proof of part 2 ) of the lemma . we are now in a position to prove theorem [ algebrafromtorustheorem ] . to prove the first part , we let @xmath151 denote the set of all translates of @xmath69 and @xmath153 by elements of @xmath12 , let @xmath345 be the involution on @xmath151 and let the relation @xmath276 be defined on @xmath151 by the condition that @xmath263 if @xmath262 is empty or the only small set of the four sets @xmath102 . lemmas [ partialorder ] and [ posatisfiesdunwoody ] show that @xmath276 is a partial order on @xmath151 and satisfies all of dunwoody s conditions ( 1)(4 ) . hence we can construct a tree @xmath193 from @xmath151 . as @xmath12 acts on @xmath151 , we have a natural action of @xmath12 on @xmath346 clearly , @xmath12 acts transitively on the edges of @xmath193 . if @xmath12 acts without inversions , then @xmath347 has a single edge and gives @xmath12 the structure of an amalgamated free product or hnn decomposition . the stabiliser of the edge of @xmath193 which corresponds to @xmath69 is the stabiliser @xmath27 of @xmath69 , so we obtain a splitting of @xmath12 over @xmath106 unless @xmath12 fixes a vertex of @xmath193 . note that as @xmath348 is finite , and @xmath106 preserves @xmath273 , it follows that @xmath106 contains @xmath13 with finite index as claimed in the theorem . if @xmath12 acts on @xmath193 with inversions , we simply subdivide each edge to obtain a new tree @xmath349 on which @xmath12 acts without inversions . in this case , the quotient @xmath350 again has one edge , but it has distinct vertices . the edge group is @xmath106 and one of the vertex groups contains @xmath106 with index two . as @xmath13 has infinite index in @xmath12 , it follows that in this case also we obtain a splitting of @xmath12 unless @xmath12 fixes a vertex of @xmath351 suppose that @xmath12 fixes a vertex @xmath339 of @xmath193 . as @xmath12 acts transitively on the edges of @xmath193 , every edge of @xmath193 must have one vertex at @xmath339 , so that all edges of @xmath193 are adjacent to each other . we will show that this can not occur . the key hypothesis here is that @xmath69 is non - trivial . let @xmath89 denote @xmath352 or @xmath353 , and note that condition 3 ) of lemma [ posatisfiesdunwoody ] shows that @xmath354 or @xmath355 . recall that there is @xmath330 such that if @xmath325 then @xmath48 lies in the @xmath287neighbourhood of @xmath69 . if @xmath323 denotes @xmath356 , and @xmath357 , it follows that @xmath358 lies in the @xmath323neighbourhood of @xmath69 . let @xmath359 denote the distance of the identity of @xmath12 from @xmath273 . then @xmath50 must lie within the @xmath360neighbourhood of @xmath69 , for all @xmath357 , so that @xmath89 itself lies in the @xmath360neighbourhood of @xmath69 . similarly , @xmath294 lies in the @xmath361neighbourhood of @xmath153 . now both @xmath69 and @xmath153 project to infinite subsets of @xmath26 , so @xmath12 can not equal @xmath89 or @xmath362 it follows that there are elements @xmath363and @xmath90 of @xmath151 such that @xmath364 , so that @xmath363and @xmath90 represent non - adjacent edges of @xmath193 . this completes the proof that @xmath12 can not fix a vertex of @xmath351 to prove the last statement of the first part of theorem [ algebrafromtorustheorem ] , we will simplify notation by supposing that the stabiliser @xmath106 of @xmath69 is equal to @xmath13 . one of the standard @xmath13almost invariant sets associated to the splitting we have obtained from the action of @xmath12 on the tree @xmath193 is the set @xmath89 in the preceding paragraph . we will show that @xmath89 is @xmath13almost equal to @xmath69 . the preceding paragraph shows that @xmath89 lies in the @xmath360neighbourhood of @xmath69 , and that @xmath294 lies in the @xmath360neighbourhood of @xmath153 . it follows that @xmath89 is @xmath13almost contained in @xmath69 and @xmath294 is @xmath13almost contained in @xmath365 , so that @xmath89 and @xmath69 are @xmath13almost equal as claimed . this completes the proof of the first part of theorem [ algebrafromtorustheorem ] . for the second part , we will simply comment on the modifications needed to the preceding proof . the statement of lemma [ smallimpliesinboundednbhd ] remains true though the proof needs a little modification . the statement and proof of lemma [ partialorder ] apply unchanged . the statement of lemma [ posatisfiesdunwoody ] remains true , though the proof needs some minor modifications . finally the proof of the first part of theorem [ algebrafromtorustheorem ] applies with minor modifications to show that @xmath12 acts on a tree @xmath193 with quotient consisting of @xmath4 edges in the required way . this completes the proof of theorem [ algebrafromtorustheorem ] . in this section , we prove our two main results about the case of zero intersection number . first we will need the following little result . [ bignormaliserimpliesnormal]let @xmath12 be a finitely generated group which splits over a subgroup @xmath13 . if the normaliser @xmath109 of @xmath13 in @xmath12 has finite index in @xmath12 , then @xmath13 is normal in @xmath25 the given splitting of @xmath12 over @xmath13 corresponds to an action of @xmath12 on a tree @xmath193 such that @xmath366has a single edge , and some edge of @xmath367has stabiliser @xmath13 . let @xmath368 denote the fixed set of @xmath13 , ie , the set of all points fixed by @xmath13 . then @xmath368 is a ( non - empty ) subtree of @xmath193 . as @xmath109 normalises @xmath13 , it must preserve @xmath368 , ie @xmath369 . suppose that @xmath370 . as @xmath109 has finite index in @xmath12 , we let @xmath371 denote a set of coset representatives for @xmath109 in @xmath12 , where @xmath372 . as @xmath12 acts transitively on @xmath193 , we have @xmath373 . edges of @xmath368 all have stabiliser @xmath13 , and so edges of @xmath374 all have stabiliser @xmath375 . as @xmath376 does not lie in @xmath109 , these stabilisers are distinct so the intersection @xmath377 contains no edges . the intersection of two subtrees of a tree must be empty or a tree , so it follows that @xmath377 is empty or a single vertex @xmath378 , for each @xmath24 . now @xmath109 preserves @xmath368 and permutes the translates @xmath374 , so @xmath379 preserves the collection of all the @xmath378 s . as this collection is finite , @xmath109 has a subgroup @xmath380 of finite index such that @xmath380 fixes a vertex @xmath339 of @xmath368 . as @xmath380 has finite index in @xmath12 , it follows that @xmath12 itself fixes some vertex of @xmath193 , which contradicts our assumption that our action of @xmath12 on @xmath193 corresponds to a splitting of @xmath12 . this contradiction shows that @xmath109 must equal @xmath12 , so that @xmath13 is normal in @xmath381as claimed . recall that if @xmath69 is a @xmath13almost invariant subset of @xmath12 associated to a splitting of @xmath12 , then the set of translates of @xmath69 and @xmath153 is nested . equivalently , for every @xmath145 , one of the four sets @xmath382 is empty . we need to consider carefully how it is possible for two of the four sets to be small , and a similar question arises when one considers two splittings of @xmath25 [ twosmallsets]let @xmath12 be a finitely generated group with two splittings over finitely generated subgroups @xmath13 and @xmath11 with associated @xmath13almost invariant subset @xmath69 of @xmath12 and associated @xmath11almost invariant subset @xmath71 of @xmath25 1 . if two of the four sets @xmath87 are small , then @xmath383 2 . if two of the four sets @xmath252 are small , then @xmath50 normalises @xmath384 our first step will be to show that @xmath13 and @xmath11 must be commensurable . without loss of generality , we can suppose that @xmath72 is small . the other small set can only be @xmath75 , as otherwise @xmath69 or @xmath385 would be small which is impossible . it follows that for each edge of @xmath386 , either it is also an edge of @xmath273 or it has ( at least ) one end in one of the two small sets . as the images in @xmath61 of @xmath387 and of each small set is finite , and as the graph @xmath53 is locally finite , it follows that the image of @xmath388 in @xmath67 must be finite . this implies that @xmath389 has finite index in the stabiliser @xmath11 of @xmath388 . by reversing the roles of @xmath13 and @xmath11 , it follows that @xmath389 has finite index in @xmath13 , so that @xmath13 and @xmath11 must be commensurable , as claimed . now let @xmath1 denote @xmath389 , so that @xmath1 stabilises both @xmath69 and @xmath71 , and consider the images @xmath44 and @xmath45 of @xmath69 and @xmath71 in @xmath390 . as @xmath391 has finite index in @xmath13 and @xmath11 , it follows that @xmath57 and @xmath221 are each finite , so that @xmath44 and @xmath45 are almost invariant subsets of @xmath392 . further , two of the four sets @xmath87 have finite image in @xmath390 , so we can assume that @xmath44 and @xmath45 are almost equal , by replacing one of @xmath69 or @xmath71 by its complement in @xmath12 , if needed . let @xmath5 denote the intersection of the conjugates of @xmath1 in @xmath13 , so that @xmath5 is normal in @xmath13 , though it need not be normal in @xmath11 . we do not have @xmath393 , but because @xmath1 has finite index in @xmath13 , we know that @xmath5 has finite index in @xmath13 and hence also in @xmath11 , which is all we need . let @xmath96 and @xmath97 denote the images of @xmath69 and @xmath71 respectively in @xmath394 , and consider the action of an element @xmath126 of @xmath13 on @xmath395 trivially @xmath396 . as @xmath96 and @xmath97 are almost equal , @xmath397 must be almost equal to @xmath398 . now we use the key fact that @xmath71 is associated to a splitting of @xmath12 so that its translates by @xmath12 are nested . thus for any element @xmath50 of @xmath12 , one of the following four inclusions holds : @xmath399 , @xmath400 , @xmath401 , @xmath402 . as @xmath403 is almost equal to @xmath97 , we must have @xmath404 or @xmath405 but @xmath126 has a power which lies in @xmath1 and hence stabilises @xmath71 . it follows that @xmath406 , so that @xmath126 lies in @xmath11 . thus @xmath118 is a subgroup of @xmath407 similarly , @xmath11 must be a subgroup of @xmath13 , so that @xmath408 . this completes the proof of part 1 of the lemma . note that it follows that @xmath409 , that @xmath410 and @xmath411 and that @xmath44 and @xmath412 are almost equal or almost complementary . in order to prove part 2 of the lemma , we apply the preceding work to the case when the second splitting is obtained from the first by conjugating by some element @xmath50 of @xmath12 . thus @xmath413 and @xmath414 which is @xmath11almost equal to @xmath93 by remark [ xgisequivalenttox ] . hence if two of the four sets @xmath252 are small , then so are two of the four sets @xmath87 small . now the above shows that @xmath415 , so that @xmath50 normalises @xmath13 . this completes the proof of the lemma . [ equalsplittings]let @xmath12 be a finitely generated group with two splittings over finitely generated subgroups @xmath13 and @xmath11 with associated @xmath13almost invariant subset @xmath69 of @xmath12 and associated @xmath11almost invariant subset @xmath71 of @xmath12 . if two of the four sets @xmath87 are small , then the two splittings of @xmath12 are conjugate . further one of the following holds : 1 . the two splittings are equivalent , or 2 . the two splittings are of the form @xmath416 , where @xmath13 has index @xmath107 in @xmath1 , and the splittings are conjugate by an element of @xmath1 , or 3 . @xmath13 is normal in @xmath12 and @xmath26 is isomorphic to @xmath417 or to @xmath418 the preceding lemma showed that the hypotheses imply that @xmath13 equals @xmath11 and also that the images @xmath44 and @xmath45 of @xmath69 and @xmath71 in @xmath26 are almost equal or almost complementary . by replacing one of @xmath69 or @xmath71 by its complement if needed , we can arrange that @xmath44 and @xmath45 are almost equal . we will show that in most cases , the two given splittings over @xmath13 and @xmath11 must be equivalent , and that the exceptional cases can be analysed separately to show that the splittings are conjugate . recall that by applying theorem [ dunwoodytreeconstruction ] , we can use information about @xmath69 and its translates to construct a @xmath12tree @xmath419 and hence the original splitting of @xmath12 over @xmath13 . similarly , we can use information about @xmath71 and its translates to construct a @xmath12tree @xmath420 and hence the original splitting of @xmath12 over @xmath11 . we will compare these two constructions in order to prove our result . as @xmath44 and @xmath45 are almost equal subsets of @xmath26 , it follows that there is @xmath421 such that , in the cayley graph @xmath53 of @xmath12 , we have @xmath69 lies in a @xmath58neighbourhood of @xmath71 and @xmath71 lies in a @xmath58neighbourhood of @xmath69 . now let @xmath422 denote one of @xmath69 or @xmath153 , let @xmath423 denote one of @xmath93 or @xmath271 and let @xmath424 and @xmath425 denote the corresponding sets obtained by replacing @xmath69 with @xmath71 . recall that @xmath426 is small if and only if its image in @xmath26 is finite . clearly this occurs if and only if @xmath423 lies in a @xmath58neighbourhood of @xmath427 , for some @xmath421 . it follows that @xmath426 is small if and only if @xmath428 is small . as @xmath69 and @xmath71 are associated to splittings , we know that for each @xmath145 , at least one of the four sets @xmath252 is empty and at least one of the four sets @xmath429 is empty . further the information about which of the four sets is empty completely determines the trees @xmath419 and @xmath420 . thus we would like to show that when we compare the four sets @xmath252 with the four sets @xmath430 , then corresponding sets are empty . note that when @xmath50 lies in @xmath13 , we have @xmath431 , so that two of the four sets @xmath382 are empty . first we consider the case when , for each @xmath432 , only one of the sets @xmath433 is small and hence empty . then only the corresponding one of the four sets @xmath429 is small and hence empty . now the correspondence @xmath434 gives a @xmath12isomorphism of @xmath419 with @xmath420 and thus the splittings are equivalent . next we consider the case when two of the sets @xmath252 are small , for some @xmath432 . part 2 of lemma [ twosmallsets ] implies that @xmath50 normalises @xmath13 . further if @xmath435 , then @xmath167is almost equal to @xmath436 or @xmath437 . let @xmath438 denote the normaliser of @xmath13 in @xmath12 , so that @xmath438 acts on the left on the graph @xmath61 and we have @xmath439 . let @xmath1 denote the subgroup of @xmath438 consisting of elements @xmath4 such that @xmath440 is almost equal to @xmath44 or @xmath52 . now we apply theorem 5.8 from @xcite to the action of @xmath441 on the left on the graph @xmath442 this result tells us that if @xmath443 is infinite , then it has an infinite cyclic subgroup of finite index . further the proof of this result in @xcite shows that the quotient of @xmath61 by @xmath441 must be finite . this implies that @xmath61 has two ends and that @xmath1 has finite index in @xmath12 . to summarise , either @xmath443 is finite , or it has two ends and @xmath1 has finite index in @xmath12 . let @xmath4 be an element of @xmath1 whose image in @xmath441 has finite order such that @xmath444 . as @xmath69 is associated to a splitting of @xmath64 , we must have @xmath445 or @xmath446 . as @xmath4 has finite order in @xmath443 , we have @xmath447 , for some positive integer @xmath23 , which implies that @xmath448 so that @xmath4 itself lies in @xmath13 . it follows that the group @xmath441 must be trivial , @xmath54 , @xmath417 or @xmath449 . in the first case , the two trees @xmath419 and @xmath420 will be @xmath12isomorphic , showing that the given splittings are equivalent . in the other three cases , @xmath450 is non - empty and we know that , for any @xmath451 , two of the four sets @xmath252 are small . thus in these cases , it seems possible that @xmath419 and @xmath420 will not be @xmath12isomorphic , so we need some special arguments . we start with the case when @xmath441 is @xmath54 . in this case , the given splitting must be an amalgamated free product of the form @xmath452 , for some group @xmath223 . if @xmath4 denotes an element of @xmath450 , then @xmath453 . thus @xmath12 acts on @xmath419 and @xmath420 with inversions . recall that either the two partial orders on the translates of @xmath170 and @xmath71 are the same under the bijection @xmath434 , or they differ only in that @xmath454 but @xmath455 , for all @xmath456 . if they differ , we replace the second splitting by its conjugate by some element @xmath456 , so that @xmath71 is replaced by @xmath457 and we replace @xmath69 by @xmath458 as @xmath459 is @xmath13almost equal to @xmath460 , the partial orders on the translates of @xmath460 and @xmath459 respectively are the same under the bijection @xmath461 except possibly when one compares @xmath460 , @xmath462 and @xmath459 , @xmath463 , where @xmath464 in this case , the inclusion @xmath454 tells us that @xmath465 , and the inclusion @xmath455 tells us that @xmath466 . we conclude that the partial orders on the translates of @xmath467 and @xmath459 respectively are exactly the same , so that @xmath468 and @xmath420 are @xmath12isomorphic , and the two given splittings are conjugate by an element of @xmath1 . now we turn to the two cases where @xmath441 is infinite , so that @xmath1 has finite index in @xmath12 and @xmath61 has two ends . as @xmath1 normalises @xmath13 , lemma [ bignormaliserimpliesnormal ] shows that @xmath13 is normal in @xmath12 . as @xmath61 has two ends , it follows that @xmath469 , so that @xmath26 is @xmath417 or @xmath470 . it is easy to check that there is only one splitting of @xmath417 over the trivial group and that all splittings of @xmath471 over the trivial group are conjugate . it follows that , in either case , all splittings of @xmath12 over @xmath13 are conjugate . this completes the proof of lemma [ equalsplittings ] . [ emptyorconjugate]let @xmath12 be a finitely generated group with two splittings over finitely generated subgroups @xmath13 and @xmath11 with associated @xmath13almost invariant subset @xmath69 of @xmath12 and associated @xmath11almost invariant subset @xmath71 of @xmath12 . let @xmath472 , and let @xmath473 and @xmath90 denote two elements of @xmath151 such that two of the four sets @xmath474 are small . then either one of the two sets is empty , or the two given splittings of @xmath12 are conjugate . recall that @xmath69 is associated to a splitting of @xmath12 over @xmath13 . it follows that @xmath93 is associated to the conjugate of this splitting by @xmath50 . thus @xmath104 and @xmath90 are associated to splittings of @xmath12 which are each conjugate to one of the two given splittings . if @xmath104 and @xmath90 are each translates of @xmath69 or @xmath274 , the nestedness of the translates of @xmath69 shows that one of the two small sets must be empty as claimed . similarly if both are translates of @xmath71 or @xmath178 , then one of the two small sets must be empty . if @xmath104 is a translate of @xmath69 or @xmath153 and @xmath90 is a translate of @xmath71 or @xmath475 , we apply lemma [ equalsplittings ] to show that the splittings to which @xmath104 and @xmath90 are associated are conjugate . it follows that the two original splittings were conjugate as required . now we come to the proof of our first main result . [ disjointsplittings]let @xmath12 be a finitely generated group with @xmath23 splittings over finitely generated subgroups . this collection of splittings is compatible up to conjugacy if and only if each pair of splittings has intersection number zero . further , in this situation , the graph of groups structure on @xmath12 obtained from these splittings has a unique underlying graph , and the edge and vertex groups are unique up to conjugacy . let the @xmath23 splittings @xmath476 of @xmath12 be over subgroups @xmath477 with associated @xmath257almost invariant subsets @xmath255 of @xmath12 , and let @xmath478 . we will start by supposing that no two of the @xmath476 s are conjugate . we will handle the general case at the end of this proof . we will apply the second part of theorem [ algebrafromtorustheorem ] to @xmath277 recall that our assumption that the @xmath476 s have intersection number zero implies that no translate of @xmath255 can cross any translate of @xmath479 , for @xmath480 . as each @xmath255 is associated to a splitting , it is also true that no translate of @xmath255 can cross any translate of @xmath481 . this means that the set @xmath151 is almost nested . in order to apply theorem [ algebrafromtorustheorem ] , we will also need to show that for any pair of elements @xmath104 and @xmath90 of @xmath151 , if two of the four sets @xmath259 are small then one is empty . now lemma [ emptyorconjugate ] shows that if two of these four sets are small , then either one is empty or there are distinct @xmath24 and @xmath482 such that @xmath476 and @xmath483 are conjugate . as we are assuming that no two of these splittings are conjugate , it follows that if two of the four sets @xmath484 are small then one is empty , as required . theorem [ algebrafromtorustheorem ] now implies that @xmath12 can be expressed as the fundamental group of a graph @xmath53 of groups whose @xmath24-th edge corresponds to a conjugate of a splitting of @xmath12 over the stabilizer @xmath260 of @xmath255 . as @xmath255 is associated to a splitting of @xmath12 over @xmath257 , its stabiliser @xmath261 must equal @xmath257 . further , it is clear from the construction that collapsing all but the @xmath24-th edge of @xmath53 yields a conjugate of @xmath476 , as the corresponding @xmath12tree has edges which correspond precisely to the translates of @xmath485 now suppose that we have a graph of groups structure @xmath486 for @xmath12 such that , for each @xmath24 , @xmath136 , collapsing all edges but the @xmath487-th yields a conjugate of the splitting @xmath476 of @xmath12 . this determines an action of @xmath12 on a tree @xmath368 without inversions . we want to show that @xmath193 and @xmath368 are @xmath12isomorphic . for this implies that @xmath488 and @xmath486 have the same underlying graph , and that corresponding edge and vertex groups are conjugate , as required . let @xmath196 denote an edge of @xmath368 , and let @xmath489 denote @xmath490 or @xmath491 . there are edges @xmath492 of @xmath368 , @xmath136 , such that the set @xmath493 of all translates of @xmath494 and @xmath495 is nested and dunwoody s construction applied to @xmath496 yields the @xmath12tree @xmath368 again . we will denote @xmath497 by @xmath498 . the hypotheses imply that there is @xmath499 such that the stabiliser @xmath500 of @xmath492 equals @xmath501 , and that @xmath498 is @xmath500almost equal to @xmath502 , where @xmath255 is one of the standard @xmath257almost invariant subsets of @xmath12 associated to the splitting @xmath476 . let @xmath503 denote @xmath504 so that @xmath503 is @xmath257almost equal to @xmath505 . now remark [ xgisequivalenttox ] shows that @xmath506 is @xmath257almost equal to @xmath255 , so that @xmath503 is @xmath257almost equal to @xmath255 . now consider the @xmath12equivariant bijection @xmath507 determined by sending @xmath255 to @xmath503 . the above argument shows that if @xmath473 is any element of @xmath151 , and @xmath508 is the corresponding element of @xmath493 , then @xmath363and @xmath508 are @xmath509almost equal . we will show that in most cases , this bijection automatically preserves the partial orders on @xmath151 and @xmath493 , implying that @xmath193 and @xmath368 are @xmath12isomorphic , as required . we compare the partial orders on @xmath151 and @xmath496 rather as in the proof of lemma [ equalsplittings ] . for any elements @xmath104 and @xmath90 of @xmath151 , let @xmath508 and @xmath510 denote the corresponding elements of @xmath493 . thus @xmath105 is small if and only if @xmath511 is small . we would like to show that when we compare the four sets @xmath102 with the four sets @xmath512 , then corresponding sets are empty , so that the partial orders are preserved by our bijection . otherwise , there must be @xmath104 and @xmath90 in @xmath151 such that two of the sets @xmath474 are small . if @xmath363and @xmath90 are translates of @xmath481 and @xmath479 , then lemma [ equalsplittings ] tells us that the splittings @xmath476 and @xmath513 are conjugate . as we are assuming that distinct splittings are not conjugate , it follows that @xmath514 . now the arguments in the proof of lemma [ equalsplittings ] show that either the splitting @xmath476 is an amalgamated free product of the form @xmath515 , with @xmath516 , or @xmath13 is normal in @xmath12 and @xmath26 is @xmath517 or @xmath518 . if the second case occurs , then there can be only one splitting in the given family , so it is immediate that @xmath53 and @xmath486 have the same underlying graph , and that corresponding edge and vertex groups are conjugate . if the first case occurs and the partial orders on translates of @xmath255 and @xmath503 do not match , we must have @xmath519 but @xmath520 , for all @xmath521 . we now pick @xmath521 and alter our bijection from @xmath151 to @xmath493 so that @xmath255 maps to @xmath522 and extend @xmath12equivariantly to the translates of @xmath481 and @xmath523 . this ensures that the partial orders on @xmath151 and @xmath496 match for translates of @xmath255 . by repeating this for other values of @xmath24 as necessary , we can arrange that the partial orders match completely , and can then conclude that @xmath193 and @xmath368 are @xmath12isomorphic as required . we end by discussing the case when some of the given @xmath23 splittings are conjugate . we divide the splittings into conjugacy classes and discard all except one splitting from each conjugacy class , to obtain @xmath4 splittings . now we apply the preceding argument to express @xmath12 uniquely as the fundamental group of a graph @xmath53 of groups with @xmath4 edges . if an edge of @xmath53 corresponds to a splitting over a subgroup @xmath13 which is conjugate to @xmath524 other splittings , we simply subdivide this edge into @xmath525 sub - edges , and label all the sub - edges and the @xmath524 new vertices by @xmath13 . this shows the existence of the required graph of groups structure @xmath526 corresponding to the original @xmath23 splittings . the uniqueness of @xmath527 follows from the uniqueness of @xmath53 , and the fact that the collection of all the edges of @xmath486 which correspond to a given splitting of @xmath12 must form an interval in @xmath486 in which all the interior vertices have valence @xmath107 . this completes the proof of theorem [ disjointsplittings ] . now we turn to the proof of theorem [ splittingsexist ] that splittings exist . it will be convenient to make the following definitions . we will use @xmath528 to denote @xmath529 if @xmath69 is a @xmath13almost invariant subset of @xmath381and @xmath71 is a @xmath11almost invariant subset of @xmath12 , and if @xmath69 and @xmath71 are @xmath13almost equal , then we will say that @xmath69 and @xmath71 are equivalent and write @xmath530 . ( note that @xmath13 and @xmath11 must be commensurable . ) if @xmath13 is a subgroup of a group @xmath12 , the commensuriser in @xmath12 of @xmath13 consists of those elements @xmath50 in @xmath12 such that @xmath13 and @xmath531 are commensurable subgroups of @xmath12 . the commensuriser is clearly a subgroup of @xmath182 and is denoted by @xmath532 or just @xmath533 , when the group @xmath12 is clear from the context . now we come to the proof of our second main result . [ splittingsexist]let @xmath12 be a finitely generated group with a finitely generated subgroup @xmath13 , such that @xmath534 . if there is a non - trivial @xmath118almost invariant subset @xmath69 of @xmath12 such that @xmath535 , then @xmath12 has a splitting over some subgroup @xmath106 commensurable with @xmath13 . further , one of the @xmath106almost invariant sets @xmath71 determined by the splitting is equivalent to @xmath253 this is the best possible result of this type , as it is clear that one can not expect to obtain a splitting over @xmath13 itself . for example , suppose that @xmath13 is carried by a proper power of a two - sided simple closed curve on a closed surface whose fundamental group is @xmath536 so that @xmath537 there are essentially only two non - trivial almost invariant subsets of @xmath538 each with vanishing self - intersection number , but there is no splitting of @xmath12 over @xmath384 the idea of the proof is much as before . we let @xmath44 denote the almost invariant subset @xmath82 of @xmath26 , and let @xmath151 denote @xmath539 . we want to apply the first part of theorem [ algebrafromtorustheorem ] . as before , the assumption that @xmath540 implies that @xmath151 is almost nested . however , in order to apply theorem [ algebrafromtorustheorem ] , we also need to know that for any pair of elements @xmath104 and @xmath90 of @xmath151 , if two of the four sets @xmath541 are small then one is empty . in the proof of theorem [ disjointsplittings ] , we simply applied lemma [ emptyorconjugate ] . however , here the situation is somewhat more complicated . lemma [ kcommensurises ] below shows that if @xmath542 and @xmath543 are both small , then @xmath50 must lie in a certain subgroup @xmath544 of @xmath545 . thus it would suffice to arrange that @xmath151 is nested with respect to @xmath544 , ie , that @xmath93 and @xmath69 are nested so long as @xmath50 lies in @xmath544 . now proposition [ almostnestedimpliesnested ] below tells us that there is a subgroup @xmath106 commensurable with @xmath13 and a @xmath106almost invariant set @xmath71 equivalent to @xmath69 such that @xmath546 is nested with respect to @xmath544 . it follows that if @xmath363and @xmath90 are any elements of @xmath493 and if @xmath319 and @xmath265 are both small , then one of them is empty . we also claim that , like @xmath151 , the set @xmath493 is almost nested . this means that if we let @xmath96 denote @xmath547 , we are claiming that @xmath548 . let @xmath549 denote @xmath550 . the fact that @xmath71 is equivalent to @xmath69 means that the pre - images in @xmath551 of @xmath44 and of @xmath552 are almost equal almost invariant sets which we denote by @xmath45 and @xmath97 . if @xmath323 denotes the index of @xmath106 in @xmath13 , then @xmath553 and similarly @xmath554 is an integral multiple of @xmath555 . as @xmath45 and @xmath97 are almost equal , it follows that @xmath556 , and hence that @xmath548 as claimed . this now allows us to apply theorem [ algebrafromtorustheorem ] to the set @xmath493 . we conclude that @xmath381splits over the stabiliser @xmath549 of @xmath71 , that @xmath549 contains @xmath106 with finite index and that one of the @xmath549almost invariant sets associated to the splitting is equivalent to @xmath460 . it follows that @xmath557 is commensurable with @xmath13 and that one of the @xmath549almost invariant sets determined by the splitting is equivalent to @xmath253 this completes the proof of theorem [ splittingsexist ] apart from the proofs of lemma [ kcommensurises ] and proposition [ almostnestedimpliesnested ] . it remains to prove the two results we just used . the proofs do not use the hypothesis that the set of all translates of @xmath69 and @xmath153 are almost nested . thus for the rest of this section , we will consider the following general situation . let @xmath12 be a finitely generated group with a finitely generated subgroup @xmath13 such that @xmath534 , and let @xmath69 denote a non - trivial @xmath13almost invariant subset of @xmath25 recall that our problem in the proof of theorem [ splittingsexist ] is the possibility that two of the four sets @xmath252 are small . as this would imply that @xmath558 or @xmath153 , it is clear that the subgroup @xmath544 of @xmath12 defined by @xmath559 or @xmath560 is very relevant to our problem . we will consider this subgroup carefully . here is the first result we quoted in the proof of theorem [ splittingsexist ] . [ kcommensurises]if @xmath559 or @xmath560 , then @xmath561 the first inclusion is clear . the second is proved in essentially the same way as the proof of the first part of lemma [ equalsplittings ] . let @xmath50 be an element of @xmath544 , and consider the case when @xmath558 ( the other case is similar ) . recall that this means that the sets @xmath562 and @xmath563 are both small . now for each edge of @xmath564 , either it is also an edge of @xmath273 or it has ( at least ) one end in one of the two small sets . as the images in @xmath61 of @xmath273 and of each small set is finite , and as the graph @xmath53 is locally finite , it follows that the image of @xmath564 in @xmath61 must be finite . this implies that @xmath565 has finite index in the stabiliser @xmath531 of @xmath564 . by reversing the roles of @xmath69 and @xmath93 , it follows that @xmath565 has finite index in @xmath13 , so that @xmath13 and @xmath566 must be commensurable , as claimed . it follows that @xmath567 , as required . another way of describing our difficulty in applying theorem [ algebrafromtorustheorem ] is to say that it is caused by the fact that the translates of @xmath69 and @xmath153 may not be nested . however , lemma [ finitenumberofdoublecosets ] assures us that `` most '' of the translates are nested . the following result gives us a much stronger finiteness result . [ finitelymanynonnested ] let @xmath12 , @xmath13 , @xmath69 , @xmath544 be as above . then @xmath568 and @xmath69 are not nested@xmath569 consists of a finite number of right cosets @xmath40 of @xmath13 in @xmath25 lemma [ finitenumberofdoublecosets ] tells us that the given set is contained in the union of a finite number of double cosets @xmath570 . if @xmath571 , we claim that the double coset @xmath327 is itself the union of only finitely many cosets @xmath40 , which proves the required result . to prove our claim , recall that @xmath572 is commensurable with @xmath13 . thus @xmath572 can be expressed as the union of cosets @xmath573 , for @xmath574 hence @xmath575 so that @xmath327 is the union of finitely many cosets @xmath40 as claimed . now we come to the key result . [ kisunionofnormalisers]let @xmath12 , @xmath13 , @xmath69 , @xmath544 be as above . then there are a finite number of finite index subgroups @xmath576 of @xmath13 , such that @xmath544 is contained in the union of the groups @xmath577 , @xmath578 , where @xmath579 denotes the normaliser of @xmath580 in @xmath25 consider an element @xmath50 in @xmath544 . lemma [ kcommensurises ] tells us that @xmath13 and @xmath531 are commensurable subgroups of @xmath12 . let @xmath1 denote their intersection and let @xmath5 denote the intersection of the conjugates of @xmath1 in @xmath13 . thus @xmath5 is of finite index in @xmath13 and @xmath528 and is normal in @xmath13 . now consider the quotient @xmath581 . let @xmath44 and @xmath45 denote the images of @xmath69 and @xmath93 respectively in @xmath582 . as before , @xmath44 and @xmath45 are almost invariant subsets of @xmath582 which are almost equal or almost complementary . now consider the action of @xmath583 on the left on @xmath582 . if @xmath126 is in @xmath13 , then @xmath584 , so that @xmath585 if @xmath586 and @xmath93 are nested , there are four possible inclusions , but the fact that @xmath587 excludes two of them . thus we must have @xmath588 or @xmath45 @xmath589 . this implies that @xmath590 as some power of @xmath126 lies in @xmath5 and so acts trivially on @xmath591 we conclude that if @xmath126 is an element of @xmath592 such that @xmath586 and @xmath93 are nested , then @xmath126 stabilises @xmath593 and so lies in @xmath531 . hence @xmath126 lies in @xmath1 . it follows that for each element @xmath126 of @xmath594 , the sets @xmath586 and @xmath93 are not nested . recall from lemma [ finitelymanynonnested ] that @xmath568 and @xmath69 are not nested@xmath569 consists of a finite number of cosets @xmath40 of @xmath13 in @xmath12 . it will be convenient to denote this number by @xmath595 . thus , for @xmath596 , the set @xmath597 and @xmath93 are not nested@xmath569 consists of @xmath595 cosets @xmath598 of @xmath531 in @xmath12 . it follows that @xmath594 lies in the union of @xmath595 cosets @xmath598 of @xmath531 in @xmath12 . as @xmath599 , it follows that @xmath594 lies in the union of @xmath595 cosets @xmath600 of @xmath1 in @xmath12 and hence that @xmath1 has index at most @xmath323 in @xmath384 a similar argument shows also that @xmath1 has index at most @xmath323 in @xmath531 . of course , the same bound applies to the index of @xmath601 in @xmath13 , for each @xmath24 . now we define @xmath602 clearly @xmath106 is a subgroup of @xmath13 which is normalised by @xmath50 . now each intersection @xmath601 has index at most @xmath323 in @xmath13 , and so @xmath603 is an intersection of subgroups of @xmath13 of index at most @xmath323 . if @xmath13 has @xmath23 subgroups of index at most @xmath323 , it follows that @xmath106 has index at most @xmath604 in @xmath13 . hence each element of @xmath544 normalises a subgroup of @xmath13 of index at most @xmath604 in @xmath13 . as @xmath13 has only finitely many such subgroups , we have proved that there are a finite number of finite index subgroups @xmath576 of @xmath13 , such that @xmath544 is contained in the union of the groups @xmath579 , @xmath578 , as required . using this result , we can prove the following . let @xmath12 , @xmath13 , @xmath69 , @xmath544 be as above . then there is a subgroup @xmath27 of finite index in @xmath13 , such that @xmath544 normalises @xmath605 we will consider how @xmath544 can intersect the normaliser of a subgroup of finite index in @xmath13 . let @xmath184 denote a subgroup of @xmath13 of finite index . we denote the image of @xmath69 in @xmath606 by @xmath44 . then @xmath44 is an almost invariant subset of @xmath606 . we consider the group @xmath607 , which we will denote by @xmath608 . then @xmath609 acts on the left on @xmath610 , and we have @xmath444 or @xmath52 , for every element @xmath4 of @xmath611 , because every element of @xmath544 satisfies @xmath612 or @xmath153 . now we apply theorem 5.8 from @xcite to the action of @xmath609 on the left on the graph @xmath613 . this result tells us that if @xmath609 is infinite , then it has an infinite cyclic subgroup of finite index . further the proof of this result in @xcite shows that the quotient of @xmath614 by @xmath609 must be finite . this implies that @xmath613 has two ends and that @xmath608 has finite index in @xmath12 . hence either @xmath609 is finite , or it has two ends and @xmath608 has finite index in @xmath25 recall that there are a finite number of finite index subgroups @xmath615 of @xmath13 , such that @xmath544 is contained in the union of the groups @xmath579 , @xmath578 . the above discussion shows that , for each @xmath24 , if @xmath616 denotes @xmath617 , either @xmath618 is finite , or it has two ends and @xmath619 has finite index in @xmath12 . we consider two cases depending on whether or not every @xmath618 is finite . suppose first that each @xmath618 is finite . we claim that @xmath544 contains @xmath13 with finite index . to see this , let @xmath620 , so that @xmath549 is a subgroup of @xmath13 of finite index , and note that @xmath544 is the union of a finite collection of groups @xmath616 each of which contains @xmath621 with finite index , so that @xmath544 is the union of finitely many cosets of @xmath549 . it follows that @xmath544 also contains @xmath549 with finite index and hence contains @xmath13 with finite index as claimed . if we let @xmath106 denote the intersection of the conjugates of @xmath13 in @xmath544 , then @xmath106 is the required subgroup of @xmath13 which is normalised by @xmath622 now we turn to the case when @xmath609 is infinite and so @xmath609 has two ends and @xmath608 has finite index in @xmath25 define @xmath106 to be @xmath623 . as @xmath544 contains @xmath608 with finite index , @xmath106 is the intersection of only finitely many conjugates of @xmath184 . as @xmath544 is contained in @xmath533 , each of these conjugates of @xmath184 is commensurable with @xmath184 . it follows that @xmath27 is a subgroup of @xmath13 of finite index in @xmath13 which is normalised by @xmath544 . this completes the proof of the lemma . the key point here is that @xmath544 normalises @xmath106 rather than just commensurises it . now we can prove the second result which we quoted in the proof of theorem [ splittingsexist ] . [ almostnestedimpliesnested]suppose that @xmath66 is a pair of finitely generated groups and that @xmath69 is a non - trivial @xmath13almost invariant subset of @xmath12 . then , there is a subgroup @xmath106 of @xmath12 which is commensurable with @xmath13 , and a non - trivial @xmath106almost invariant set @xmath71 equivalent to @xmath69 such that @xmath624 is nested with respect to the subgroup @xmath559 or @xmath560 of @xmath64 . the previous lemma tells us that there is a subgroup @xmath106 of finite index in @xmath13 such that @xmath544 normalises @xmath106 . let @xmath44 denote the almost invariant subset @xmath82 of @xmath26 , and let @xmath96 denote the almost invariant subset @xmath625 of @xmath626 suppose that the index of @xmath106 in @xmath544 is infinite . recall from the proof of the preceding lemma that @xmath627 has two ends and that @xmath544 has finite index in @xmath12 . we construct a new non - trivial @xmath549almost invariant set @xmath71 as follows . since the quotient group @xmath628 has two ends , @xmath629 splits over a subgroup @xmath549 which contains @xmath27 with finite index . thus there is a @xmath630almost invariant set @xmath631 in @xmath544 which is nested with respect to @xmath544 . further , @xmath549 is normal in @xmath629 and the quotient group must be isomorphic to @xmath417 or @xmath449 . let @xmath632 be coset representatives of @xmath544 in @xmath12 so that @xmath633 . we take @xmath634 . it is easy to check that @xmath71 is @xmath549almost invariant and that @xmath635 is nested with respect to @xmath622 now suppose that the index of @xmath106 in @xmath544 is finite . we will define the subgroup @xmath636 of @xmath637 . the index of @xmath638 in @xmath544 is at most two . first we consider the case when @xmath639 . we define @xmath640 to be the intersection of the translates of @xmath96 under the action of @xmath628 . thus @xmath641 is invariant under the action of @xmath627 . as all the translates of @xmath96 by elements of @xmath642 are almost equal to @xmath96 , it follows that @xmath643 so that @xmath644 is also an almost invariant subset of @xmath645 . let @xmath71 denote the inverse image of @xmath644 in @xmath12 , so that @xmath71 is invariant under the action of @xmath544 . in particular , @xmath624 is nested with respect to @xmath544 , as required . now we consider the general case when @xmath646 . we can apply the above arguments using @xmath638 in place of @xmath637 to obtain a subgroup @xmath549 of @xmath12 and a @xmath557almost invariant subset @xmath71 of @xmath12 which is equivalent to @xmath69 , and whose translates are nested with respect to @xmath638 . we also know that @xmath647 is @xmath638invariant . let @xmath45 denote the image of @xmath71 in @xmath648 , let @xmath4 denote an element of @xmath649 and consider the involution of @xmath650 induced by @xmath4 . then @xmath45 is a non - trivial almost invariant subset of @xmath648 and @xmath651 . define @xmath652 , so that @xmath653 and let @xmath48 denote the pre - image of @xmath436 in @xmath12 . we claim that the translates of @xmath48 and @xmath343 are nested with respect to @xmath544 . first we show that they are nested with respect to @xmath654 , by showing that @xmath655 is @xmath638invariant . for @xmath656 , we have @xmath657 as @xmath638 must be normal in @xmath544 . it follows that @xmath658 . as @xmath659 , we see that @xmath48 is @xmath638invariant as required . in order to show that the translates of @xmath48 and @xmath343 are nested with respect to @xmath544 , we will also show that @xmath660 is empty . this follows from the fact that @xmath661 which is clearly empty . this completes the proof of proposition [ almostnestedimpliesnested ] . let @xmath12 be a finitely generated group and let @xmath13 and @xmath11 be subgroups of @xmath12 . let @xmath69 be a non - trivial @xmath13almost invariant subset of @xmath12 and let @xmath71 be a non - trivial @xmath11almost invariant subset of @xmath25 in section 1 , we discussed what it means for @xmath69 to cross @xmath71 and the fact that this is symmetric . as mentioned in the introduction , there is an alternative way to define crossing of almost invariant sets . recall that , in section 1 , we introduced our definition of crossing by discussing curves on surfaces . thus it seems natural to discuss the crossing of @xmath69 and @xmath71 in terms of their boundaries . we call this strong crossing . however , this leads to an asymmetric intersection number . in this section , we define strong crossing and discuss its properties and some applications . we consider the cayley graph @xmath53 of @xmath12 with respect to a finite system of generators . we will usually assume that @xmath13 and @xmath11 are finitely generated though this does not seem necessary for most of the definitions below . we will also think of @xmath273 as a set of edges in @xmath53 or as a set of points in @xmath12 , where the set of points will simply be the collection of endpoints of all the edges of @xmath662 [ strongcrossingimpliescrossing]this definition is independent of the choice of generators for @xmath12 which is used to define @xmath53 . clearly , if @xmath665 crosses @xmath69 strongly , then @xmath71 crosses @xmath69 . strong crossing is not symmetric . for an example , one need only consider an essential two - sided simple closed curve @xmath2 on a compact surface @xmath0 which intersects a simple arc @xmath1 transversely in a single point . let @xmath12 denote @xmath666 , and let @xmath13 and @xmath11 respectively denote the subgroups of @xmath12 carried by @xmath2 and @xmath1 , so that @xmath13 is infinite cyclic and @xmath11 is trivial . then @xmath2 and @xmath1 each define a splitting of @xmath12 over @xmath10and @xmath11 respectively . let @xmath69 and @xmath71 denote associated standard @xmath13almost invariant and @xmath11almost invariant subsets of @xmath12 . these correspond to submanifolds of the universal cover of @xmath0 bounded respectively by a line @xmath667 lying above @xmath2 and by a compact interval @xmath668 lying above @xmath1 , such that @xmath669 meets @xmath668 transversely in a single point . clearly , @xmath69 crosses @xmath71 strongly but @xmath71 does not cross @xmath69 strongly . however , a strong intersection number can be defined as before . it is usually asymmetric , but we will be particularly interested in the case of self - intersection numbers when this asymmetry will not arise . the strong intersection number @xmath670 is defined to be the number of double cosets @xmath42 such that @xmath93 crosses @xmath71 strongly . in particular , @xmath671 if and only if at least one of @xmath672 and @xmath673 is @xmath13finite , for each @xmath145 . if @xmath35 and @xmath674 are splittings of a group @xmath12 over subgroups @xmath13 and @xmath11 , with associated almost invariant subsets @xmath69 and @xmath71 of @xmath12 , it is natural to say that @xmath35 crosses @xmath674 strongly if @xmath675 . it is easy to show that this is equivalent to the idea introduced by sela @xcite that @xmath35 is hyperbolic with respect to @xmath674 . remark [ strongcrossingimpliescrossing ] shows that @xmath676 . recall that theorem [ splittingsexist ] shows that if @xmath677 , then @xmath12 splits over a subgroup @xmath106 commensurable with @xmath13 . thus the vanishing of the strong self - intersection number may be considered as a first obstruction to splitting @xmath12 over some subgroup related to @xmath13 . we will show in corollary [ canmakenested ] that the vanishing of the strong self - intersection number has a nice algebraic formulation . this is that when @xmath678 vanishes , we can find a subgroup @xmath11 of @xmath12 , commensurable with @xmath13 , and a @xmath11almost invariant subset @xmath71 of @xmath12 which is nested with respect to @xmath679 . however , @xmath71 may be very different from @xmath69 . this leads to some splitting results when we place further restrictions on @xmath13 . [ conditionforsi=0]let @xmath12 be a finitely generated group with finitely generated subgroup @xmath13 , and let @xmath69 be a non - trivial @xmath13almost invariant subset of @xmath12 . then @xmath671 if and only if there is a subset @xmath71 of @xmath12 which is @xmath13almost equal to @xmath69 ( and hence @xmath13almost invariant ) such that @xmath680 . suppose that there exists a subset @xmath71 of @xmath12 which is @xmath13almost equal to @xmath170 , such that @xmath680 . we have @xmath681 as @xmath69 and @xmath71 are @xmath13almost equal . so , it is enough to show that for every @xmath145 , either @xmath682 or @xmath683 is @xmath13finite . suppose that @xmath684 . consider @xmath685 which is a union of a finite number of right cosets @xmath686 , @xmath136 . since @xmath684 , @xmath687 . for any @xmath123 , @xmath688 . thus @xmath689 is at a bounded distance from @xmath71 and hence @xmath683 has finite image in @xmath26 . similarly , if @xmath690 , @xmath691 projects to a finite set in @xmath26 . for the converse , suppose that @xmath671 and let @xmath692 denote the projection from @xmath12 to @xmath26 . by hypothesis , @xmath693 or @xmath694 is finite . the proof of lemma [ posatisfiesdunwoody ] tells us that there is a positive number @xmath323 such that , for every @xmath145 , the set @xmath695 is contained in a @xmath323neighbourhood of @xmath69 or @xmath153 . let @xmath696 , the @xmath323neighbourhood of @xmath69 and let @xmath697 . if @xmath684 and @xmath123 , then @xmath698 and thus @xmath699 . if @xmath684 and @xmath123 , then @xmath700 and thus @xmath701 . it only remains to show that @xmath71 is @xmath13almost equal to @xmath253 this is essentially shown in the third and fourth paragraphs of the proof of theorem [ algebrafromtorustheorem ] . we will say that a pair of finitely generated groups @xmath66 is of _ surface type _ if @xmath702 for every subgroup @xmath106 of finite index in @xmath13 and @xmath703 for every subgroup @xmath106 of infinite index in @xmath13 . this terminology is suggested by the dichotomy in @xcite . note that for such pairs any two non - trivial @xmath13almost invariant sets in @xmath12 are @xmath13almost equal or @xmath13almost complementary . we will see that for pairs of surface type , strong and ordinary intersection numbers are equal . [ crossingiffstrongcrossing]let @xmath66 be a pair of surface type , let @xmath704 be a non - trivial @xmath13almost invariant subset of @xmath12 and let @xmath71 be a non - trivial @xmath11almost invariant subset of @xmath12 for some subgroup @xmath11 of @xmath12 . then @xmath71 crosses @xmath69 if and only if @xmath71 crosses @xmath69 strongly . let @xmath53 be the cayley graph of @xmath12 with respect to a finite system of generators and let @xmath705 . as in the proof of lemma [ finitenumberofdoublecosets ] , for a set @xmath2 of vertices in a graph , we let @xmath706 denote the maximal subgraph with vertex set equal to @xmath2 . we will show that exactly one component of @xmath242 has infinite image in @xmath61 . note that @xmath707 has exactly one infinite component as @xmath61 has only two ends . let @xmath45 denote the set of vertices of the infinite component of @xmath707 and let @xmath89 denote the inverse image of @xmath45 in @xmath12 . if @xmath708 has components with vertex set @xmath22 , then we have @xmath709 . let @xmath1 denote the vertex set of a component of @xmath708 , and let @xmath710 be the stabilizer in @xmath13 of @xmath1 . since @xmath221 is finite , we see that @xmath711 is finite . hence @xmath712 has more than one end . now our hypothesis that @xmath66 is of surface type implies that @xmath710 has finite index in @xmath13 and thus @xmath713 is finite . if @xmath714 , we see that @xmath713 divides @xmath715 into at least three infinite components . thus @xmath716 and so @xmath708 is connected . the other components of @xmath242 have finite image in @xmath717 . similarly , exactly one component of @xmath718 has infinite image in @xmath61 . the same argument shows that for any finite subset @xmath227 of @xmath61 containing @xmath57 , the two infinite components of @xmath719 and@xmath720 have connected inverse images in @xmath70 . recall that if @xmath71 crosses @xmath69 strongly , then @xmath71 crosses @xmath69 . we will next show that if @xmath71 does not cross @xmath69 strongly , then @xmath71 does not cross @xmath69 . suppose that @xmath721 projects to a finite set in @xmath67 . take a compact set @xmath227 in @xmath61 large enough to contain @xmath721 and @xmath57 . by the argument above , if @xmath436 is the infinite component of @xmath719 , then its inverse image @xmath48 is connected and is contained in @xmath242 . any two points in @xmath48 can be connected by a path in @xmath48 and thus the path does not intersect @xmath388 . thus @xmath48 is contained in @xmath71 or @xmath178 . hence @xmath722 or @xmath723 is empty . suppose that @xmath724 is empty . then @xmath725 . since @xmath726 projects to a finite set , we see that @xmath727 projects to a finite set . similarly , if @xmath728 is empty , then @xmath729 projects to a finite set in @xmath83 . thus , we have shown that if @xmath721 projects to a finite set , then either @xmath727 or @xmath729 projects to finite set . thus @xmath71 does not cross @xmath69 . if @xmath66 and @xmath730 are both of surface type and @xmath69 is a non - trivial @xmath13almost invariant set in @xmath12 , and @xmath71 is a non - trivial @xmath11almost invariant set in @xmath12 then @xmath731 . in particular @xmath677 if and only if @xmath732 . let @xmath11 be a poincar duality group of dimension @xmath733 which is a subgroup of a poincar duality group @xmath12 of dimension @xmath23 . thus the pair @xmath730 is of surface type . in @xcite , kropholler and roller defined an obstruction @xmath734 to splitting @xmath12 over a subgroup commensurable with @xmath11 . their main result was that @xmath734 vanishes if and only if @xmath12 splits over a subgroup commensurable with @xmath11 . at an early stage in their proof , they showed that @xmath734 vanishes if and only if there is a @xmath11almost invariant subset @xmath71 of @xmath12 such that @xmath735 . starting from this point , proposition [ conditionforsi=0 ] , the above corollary and then theorem [ splittingsexist ] give an alternative proof of their splitting result . thus theorem [ splittingsexist ] may be considered as a generalization of their splitting theorem . we next reformulate in our language a conjecture of kropholler and roller @xcite : [ krconjecture]if @xmath12 is a finitely generated group with a finitely generated subgroup @xmath13 , and if @xmath69 is a non - trivial @xmath13almost invariant subset of @xmath12 such that @xmath671 , then @xmath12 splits over a subgroup commensurable with a subgroup of @xmath13 . note that theorem [ splittingsexist ] has a stronger hypothesis than this conjecture , namely the vanishing of the self - intersection number @xmath736 , rather than the vanishing of the strong self - intersection number , and it has a correspondingly stronger conclusion , namely that @xmath12 splits over a subgroup commensurable with @xmath13 itself . a key difference between the two statements is that , in the above conjecture , one does not expect the almost invariant set associated to the splitting of @xmath12 to be at all closely related to @xmath69 . dunwoody and roller proved this conjecture when @xmath13 is virtually polycyclic @xcite , and sageev @xcite proved it for quasiconvex subgroups of hyperbolic groups . the paper of dunwoody and roller @xcite contains information useful in the general case . the second step in their proof , which uses a theorem of bergman @xcite , proves the following result , stated in our language . ( there is an exposition of bergman s argument and parts of @xcite in the later versions of @xcite . ) let @xmath66 be a pair of finitely generated groups , and let @xmath69 be a @xmath13almost invariant subset of @xmath12 . if @xmath671 , then there is a subgroup @xmath106 commensurable with @xmath13 , and a non - trivial @xmath106almost invariant set @xmath71 with @xmath737 such that the set @xmath635 is almost nested with respect to @xmath738 . [ canmakenested]with the hypotheses of the above theorem we can choose @xmath27 and a non - trivial @xmath106almost invariant set @xmath71 with @xmath739 such that @xmath740 is almost nested with respect to @xmath532 and is nested with respect to the subgroup @xmath559 or @xmath741 of @xmath532 . if @xmath12 , @xmath13 are finitely generated groups with @xmath68 , if @xmath742 for every subgroup @xmath11 commensurable with a subgroup of infinite index in @xmath13 , and if @xmath69 is a @xmath13almost invariant subset of @xmath12 such that @xmath743 , then @xmath12 splits over a subgroup commensurable with @xmath13 . observe that corollary [ canmakenested ] shows that , by changing @xmath13 up to commensurability , and changing @xmath69 , we may assume that the translates of @xmath69 are almost nested with respect to @xmath532 and nested with respect to @xmath744 or @xmath560 . if we do not have almost nesting for all translates of @xmath69 , then there is @xmath50 outside @xmath532 such that none of @xmath252 is @xmath13finite . in particular , none of these sets is @xmath745finite . but these four sets are each invariant under @xmath565 and the fact that the strong intersection number vanishes shows that at least one of them has boundary which is @xmath745finite . since @xmath50 is not in @xmath532 , we have a contradiction to our hypothesis that @xmath742 with @xmath746 . this completes the proof . we note another application of groups of surface type which provides an approach to the algebraic torus theorem @xcite similar to ours in @xcite . we will omit a complete discussion of this approach , but will prove the following proposition to illustrate the ideas . [ bigcommensuriserimpliesvirtuallynormal]if @xmath66 is of surface type and if @xmath13 has infinite index in @xmath532 , then there is a subgroup @xmath27 of finite index in @xmath13 such that the normalizer @xmath747 of @xmath106 is of finite index in @xmath12 and @xmath748 is virtually infinite cyclic . in particular , if @xmath13 is virtually polycyclic , then @xmath12 is virtually polycyclic . let @xmath69 be a non - trivial @xmath13almost invariant subset of @xmath12 , let @xmath50 be an element of @xmath532 and let @xmath749 , so that @xmath172has stabiliser @xmath531 . let @xmath106 denote the intersection @xmath565 which has finite index in both @xmath13 and in @xmath531 because @xmath50 lies in @xmath532 . thus @xmath750 and @xmath547 are both almost invariant subsets of @xmath645 . as @xmath66 is of surface type , the pair @xmath751 has two ends so that @xmath752 and @xmath547 are almost equal or almost complementary . it follows that @xmath69 is @xmath13almost equal to @xmath71 or @xmath178 , ie , @xmath753 or @xmath754 . recall from lemma [ kcommensurises ] , that if @xmath629 denotes @xmath755 or @xmath756 , then @xmath757 . it follows that in our present situation @xmath629 must equal @xmath532 . by lemma [ kisunionofnormalisers ] , we see that there are a finite number of subgroups @xmath576 of finite index in @xmath13 such that @xmath544 is contained in the union of the normalizers @xmath579 . as @xmath13 has infinite index in @xmath758 , one of the @xmath257 , say @xmath184 , has infinite index in its normalizer @xmath759 . as @xmath66 is of surface type , the pair @xmath760 has two ends , so we can apply theorem 5.8 from @xcite to the action of @xmath761 on the left on the graph @xmath614 . this result tells us that @xmath761 is virtually infinite cyclic . further the proof of this result in @xcite shows that the quotient of @xmath613 by @xmath761 must be finite so that @xmath762 has finite index in @xmath12 . [ toomanyends ] let @xmath66 be a pair of finitely generated groups with @xmath118 virtually polycyclic and suppose that @xmath12 does not split over a subgroup commensurable with a subgroup of infinite index in @xmath13 . if for some subgroup @xmath11 of @xmath13 , @xmath763 , then @xmath12 splits over a subgroup commensurable with @xmath13 . we end this section with an interpretation of intersection numbers in the case when the strong and ordinary intersection numbers are equal . this corrects a mistake in @xcite . suppose that a group @xmath12 splits over subgroups @xmath13 and @xmath11 and let the corresponding @xmath13almost and @xmath11almost invariant subsets of @xmath12 be @xmath69 and @xmath71 . let @xmath193 denote the bass serre tree corresponding to the splitting of @xmath12 over @xmath11 and consider the action of @xmath13 on @xmath193 . let @xmath368 denote the minimal @xmath13invariant subtree of @xmath193 , and let @xmath764 denote the quotient graph @xmath765 similarly , we get a graph @xmath231 by considering the action of @xmath11 on the bass serre tree corresponding to the splitting of @xmath12 over @xmath13 . we have : the proof of theorem 3.1 of @xcite goes through because of our assumption that @xmath767 . the mistake in @xcite occurs in the proof of lemma 3.6 of @xcite where it is implicitly assumed that if @xmath69 crosses @xmath71 , then it crosses @xmath71 strongly . since we have assumed that the two intersection numbers are equal , the argument is now valid .	we prove algebraic analogues of the facts that a curve on a surface with self - intersection number zero is homotopic to a cover of a simple curve , and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint . in this paper , we will discuss an algebraic version of intersection numbers which was introduced by scott in @xcite . first we need to discuss intersection numbers in the topological setting . let @xmath0 denote a surface and let @xmath1 and @xmath2 each be a properly immersed two - sided circle or compact arc in @xmath0 . here ` properly ' means that the boundary of the @xmath3manifold lies in the boundary of @xmath0 . one can define the intersection number of @xmath1 and @xmath2 to be the least number of intersection points obtainable by homotoping @xmath1 and @xmath2 transverse to each other . ( the count is to be made without any signs attached to the intersection points . ) it is obvious that this number is symmetric in the sense that it is independent of the order of @xmath1 and @xmath2 . it is also obvious that @xmath1 and @xmath2 have intersection number zero if and only if they can be properly homotoped to be disjoint . it seems natural to define the self - intersection number of an immersed two - sided circle or arc @xmath1 in @xmath0 to be the least number of transverse intersection points obtainable by homotoping @xmath1 into general position . with this definition , @xmath1 has self - intersection number zero if and only if it is homotopic to an embedding . however , in light of later generalisations , it turns out that this definition should be modified a little in order to ensure that the self - intersection number of any cover of a simple closed curve is also zero . no modification is needed unless @xmath1 is a circle which can be homotoped to cover another immersion with degree greater than @xmath3 . in this case , suppose that the maximal degree of covering which can occur is @xmath4 and that @xmath1 covers @xmath5 with degree @xmath4 . then we define the self - intersection number of @xmath1 to be @xmath6 times the self - intersection number of @xmath5 . with this modified definition , @xmath7has self - intersection number zero if and only if it can be homotoped to cover an embedding . in @xcite , freedman , hass and scott introduced a notion of intersection number and self - intersection number for two - sided @xmath8injective immersions of compact surfaces into @xmath9manifolds which generalises the preceding ideas . their intersection number can not be described as simply as for curves on a surface , but it does share some important properties . in particular , it is a non - negative integer and it is symmetric , although this symmetry is not obvious from the definition . further , two surfaces have intersection number zero if and only if they can be homotoped to be disjoint , and a single surface has self - intersection number zero if and only if it can be homotoped to cover an embedding . these two facts are no longer obvious consequences of the definition , but are non - trivial applications of the theory of least area surfaces . in @xcite , scott extended the ideas of @xcite to define intersection numbers in a purely group theoretic setting . the details will be discussed in the first section of this paper , but we give an introduction to the ideas here . it seems clear that everything discussed in the preceding two paragraphs should have a purely algebraic interpretation in terms of fundamental groups of surfaces and @xmath9manifolds , and the aim is to find an interpretation which makes sense for any group . it seems natural to attempt to define the intersection number of two subgroups @xmath10and @xmath11 of a given group @xmath12 . this is exactly what the topological intersection number of simple closed curves on a surface does when @xmath12 is the fundamental group of a closed orientable surface and we restrict attention to infinite cyclic subgroups @xmath13 and @xmath11 . however , if one considers two simple arcs on a surface @xmath0 with boundary , they each carry the trivial subgroup of @xmath14 , whereas we know that some arcs have intersection number zero and others do not . thus intersection numbers are not determined simply by the groups involved . we need to look a little deeper in order to formulate the algebraic analogue . first we need to think a bit more about curves on surfaces . let @xmath1 be a simple arc or closed curve on an orientable surface @xmath0 , let @xmath12 denote @xmath15 and let @xmath13 denote the image of @xmath16 in @xmath12 . if @xmath1 separates @xmath0 then , in most cases , it gives @xmath12 the structure of an amalgamated free product @xmath17 , and if @xmath1 is non - separating , it gives @xmath12 the structure of a hnn extension @xmath18 . in order to avoid discussing which of these two structures @xmath12 has , it is convenient to say that a group @xmath12 _ splits over a subgroup _ @xmath13 if @xmath12 is isomorphic to @xmath18 or to @xmath17 , with @xmath19 . ( note that the condition that @xmath19 is needed as otherwise any group @xmath12 would split over any subgroup @xmath13 . for one can always write @xmath20 . ) thus , in most cases , @xmath1 determines a splitting of @xmath14 . usually one ignores base points , so that the splitting of @xmath12 is only determined up to conjugacy . in @xcite , scott defined the intersection number of two splittings of any group @xmath12 over any subgroups @xmath13 and @xmath11 . in the special case when @xmath12 is the fundamental group of a compact surface @xmath0 and these splittings arise from embedded arcs or circles on @xmath0 , the algebraic intersection number of the splittings equals the topological intersection number of the corresponding @xmath3manifolds . the analogous statement holds when @xmath12 is the fundamental group of a compact @xmath9manifold and these splittings arise from @xmath8injective embedded surfaces . in general , the algebraic intersection number shares some properties of the topological intersection number . algebraic intersection numbers are symmetric , and if @xmath12 , @xmath13 and @xmath11 are finitely generated , the intersection number of splittings of @xmath12 over @xmath13 and over @xmath11 is a non - negative integer . the first main result of this paper is a generalisation to the algebraic setting of the fact that two simple arcs or closed curves on a surface have intersection number zero if and only if they can be isotoped apart . of course , the idea of isotopy makes no sense in the algebraic setting , so we need some algebraic language to describe multiple disjoint curves on a surface . let @xmath21 be disjoint simple arcs or closed curves on a compact orientable surface @xmath0 with fundamental group @xmath12 , such that each @xmath22 determines a splitting of @xmath12 . together they determine a graph of groups structure on @xmath12 with @xmath23 edges . we say that a collection of @xmath23 splittings of a group @xmath12 is _ compatible _ if @xmath12 can be expressed as the fundamental group of a graph of groups with @xmath23 edges , such that , for each @xmath24 , collapsing all edges but the @xmath24-th yields the @xmath24-th splitting of @xmath25 we will say that the splittings are _ compatible up to conjugacy _ if collapsing all edges but the @xmath24-th yields a splitting of @xmath12 which is conjugate to the @xmath24-th given splitting . clearly disjoint essential simple arcs or closed curves on @xmath0 define splittings of @xmath12 which are compatible up to conjugacy . the precise statement we obtain is the following . * theorem [ disjointsplittings ] * so far , we have not discussed any algebraic analogue of non - embedded arcs or circles on surfaces . there is such an analogue which is the idea of an almost invariant subset of the quotient @xmath26 , where @xmath13 is a subgroup of @xmath12 . this generalises the idea of an immersed curve in a surface or of an immersed @xmath8injective surface in a 3manifold which carries the subgroup @xmath13 of @xmath12 . we give the definitions in section 1 . there is also an idea of intersection number of such things , which we give in definition [ defnofintersectionnumber ] . this too was introduced by scott in @xcite . our second main result , theorem [ splittingsexist ] , is an algebraic analogue of the fact that a singular curve on a surface or a singular surface in a @xmath9manifold which has self - intersection number zero can be homotoped to cover an embedding . it asserts that if @xmath26 has an almost invariant subset with self - intersection number zero , then @xmath12 has a splitting over a subgroup @xmath27 commensurable with @xmath13 . we leave the precise statement until section 2 . in a separate paper @xcite , we use the ideas about intersection numbers of splittings developed in @xcite and in this paper to study jsj decompositions of haken @xmath9manifolds . the problem there is to recognize which splittings of the fundamental group of such a manifold arise from the jsj decomposition ( see @xcite and @xcite ) . it turns out that a class of splittings which we call canonical can be defined using intersection numbers and we use this to show that the jsj decomposition for haken @xmath9manifolds depends only on the fundamental group . this leads to an algebraic proof of johannson deformation theorem . it seems very likely that similar ideas apply to sela s jsj decompositions @xcite of hyperbolic groups and thus provide a common thread to the two types of jsj decomposition . thus , the use of intersection numbers seems to provide a tool in the study of diverse topics in group theory and this paper together with @xcite provides some of the foundational material . this paper is organised as follows . in section 1 , we recall from @xcite the basic definitions of intersection numbers in the algebraic context . we also prove a technical result which was essentially proved by scott @xcite in 1980 . however , scott s results were all formulated in the context of surfaces in @xmath9manifolds , so we give a complete proof of the generalisation to the purely group theoretic context . section 2 is devoted to the proofs of our two main results discussed above . there is a second natural idea of intersection number , which we discuss in section 3 . we call it the strong intersection number . it is not symmetric in general , but this is not a problem when one is considering self - intersection numbers . we also discuss when the two kinds of intersection number are equal , which then forces the strong intersection number to be symmetric . we use these ideas to give a new approach to a result of kropholler and roller @xcite on splittings of poincar duality groups . we also discuss applications of our ideas to prove a special case of a conjecture of kropholler and roller @xcite on splittings of groups in general . we point out that these ideas lead to an alternative approach to the algebraic torus theorem @xcite . we end the section with a brief discussion of an error in @xcite . in section 3 of that paper , scott gave an incorrect interpretation of the intersection number of two splittings . his error was caused by confusing the ideas of strong and ordinary intersection . however , the arguments in @xcite work to give a nice interpretation of the intersection number in the case when it is equal to the strong intersection number . without this condition , finding nice interpretations of the two intersection numbers is an open problem .	35,290	3,204
in this paper , the novel statistical problem of deriving asymptotic results for nested random sequences of statistical descriptors for data in a non - euclidean space is considered . it can be viewed as a generalization of classical pca s asymptotics , e.g. by @xcite , where , as a consequence of pythagoras theorem , nestedness of approximating subspaces is trivially given and thus requires no special attention . for pca analogs for data in non - euclidean spaces , due to curvature , nestedness considerably complicates design of descriptors and , to the best knowledge of the authors , has hindered any asymptotic theory to date . for dimension reduction of non - euclidean data , _ procrustes analysis _ by @xcite and later _ principal geodesic analysis _ by @xcite are approaches to mimic pca on shape spaces and riemannian manifolds , respectively . both build on the concept of a frchet mean , a minimizer of expected squared distance , around which classical pca is conducted for the data mapped to a suitable tangent space . asymptotics for such means have been subsequently provided , among others , by @xcite , allowing for inferential methods such as two - sample tests . asymptotics for these _ tangent space pca _ methods , however , reflecting the _ forward nestedness _ due to random basepoints ( i.e. corresponding means ) of tangent spaces with random pcs therein , remain open to date . moreover , these tangent space pca methods are in no way canonical . not only may statistical outcomes depend on specific choices of tangent space coordinates , more severely , given curvature , no tangent space coordinates can correctly reflect mutual data distances . for this reason , among others , _ geodesic principal component analysis _ ( gpca ) has been introduced by @xcite , _ iterated frame bundle development _ by @xcite and _ barycentric subspaces _ by @xcite . as the following example teaches , nestedness may be lost . [ intro.ex ] consider data on a two - sphere that is confined to its equator and nearly uniformly spread out on it . then the best @xmath0 approximating geodesic is the equator and far away there are two ( due to symmetry ) intrinsic frchet means , each close to one of the poles , see @xcite . let us now detail our ideas , first by elucidating the following . * classical pca from a geometric perspective . * given data on @xmath1 , for every @xmath2 a unique affine subspace @xmath3 of dimension @xmath4 is determined by equivalently minimizing residual sums of squares or , among those containing the classical mean @xmath5 , maximizing the projected variance . also equivalently , these subspaces have representations as @xmath6 , the affine translates of spans from an eigenvector decomposition @xmath7 of the data s covariance matrix with descending eigenvalues . in consequence , one may either start from the zero dimensional mean and subsequently add most descriptive dimensions ( forward ) or start from the full dimensional space and remove least descriptive dimensions ( backward ) to obtain the same forward and backward nested sequence of subspaces @xmath8 for non - euclidean data , due to failure of pythagoras theorem , this canonical decomposition of data variance is no longer possible . for a detailed discussion see @xcite . * nestedness of non - euclidean pca * is highly desirable , when due to curvature and data spread , intrinsic frchet means are away from the data . for instance in example [ intro.ex ] , in order to have a mean on the equator , also in this case , @xcite devised _ principal arc analysis _ with the _ backward nested mean _ confined to the best approximating circle . this method and its generalization _ backward nested sphere analysis _ ( pns ) by @xcite give a tool for descriptive shape analysis that often strikingly outperforms tangent space pca , e.g. @xcite . here , the data space is a unit sphere @xmath9 of dimension @xmath10 , say , and in each of the @xmath3 is a @xmath4-dimensional ( small ) subsphere for pns and for _ principal nested great spheres _ ( pngs ) it is a @xmath4-dimensional great subsphere . in passing we note that pns is _ higher dimensional _ in the sense of having higher dimensional descriptor spaces than classical pca and pngs which are equally high dimensional , cf . @xcite . to date , however , there is no asymptotic theory for pns available , in particular there are no inferential tools for backward nested means , say . asymptotic results for non - data space valued descriptors , geodesics , say , are only available for single descriptors ( cf . @xcite ) that are directly defined as minimizers , not indirectly as a nested sequence of minimizers . * challenges for and results of this paper . * it is the objective of this paper to close this gap by providing asymptotic results for rather general random _ backward nested families of descriptors _ ( bnfds ) on rather general spaces . the challenge here is that random objects that are constrained by other random objects are to be investigated , requiring an elaborate setup . into this setup , we translate strong consistency arguments of @xcite and @xcite , and introducing a _ constrained _ m - estimation technique , we show joint asymptotic normality of an entire bnfd . in the special case of nested subspaces , bnfds may terminate at any dimension and @xmath11 is not required . as we minimize a functional under the constraining conditions that other functionals are minimized as well , our approach can be called _ constrained m - estimation_. in the literature , this term _ constrained m - estimation _ has been independently introduced by @xcite who robustify m - estimators by introducing constraining conditions and by @xcite , who consider m - estimators that are confined to closed subsets of a euclidean space with specifically regular boundaries . it seems that our m - estimation problem , which is constrained to satisfying other m - estimation problems has not been dealt with before . we solve it using a random lagrange multiplier approach . furthermore , in order to obtain asymptotic normality of each single sequence element , in particular for the last , we require the rather technical concept of _ factoring charts_. our very general setup will be illustrated , still with some effort , by example of pns , pngs and the _ intrinsic mean on a first geodesic principal component _ ( imo1gpc ) . in order to exploit nested asymptotic normality for a _ nested two - sample test _ , we utilize bootstrapping techniques . while for frchet means , as they are descriptors assuming values in the data space , one can explicitly model the dependence of the random base points of the tangent spaces as in @xcite , so that suitable statistics can be accordingly directly approximated , this modeling and approximation can be avoided using the bootstrap as in @xcite . for our application at hand , as data space and descriptor space are different , we can not approximate the distribution of random descriptors and we fall back on the bootstrap . * suggestions for live imaging of stem cell differentiation . * after illustrations of our nested two - sample test by simulations for pns and pngs , we apply it to a cutting edge application in adult human stem cell differentiation research . `` rooted in a line of experimentation originating in the 1960s '' ( from @xcite ) , the promise that stem cells taken from a patient s bone marrow may be used to rebuild specific , previously lost , patient s tissue is currently undergoing an abundance of clinical trials . although the underlying mechanisms are , to date , not fully understood , it is common knowledge that early stem cell differentiation is triggered by biomechanical cues , e.g. @xcite , which result in specific ordering of the cellular _ actin - myosin filament skeleton_. in collaboration with the third institute of physics at the university of gttingen we map fluorescence images of cell structures to two - spheres , where each point stands for a specific ordering . with our 2d pngs two - sample test we can track the direction of increased ordering over the first 24 hours . we find , however , a consistent reversal of ordering between hours 16 to 20 which hint toward the effect of cell division . this effect suggests that the commonly used time point of 24 hours for fixated hmscs imaging , e.g. as in @xcite , may not be ideal for cell differentiation detection . in fact , our method can be used to direct more elaborate and refined imaging techniques , such as time resolved in - vivo cell imaging , using @xcite , say , to investigate specifically discriminatory time intervals in detail . * we conclude our introduction * by noting that our setup of bnfds has a canonical form on a riemannian manifold with @xmath3 in ( [ backward - nested - subspaces : eq ] ) being a totally geodesic submanifold , not necessarily of codimension one in @xmath12 , however . for example for kendall s shape spaces @xmath13 which is a complex projective space of real dimension @xmath14 , cf . @xcite , we have a sequence of @xmath15 where the numbers below the inclusions denote the corresponding co - dimensions . more generally , we believe that our setup can be generalized to riemann stratified spaces . for example , ( [ nested - shape - spaces : eq ] ) generalizes at once to @xmath16 ( the shape space of @xmath17-dimensional @xmath18 landmark configurations which has dimension @xmath19 ) with @xmath20 , cf . @xcite , now with @xmath21 indeed , in section [ scn : kendalls - shape - spaces ] we illustrate the generalization to the sequence @xmath22 , giving the imo1gpc for arbitrary @xmath16 . our setup may also generalize to phylogenetic tree spaces as introduced by @xcite , cf . also @xcite , or torus - pca and the more general polysphere - pca , cf . moreover , our setup may be applied to flags of barycentric subspaces as introduced by @xcite . * our paper is organized as follows . * in the following section we introduce the abstract setup of bnfds and show that the essential assumptions are fulfilled for pns , pngs and imo1gpcs for riemannian manifolds and kendall s shape spaces . in the section to follow we will develop a set of assumptions necessary for the main results on asymptotic strong consistency and normality which are stated in section [ main - results : scn ] . also in section [ main - results : scn ] , we give our nested bootstrap two - sample test . the elaborate proof of asymptotic strong consistency is deferred to the appendix . in section [ application : scn ] we show simulations and our applications to stem cell differentiation . in this section we first introduce the general framework including the fundamental assumption of _ factoring charts _ which is essential to prove asymptotic normality of single nested descriptors in section [ main - results : scn ] . then we give examples : the intrinsic mean on a first geodesic principal component ( imo1gpc ) for riemannian manifolds , principal nested spheres ( pns ) as well as principal nested great spheres ( pngs ) and finally we give an example for the imo1gpcs also on non - manifold kendall s shape spaces . the first example is rather straightforward , the last is slightly more involved and the second and third are much more involved . the differential geometry used here can be found in any standard textbook , e.g. @xcite . first , let us quickly sketch the ideas in case of imo1gpcs on a riemannian manifold @xmath23 . there , we have the space @xmath24 of point sets of geodesics on @xmath23 which is the first non - trivial descriptor space `` below '' the space @xmath25 . in order to show strong asymptotic consistency in theorem [ sc : thm ] , on @xmath26 we require the concept of a _ loss function _ @xmath27 that has some properties of a distance between two ( point sets of ) geodesics . in order to model nestedness , given a geodesic @xmath28 we require the set @xmath29 of lower dimensional descriptors in @xmath30 which lie on @xmath31 . these are the candidate nested means on @xmath31 , and in this case , @xmath32 . further , we need the data projection @xmath33 and we measure the distance @xmath34 of the projected data to a candidate nested mean @xmath35 . then every @xmath36 with @xmath37 will be a _ backward nested family of descriptors _ ( bnfd ) and the set @xmath38 with @xmath37 carries a natural manifold structure . it is the objective of _ factoring charts _ to represent this manifold locally as a direct product of arbitrary variable offsets @xmath39 times a suitable space parametrizing directions of geodesics , parametrized independently from the offset @xmath40 , cf . figure [ chartfactoring : fig ] . we will see that is precisely the geometry of the projective bundle . once we establish asymptotic normality of the backward nested descriptor @xmath38 , asymptotic normality follows at once also for @xmath41 , because , under factoring charts , @xmath41 is given by some coordinates of a gaussian vector as reasoned in the proof of theorem [ clt : thm ] . with a silently underlying probability space @xmath42 , _ random elements _ on a topological space @xmath23 are mappings @xmath43 that are measurable with respect to the borel @xmath44-algebra of @xmath23 . in the following , _ smooth _ refers to existing continuous 2nd order derivatives . for a topological space @xmath23 we say that a continuous function @xmath45 is a _ loss function _ if @xmath46 if and only if @xmath47 . we say that a set @xmath48 is @xmath49-bounded if @xmath50 . moreover , we say that @xmath51 is _ @xmath49-heine borel _ if all closed @xmath49-bounded subsets of @xmath52 are compact . [ bnfd : def ] a separable topological space @xmath23 , called the _ data space _ , admits _ backward nested families of descriptors _ ( bnfds ) if a. there is a collection @xmath53 ( @xmath54 ) of topological separable spaces with loss functions @xmath55 ; b. @xmath56 ; c. every @xmath57 ( @xmath58 ) is itself a topological space and gives rise to a topological space @xmath59 which comes with a continuous map @xmath60 d. for every pair @xmath57 ( @xmath58 ) and @xmath61 there is a measurable map called _ projection _ @xmath62 for @xmath63 and @xmath64 call a family @xmath65 a _ backward nested family of descriptors ( bnfd ) from @xmath53 to @xmath66_. the space of all bnfds from @xmath53 to @xmath66 is given by @xmath67 for @xmath68 , given a bnfd @xmath69 set @xmath70 which projects along each descriptor . for another bnfd @xmath71 set @xmath72 in case of pns , the nested projection @xmath73 is illustrated in figure [ pns_illustration : fig ] ( a ) . random elements @xmath74 on a data space @xmath23 admitting bnfds give rise to _ backward nested population _ and _ sample means _ ( abbreviated as bn means ) @xmath75 recursively defined via @xmath76 , i.e. @xmath77 and @xmath78 , & f^j & = \{p^k\}_{k = j}^{m } \\ e^{f_n^{j-1}}_n & = \operatorname{\mbox{\rm argmin}}_{s\in s_{p^{j}_n } } \sum_{i=1}^n\rho_{p_n^{j}}(\pi_{f_n^{j}}\circ x_i , s)^2 , & f_n^j & = \{p_n^k\}_{k = j}^{m } \ , . \end{aligned}\ ] ] where @xmath79 and @xmath80 is a measurable choice for @xmath58 . we say that a bnfd @xmath81 gives _ unique _ bn population means if @xmath82 with @xmath83 for all @xmath54 . each of the @xmath84 and @xmath85 is also called a _ generalized frchet mean_. note that by definition there is only one @xmath86 . for this reason , for notational simplicity , we ignore it from now on and begin all bnfds with @xmath87 and consider thus the corresponding @xmath88 . [ def : factoring ] let @xmath89 . if @xmath90 and @xmath91 carry smooth manifold structures near @xmath92 and @xmath93 , respectively , with open @xmath94 , @xmath95 such that @xmath96 , @xmath97 , and with local charts @xmath98 we say that the _ chart @xmath99 factors _ , cf . figure [ chartfactoring : fig ] ( a ) and ( b ) , if with the projections @xmath100 we have @xmath101 @xmath102{\includegraphics[width=0.27\textwidth , clip = true , trim=8 cm 4 cm 3.5 cm 1.5cm]{"bnfd_illustration_family " } } } } \scalebox{2}{$\stackrel{\psi}{\to}$}~~ \vcenter{\hbox{\subcaptionbox{coordinates}[0.27\textwidth]{\includegraphics[width=0.27\textwidth , clip = true , trim=8 cm 4 cm 3.5 cm 1.5cm]{"bnfd_illustration_chart " } } } } \hspace{0.02\textwidth } \vcenter{\hbox{\subcaptionbox{projective bundle}[0.35\textwidth]{\includegraphics[width=0.35\textwidth , clip = true , trim=4 cm 2.5 cm 3 cm 1.5cm]{projective_bundle}}}}$ ] suppose that @xmath103 are random variables assuming values on a riemannian manifold @xmath23 with riemannian norm @xmath104 for the tangent spaces @xmath105 ( @xmath106 ) , induced metric @xmath107 , _ projective tangent bundle _ @xmath108 and space of classes of geodesics given by their point sets @xmath109 : ( q,\{v ,- v\})\in pq\ } , ~ [ \gamma_{q , v } ] = \{\gamma_{r , w } : \gamma_{q , v}(t ) = r,~ \dot\gamma_{q , v}(t)= w\mbox { for some } t\}\end{aligned}\ ] ] where @xmath110 denotes the unique maximal geodesic through @xmath111 with unit speed velocity @xmath112 , @xmath113 . then consider @xmath114 there is a well defined distance between a point @xmath40 and a class of geodesics determined by @xmath115\big ) \mapsto \inf_{t}d\big(s,\gamma_{q , v}(t))\,.\end{aligned}\ ] ] then every class of geodesics determined by @xmath116\big)^2\big]\mbox { or } \operatorname{\mbox{\rm argmin}}_{(q , v)\in tq}\sum_{k=1}^n \rho\big(x_k,[\gamma_{q , v}]\big)^2\end{aligned}\ ] ] is called a _ first population principal component geodesic _ or a _ first sample principal component geodesic _ , respectively , cf . @xcite . moreover , given a first population principal component geodesic @xmath117 $ ] and a first sample principal component geodesic @xmath118 $ ] , with the orthogonal projection @xmath119 , q'\mapsto \operatorname{\mbox{\rm argmin}}_{\gamma_{q , v}(t ) } d(q',\gamma_{q , v}(t))\end{aligned}\ ] ] which is well defined outside a set of zero riemannian volume , e.g. ( * ? ? ? * theorem 2.6 ) , we have the _ intrinsic population means on @xmath31 _ and _ intrinsic sample means on @xmath120 _ determined by @xmath121\mbox { or } \operatorname{\mbox{\rm argmin}}_{\gamma_{q , v}(t)}\sum_{k=1}^n \rho_p\big(\pi_{q , p}\circ x_k,\gamma_{q , v}(t)\big)^2\,,\end{aligned}\ ] ] respectively , where @xmath122 for @xmath123 in @xmath31 . in particular , we have the space of _ backward nested descriptors _ @xmath124 \in p_1,~ s \in p\}\end{aligned}\ ] ] which carries the natural manifold structure of the projective tangent bundle @xmath125 conveyed by the identity @xmath126,s ) \mapsto ( s,\{w ,- w\})\end{aligned}\ ] ] where @xmath127 , @xmath128 , if @xmath129 . recall that the tangent bundle @xmath130 admits _ local trivializations _ , i.e. every @xmath106 has a local neighborhood @xmath131 with a smooth one - to - one mapping @xmath132 where the first coordinate satisfies @xmath133 for all @xmath134 , @xmath135 and the second coordinate @xmath136 is a vector space isomorphism . in consequence , for a given @xmath137 , with local charts @xmath138 of @xmath23 around @xmath139 , and @xmath140 of the real projective space @xmath141 of dimension @xmath142 around @xmath143 , @xmath144 open , and the open set @xmath145 the mapping @xmath146 yields a local chart that factors as in definition [ def : factoring ] . this scenario is sketched in figure [ chartfactoring : fig ] ( c ) . the ( nested ) projections detailed below are illustrated in figure [ pns_illustration : fig ] . notation . * consider the standard inner product @xmath147 on @xmath148 with norm @xmath149 and the @xmath150-dimensional unit sphere @xmath151 , with interior @xmath152 . for any matrix @xmath153 we have the frobenius norm @xmath154 and the inner product @xmath155 for @xmath156 . the @xmath157 dimensional unit matrix is @xmath158 and @xmath159 is the orthogonal group . moreover , e.g. @xcite , @xmath160 denotes the stiefel manifold of orthonormal @xmath4-frames in @xmath161 . for every such orthonormal @xmath4-frame we have a non - unique _ orthonormal complement _ @xmath162 such that @xmath163 . in pns and pngs , for a top sphere @xmath164 a sequence of nested subspheres is sought for . only in the scenario of pngs it is required that each subsphere is a great subsphere . in general , every @xmath18-dimensional subsphere ( @xmath165 ) is the intersection of a @xmath166 dimensional affine subspace of @xmath148 with @xmath167 . recall that every @xmath166 dimensional affine subspace @xmath168 is determined by a matrix @xmath169 of @xmath170 orthonormal column vectors that are orthogonal to @xmath171 and a vector of signed distances from the origin @xmath172 in particular @xmath173 ensures that @xmath171 intersects with @xmath167 in a @xmath18 dimensional subsphere . obviously , @xmath171 determines @xmath174 and @xmath175 up to an action of @xmath176 , i.e. with every @xmath177 , @xmath178 and @xmath179 determine the same @xmath171 . in pngs only great subspheres are allowed as intersections . hence all affine spaces under consideration above pass through the origin , i.e. @xmath180 above . * the parameter space . * in consequence we have that the family of @xmath18-dimensional subspheres ( @xmath181 is given by the smooth manifold @xmath182 : z \in m_j\right\}\mbox { with } [ z ] = \left\{z r : r\in o(m - j)\right\}\end{aligned}\ ] ] where @xmath183 with @xmath184 above , for compatibility , becoming clear in the considerations below . indeed , the smooth action from the right of the compact lie group @xmath185 on @xmath186 is free ( for @xmath187 and @xmath188 , @xmath189 implies that @xmath190 ) , giving rise to a smooth quotient manifold , e.g. ( * ? ? ? * theorem 7.10 ) , of dimension @xmath191 for pns and of dimension @xmath192 for pngs . notably , in the latter case , @xmath53 is just a grassmannian . in the above setup , we had excluded the cases @xmath193 . for @xmath194 , for pns and pngs it is natural to set , as in definition [ bnfd : def ] , @xmath195 in case of @xmath196 , the setup above would yield pairs of points ( the intersections of suitable lines with @xmath167 give topologically zero - dimensional spheres @xmath197 ) . in order to have a single nested mean as a zero dimensional descriptor only , for pns and pngs ( in order to represent nestedness ) we use the convention @xmath198 * distance between subspheres of equal dimension . * for pns and pngs , on @xmath186 we have the _ extrinsic metric _ @xmath199 since the extrinsic metric is invariant under the action of @xmath185 it gives rise to the well defined quotient metric , called the _ ziezold metric _ , cf . @xcite ) on @xmath53 given by @xmath200 for arbitrary representatives @xmath201 of @xmath202 , respectively , as the following lemma teaches . [ lem : ziezold - metric ] the mapping @xmath203 satisfies the triangle inequality and it is definite , i.e. @xmath204 implies @xmath205 . suppose that @xmath206 are representatives of @xmath207 with the property , w.l.o.g . , that @xmath208 . then the usual triangle inequality yields @xmath209 moreover , we have @xmath210 where @xmath211 and @xmath212 with equality if and only if @xmath213 and @xmath214 with @xmath215 , yielding that the above vanishes if and only if this is the case with @xmath216 . * backwards nesting . * in pns , the space of subspheres @xmath217 within a given subsphere @xmath218^t\in p_j$ ] ( @xmath219 ) can be given the following structure . @xmath220 indeed , if @xmath221 , @xmath188 is another representative for @xmath31 , then @xmath222 if and only if @xmath223 . in case of pngs , the condition on entries of @xmath175 above is simply @xmath224 because @xmath225 . according to our convention of @xmath226 , in case of @xmath227 the inequality above needs to be replaced with @xmath228 ( for pns ) and with @xmath229 for ( pngs ) . * projections . * for all @xmath230 we have the intrinsic orthogonal projection onto @xmath218^t \in p_j$ ] @xmath231 which is independent of the representative @xmath232 chosen and independent of the specific orthogonal complement @xmath233 of @xmath174 chosen ; and it is well defined except for a set @xmath234 of spherical measure zero . note that we have @xmath235 for @xmath196 and hence the constant mapping @xmath236 . * nested projections . * more generally , if @xmath237 are from a family of backward nested subspheres @xmath238 , @xmath239 , we may choose representatives @xmath232 of @xmath240 and @xmath241 of @xmath242 such that @xmath243 with ( [ eq : blow - down ] ) and the arbitrary but fixed complement @xmath244 of @xmath174 chosen , embed @xmath240 in @xmath245 ( first a translation , possible in pns , and then a blow up ) , depending on the specific choice of @xmath244 , via @xmath246 now , @xmath247 embeds @xmath242 as a @xmath248-dimensional ( possibly small ) subsphere @xmath249 in @xmath245 , given by @xmath250^t = p'_{j'}$ ] , for suitable @xmath251 , and there is an orthogonal complement @xmath252 of @xmath253 , such that @xmath254 this gives the definition of the projection , independent of the specific orthogonal complements chosen , @xmath255 plugging in ( [ eq : blow - down ] ) into the above equality and taking into account that @xmath256 , cf . ( [ eq : blow - up ] ) , yields at once the following proposition which asserts that projections along nested subspheres only depend on the final subsphere at which it ends . recall that @xmath225 ( @xmath257 if @xmath258 ) in case of pngs . [ pns - bnds : prop ] with the above notation @xmath259 . * distance between projected data and next subsphere . * here , we compute the intrinsic geodesic distance @xmath260 between @xmath261 and @xmath262^t\in s_{p^j}$ ] in a subsphere @xmath263^t \in p_j$ ] ( @xmath264 . note that only in case of @xmath240 being a great subsphere , this distance agrees with the spherical distance @xmath265 in the top sphere @xmath167 . if @xmath240 is a proper subsphere ( @xmath266 ) , assuming w.l.o.g . that @xmath267 and @xmath268 we have @xmath269 indeed with @xmath270 and the notation from ( [ eq : blow - down ] ) to ( [ eq : projections ] ) and orthogonal complements @xmath271 of @xmath174 and @xmath272 of @xmath273 , respectively , we have @xmath274 since @xmath275 due to ( [ eq : blow - down ] ) , @xmath276 and @xmath277 . * optimal positioning . * on @xmath186 we have the extrinsic metric due to its embedding in @xmath278 , cf . ( [ extrinsic - metric : eq ] ) , and its thus induced riemannian metric . obviously @xmath185 acts on @xmath186 isometrically w.r.t . the extrinsic metric . it also acts isometrically w.r.t . the riemannian metric because the action also preserves geodesics on @xmath279 , e.g. @xcite . in consequence , for both metrics , we say for given @xmath280 that @xmath188 puts @xmath281 i.o.p . ( _ in optimal position _ ) to @xmath282 , if @xmath283 [ lem : smooth - op ] let @xmath284 and @xmath201 be sufficiently close . then , there is a unique @xmath285 such that a. @xmath286 puts @xmath281 uniquely i.o.p . @xmath287 to @xmath282 , b. if @xmath288 is symmetric then @xmath289 . in order that @xmath188 minimizes the r.h.s of ( [ extrinsic - metric : eq ] ) , given by @xmath290 with @xmath291 , it maximizes @xmath292 . this is so if @xmath293 for a singular value decomposition ( svd ) @xmath294 stemming from a spectral decomposition @xmath295 . since for @xmath296 we have that @xmath297 is of full rank for all @xmath298 , there is locally a full rank svd , which is unique up to @xmath299 and @xmath300 for any @xmath301 with @xmath302 . however , @xmath303 is unique under actions of such @xmath304 , yielding ( i ) . ( ii ) : the symmetry @xmath305 allows to choose @xmath306 , i.e. @xmath307 . this lemma has the following immediate consequence . if @xmath201 are sufficiently close then @xmath308 thus , in further consequence , for every @xmath309 there is @xmath310 such that @xmath311 is a smooth manifold and every @xmath312 has a neighborhood @xmath313 with a smooth diffeomorphism @xmath314 where @xmath315 is a fixed representative of @xmath316 and @xmath317 is an arbitrary representative of @xmath31 , and @xmath281 is a representative i.o . p to @xmath282 . this follows from the fact that locally a point is i.o.p to @xmath282 if and only if it can be reached by a _ horizontal _ geodesic from @xmath282 , and from the fact that all geodesics in @xmath53 through @xmath316 lift to horizontal geodesics in @xmath186 through @xmath282 and all horizontal geodesics in @xmath186 project to geodesics in @xmath53 . for a detailed discussion e.g. @xcite . * representing spaces of nested subspheres : factoring charts . * for @xmath318 recall @xmath319 and that if @xmath320 is a representative of @xmath321 , then every @xmath61 is represented by some @xmath322 with suitable choices @xmath323 , @xmath324 and @xmath325 ( for pns , @xmath326 ) , @xmath327 ( for pngs , @xmath326 ) and @xmath328 ( for @xmath227 ) . vice versa , from every representative @xmath329 of @xmath330 , all of the @xmath57 with @xmath61 are determined by choosing @xmath331 orthonormal vectors from the column span of @xmath253 ( the columns of @xmath332 below ) with suitable @xmath331 distances ( given by the vector @xmath333 below ) , i.e. every such @xmath31 has a representation @xmath334\end{aligned}\ ] ] as @xmath52 ranges over @xmath335 . in case of pngs and @xmath227 , for compatibility we set @xmath336 . of course , different @xmath52 may give same the @xmath31 . more precisely , here is the central observation . given @xmath337\in p^{j-1}$ ] , every @xmath338 with @xmath61 is uniquely determined as @xmath339 $ ] by choice of @xmath340 , where @xmath52 is an arbitrary orthonormal complement of @xmath341 in @xmath342 , i.e. @xmath343 . this means that @xmath344 uniquely parametrizes @xmath345 . the reason is the well known fact that the grassmannian @xmath346 is diffeomorphic with @xmath344 . again , in case of pngs and @xmath227 , for compatibility we set @xmath347 . this observation gives rise to _ factoring charts _ as introduced in definition [ def : factoring ] . here and below in theorem [ th : several - nested - subspheres ] , @xmath348 assumes the role of @xmath349 from definition [ def : factoring ] . [ lem : two - nested - subspheres ] every @xmath350 has a neighborhood @xmath351 and a smooth diffeomorphism @xmath352 where @xmath348 is from ( [ eq : chart - pj ] ) , @xmath353 is a neighborhood of @xmath354 and @xmath355 are representatives of @xmath356 and @xmath316 respectively . in particular , @xmath357\mbox { and } p = [ zb ] \end{aligned}\ ] ] with arbitrary orthonormal complement @xmath52 of @xmath341 . let @xmath358 $ ] for some @xmath359 be sufficiently close to @xmath356 . then , according to lemma [ lem : smooth - op ] , @xmath41 has a unique representative @xmath360 in optimal position to @xmath282 , varying smoothly in @xmath361 , hence smoothly in @xmath41 . moreover , as elaborated above , every @xmath57 with @xmath61 depends uniquely upon the choice of @xmath340 and any orthonormal complement @xmath52 , i.e. @xmath362 $ ] . although @xmath363 determines the same @xmath331 dimensional subspace , a one - to - one mapping is obtained choosing the neighborhood @xmath313 of @xmath364 suitably small not containing antipodals . the following is a straightforward generalization to spaces of sequences of several nested subspheres . [ th : several - nested - subspheres ] let @xmath365 and @xmath366 . then every @xmath367 has a neighborhood @xmath94 and a smooth diffeomorphism @xmath368 where @xmath369 is from ( [ eq : chart - pj ] ) , each @xmath370 is a neighborhood of @xmath371 , @xmath372 are representatives of @xmath373 and @xmath374 respectively ( @xmath375 ) . in particular , @xmath376\mbox { and } p^{j - r+1 } = [ zb_k\cdots b_{r } ] \end{aligned}\ ] ] with arbitrary orthonormal complements @xmath377 in @xmath378 of @xmath379 ( @xmath375 ) . * a joint representation . * here we assume that @xmath380 and @xmath381 are two sufficiently close families of backward nested spheres with representatives @xmath382 such that @xmath383 are in o. @xmath384-p . notably , there are then uniquely determined @xmath385 complements @xmath386 and @xmath387 respectively . for a sufficiently small neighborhood @xmath388 of @xmath389 and neighborhoods @xmath390 of @xmath391 in @xmath392 ( @xmath375 ) we have then a smooth mapping @xmath393 determined by the following algorithm ( here is only the pngs version ) . for arbitrary @xmath394 , @xmath395 denotes the orthogonal projection to the complement of @xmath174 . * @xmath396 is the unique element such that @xmath397 . * @xmath398 is the unique element such that @xmath399 . here , @xmath400 is obtained from @xmath401 by removing one column such that @xmath402 is in the span of the rest ( usually @xmath403 ) . * @xmath404 * @xmath405 is the unique element such that @xmath406 . here , @xmath407 is obtained from @xmath408 by removing one column such that @xmath409 is in the span of the other two . * local charts . * for @xmath53 and @xmath90 , we have derived in and theorem [ th : several - nested - subspheres ] , respectively , local smooth diffeomorphic representations in a euclidean space of suitable dimension , @xmath17 , say , that can be generically written as @xmath410 with a neighborhood @xmath411 of some @xmath412 and a smooth mapping @xmath413 , @xmath414 , @xmath415 with full rank derivative @xmath416 at @xmath417 . here , the corresponding matrices are viewed as vectorized by stacking their columns on top of one another . with @xmath418 , the orthogonal projection to the row space of the @xmath419 matrix of the derivative at @xmath420 and orthogonal @xmath17-vectors @xmath421 , spanning the kernel of @xmath416 of dimension @xmath422 , obtain the local chart @xmath423 another application is given by the intrinsic mean on a geodesic principal component on a quotient space @xmath424 due to an isometric action of a lie group @xmath425 on a riemannian manifold @xmath426 . we treat here the prominent application of kendall s shape spaces @xmath427:x\in { \mathbb s}^{m(k-1)-1}\},\quad , [ x]=\{gx : g\in so(m)\}\end{aligned}\ ] ] where the space @xmath428 of unit size @xmath429 matrices corresponding to normed and centered configuration of @xmath4 landmarks in @xmath150-dimensional euclidean space is taken modulo the group of rotations in @xmath150-dimensional space . this space models all @xmath150-dimensional configurations of @xmath4 landmarks , not all coinciding , modulo similarity transformations , cf . @xcite . for @xmath430 this space is no longer a manifold but decomposes into strata of manifolds of different dimensions , cf . geodesics on the unit sphere @xmath428 , i.e. great circles , orthogonal to the the orbits @xmath431 $ ] , called _ horizontal great circles _ , project to geodesics in @xmath432 such that the space @xmath24 of geodesics of @xmath23 can be given the quotient structure of a stratified space @xmath433 with the horizontal stiefel manifold @xmath434 and the orbits @xmath435 = \ { ( gx , gv ) h : g\in so(m ) , h \in o(2)\}\,\end{aligned}\ ] ] for @xmath436 , cf . * theorem 5.2 ) . the action of @xmath437 is not free for @xmath430 , giving rise to a non - trivial stratified structure . as before in section [ imo1gpc : scn ] , we set @xmath438 having geodesics , orthogonal projections @xmath439 can be defined , which are unique outside a set of intrinsic measure zero , cf . * theorem 2.6 ) . the geodesic distance @xmath49 on @xmath23 gives rise to the geodesic distance @xmath440 between a datum @xmath139 and a geodesic @xmath31 . similarly the induced intrinsic distance on @xmath31 gives rise to @xmath441 where @xmath442 can be identified with @xmath31 . for @xmath27 and @xmath443 , for simplicity , not the canonical intrinsic distances on @xmath24 and @xmath444 but quotient distances due to the embedding of @xmath445 and @xmath446 , respectively , can be used , called the ziezold distances , cf . @xcite . the generalized frchet mean corresponding to @xmath447 is the first geodesic principal component ( gpc ) @xmath31 , the generalized frchet mean corresponding to @xmath448 is the the intrinsic mean on the first gpc , cf . the latter is again a nested mean . with the _ horizontal projective bundle _ over the unit sphere @xmath449 we have the space @xmath450,[x ] ) : x\in { \mathbb s}^{m(k-1)-1 } , \{v ,- v\ } \in p^h_x { \mathbb s}^{m(k-1)-1}\}\,.\end{aligned}\ ] ] the principal orbit theorem ( e.g. @xcite states in particular , that @xmath444 and @xmath24 have open and dense subsets @xmath451 that are manifolds , in our case smooth manifolds . this gives rise to the manifold @xmath452 with local smooth coordinates near @xmath453 @xmath454 where @xmath455 and @xmath456 is a representative of @xmath31 i.o.p . to @xmath457 , under the action of @xmath458 , with @xmath459 . here @xmath457 is an arbitrary but fixed representative of @xmath316 such that @xmath417 is a representative of @xmath356 . along the lines of lemma [ lem : smooth - op ] one can show that optimal positioning is unique if @xmath460 has rank @xmath461 , which may be assumed for most realistic data scenarios . arguing as in section [ imo1gpc : scn ] , with every local trivialization of the horizontal bundle @xmath462 comes a factoring chart . in this section we are back in the general scenario described in section [ general_framework : scn ] . we develop a set of assumptions necessary for the general results on asymptotic consistency and asymptotic normality in section [ asymptotics : scn ] . we then show that they are fulfilled in case of pns / pngs and the imo1gpc of kendall s shape spaces . for the following assumptions suppose that @xmath463 . [ ass - meas - exists : as ] for a random element @xmath464 in @xmath23 , assume that @xmath465<\infty$ ] for all bnfds @xmath466 ending at @xmath240 , @xmath467 . in order to measure a difference between @xmath61 and @xmath468 for @xmath469 define the orthogonal projection of @xmath61 onto @xmath470 as @xmath471 in case of pns this is illustrated in figure [ pns_illustration : fig ] ( a ) . [ projection - unique : as ] for every @xmath61 there is @xmath472 such that @xmath473 whenever @xmath202 with @xmath474 . [ 0.48 ] [ 0.48 ] for @xmath61 and @xmath202 sufficiently close let @xmath475 be the unique element . note that in general @xmath476 in the following assumption , however , we will require that they will uniformly not differ too much if @xmath31 is close to @xmath316 . also , we require that @xmath477 and @xmath41 be close . [ projection - close : as ] for @xmath310 there is @xmath472 such that @xmath478 whenever @xmath202 with @xmath474 . we will also require the following assumption , which , in conjunction with assumption [ projection - close : as ] , is a consequence of the triangle inequality , if @xmath479 is a metric . [ almost - triangle : as ] suppose that @xmath480 and @xmath481 with @xmath482 and @xmath483 . then also @xmath484 moreover , we require uniformity and coercivity in the following senses . [ uniform - link : as ] for all @xmath310 there are @xmath485 such that @xmath486 for all bnfds @xmath487 ending in @xmath202 , respectively , with @xmath488 and @xmath489 with @xmath490 . [ coercivity : as ] if @xmath491 and @xmath492 with @xmath493 , then for every @xmath494 we have that @xmath495 for every @xmath106 with @xmath496 and bnfds @xmath497 ending at @xmath498 respectively . [ compact : rm ] due to continuity , assumptions [ ass - meas - exists : as ] and [ uniform - link : as ] hold if @xmath23 is compact and assumption [ coercivity : as ] if each @xmath53 is compact . assumptions [ ass - meas - exists : as ] [ coercivity : as ] hold for pns and pngs for all @xmath499 . moreover , each @xmath53 is @xmath500-heine borel for @xmath54 . recall from proposition [ pns - bnds : prop ] that @xmath73 only depends on the final descriptor at which @xmath466 ends . for pns and pngs we use the notation introduced in section [ setup : scn ] and show first assumption [ projection - unique : as ] . let @xmath232 and @xmath501 be representatives of @xmath202 in optimal position and @xmath502^t\in s_p$ ] with @xmath503 and @xmath504 . moreover consider a candidate element @xmath505 and its squared distance to @xmath41 , @xmath506 with @xmath507 with the continuous function @xmath508 , we have @xmath509 , which , by lemma [ lem : ziezold - metric ] , is uniquely assumed for @xmath510 and @xmath511 . due to continuity , for @xmath512 sufficiently close to @xmath513 all minimizers @xmath514 of @xmath515 are in a neighborhood of @xmath516 and there , @xmath144 is full rank and hence , arguing as in the proof of ( i ) of lemma [ lem : smooth - op ] , the extremal @xmath188 is uniquely determined . let us write @xmath517 to obtain @xmath518 as @xmath341 is unique , minimizing the above for @xmath519 is equivalent to minimizing @xmath520 under the constraining conditions @xmath521 and @xmath522 which yields the necessary equation @xmath523 right multiplication with @xmath524 yields at once @xmath525 such that @xmath519 is uniquely determined by @xmath526 indeed , @xmath519 is well defined because it is in a neighborhood of @xmath527 and hence @xmath341 in a neighborhood of @xmath391 . more simply , without lagrange minimization , we obtain @xmath528 . for @xmath529 sufficiently close , this yields a unique @xmath530 minimizing @xmath531 over @xmath532 , yielding at once assumptions [ projection - unique : as ] , [ projection - close : as ] and [ almost - triangle : as ] ( because @xmath500 is a metric ) . due to remark [ compact : rm ] , assumptions [ ass - meas - exists : as ] , [ uniform - link : as ] and [ coercivity : as ] hold . because each @xmath53 is a finite dimensional manifold and @xmath500 is a topologically compatible metric , @xmath53 is @xmath500-heine borel . for imo1gpcs on kendall s shape spaces @xmath432 , @xmath533 , assumptions [ ass - meas - exists : as ] [ coercivity : as ] hold for @xmath227 . moreover , @xmath53 is @xmath534 heine - borel for @xmath535 . assumption [ projection - unique : as ] follows at once from the compactness of @xmath23 , hence the geodesics @xmath536 are also compact and the proof of ( * ? ? ? * theorem a.5 ) , as there , in claim ii , a neighborhood of a geodesic @xmath31 is constructed , restricted to which the orthogonal projection @xmath537 is well defined and continuous in @xmath31 . compactness and continuity also imply assumptions [ projection - close : as ] and [ almost - triangle : as ] . assumptions [ ass - meas - exists : as ] , [ uniform - link : as ] and [ coercivity : as ] follow from remark [ compact : rm ] . again , let @xmath538 . [ as : clt - uniform - derivatives ] assume that @xmath539 carries a smooth manifold structure near the unique bn population mean @xmath540 such that there is an open set @xmath541 , @xmath542 and a local chart @xmath543 further , assume that for every @xmath544 the mapping @xmath545 has first and second derivatives , such that for all @xmath544 , @xmath546\mbox { , and } \mathbb e\big[{\mbox{\rm hess\,}}_\eta \tau^l(\eta',x)\big]\ , \end{aligned}\ ] ] exist and are in expectation continuous near @xmath547 , i.e. for @xmath548 we have @xmath549\:\to\:0\ , , \\ { \mathbb e}\left[\mathop{\sup}_{\\|\eta-\eta'\\|<\delta}\left\\|{\mbox{\rm hess\,}}_\eta \tau^l(\eta , x ) - { \mbox{\rm hess\,}}_\eta \tau^l(\eta',x)\right\\|\right]\:\to\:0\ , . \end{aligned}\ ] ] finally , assume that the vectors @xmath550,\ldots , { \mathbb e}\big[{\mbox{\rm grad}}_\eta \tau^{m}(\eta',x)\big]$ ] are linearly independent . for pns and pngs a global , manifold structure has been derived in section [ sec : pns - framework ] with projections ( [ eq : blow - down ] ) ( see also proposition [ pns - bnds : prop ] ) and distances ( [ rho - p - pns : eq ] ) smooth away from singularity sets . for imo1gpcs on kendall s shape spaces , this has been provided in section [ scn : kendalls - shape - spaces ] , cf . also @xcite . in general , however , it is unclear under which circumstances ( if the second derivatives are continuous in both arguments where @xmath464 is supported in a compact set , then convergence to zero holds not only in expectation but also a.s . ) the three assumptions above , uniqueness , existence of first and second moments of second and first derivatives and their continuity in expectation are valid . even for the much simpler case of intrinsic means on manifolds this is only very partially known , cf . the discussion in @xcite . it seems that only for the most simple non - euclidean case of intrinsic means on circles the full picture is available ( @xcite ) . recently , rather generic conditions for densities have been derived by @xcite , ensuring @xmath551-gaussian asymptotic normality . the condition on linear independence is rather natural for realistic scenarios where each constraining condition adds a new constraint , not covered by the previous , as introduced after corollary [ cor : lambda - lem2 ] . for example , if charts factor , then with decreasing @xmath552 , every constraning condition results in conditions on new coordinates . [ sc : thm ] let @xmath553 and consider random data @xmath554 on a data space @xmath23 admitting bn descriptor families from @xmath555 to @xmath556 , unique bn population means @xmath557 and bn sample means @xmath558 due to a measurable selection @xmath559 giving rise to bnfds @xmath560 , @xmath561 . if assumptions [ ass - meas - exists : as ] [ coercivity : as ] are valid for all @xmath562 , and every @xmath563 is a.s . @xmath500-heine borel ( @xmath561 ) then @xmath558 converges a.s . to @xmath557 in the sense that @xmath564 measurable with @xmath565 such that for all @xmath561 , @xmath310 and @xmath566 , @xmath567 with @xmath568 we proceed by backward induction on @xmath18 . the case @xmath194 is trivial and the case @xmath569 has been covered by theorems a.3 and a.4 from @xcite . now suppose that ( [ bp - sc : eq ] ) have been established for @xmath570 . set @xmath571 and for an arbitrary bnfd @xmath389 ending at @xmath572 @xmath573,~s\in s_p & f_{n , f'}(s ) & = \frac{1}{n}\sum_{i=1}^n \rho_{p'}(\pi_{f'}\circ x_i , s)^2,~s\in s_{p'}\\ { \mbox{\handw \symbol{96}}}_f & = \inf_{s\in s_p } f_f(s ) , & { \mbox{\handw \symbol{96}}}_{n , f ' } & = \inf_{s\in s_{p ' } } f_{n , f'}(s ) \end{aligned}\ ] ] then , @xmath574 for all @xmath61 , by hypothesis , and with @xmath575 , @xmath576 to complete the proof we first show in the appendix @xmath577 this is ziezold s version of strong consistency ( cf . further , we show that this implies the bhattacharya - patrangenaru version ( cf . @xcite ) of strong consistency which takes here the form ( [ bp - sc : eq ] ) . careful inspection of the proof yields that we have only used that the `` distances '' @xmath500 vanish on the diagonal @xmath578 for all @xmath57 ; they need not be definite , i.e. it is not necessary that @xmath579 . moreover , note that the @xmath500-heine borel property holds trivially in case of unique sample descriptors . [ cor : lambda - lem2 ] suppose that ( [ bp - sc : eq ] ) holds together with assumption [ as : clt - uniform - derivatives ] . then we have for @xmath580 the following convergence in probability @xmath581\ , , & \frac{1}{n}\sum_{k=1}^n { \mbox{\rm hess\,}}_\eta\tau^l(\eta_n , x_k ) & \:\stackrel{{\mathbb p}}{\to}\ : { \mathbb e}\left [ { \mbox{\rm hess\,}}_\eta\tau^l(\eta',x)\right]\ , . \end{aligned}\ ] ] let @xmath310 . then by assumption [ as : clt - uniform - derivatives ] , chebyshev s inequality and ( [ bp - sc : eq ] ) , there is a sequence @xmath582 such that @xmath583 and @xmath584\right\\| \geq \epsilon \right\ } \\ & \leq { \mathbb p}\left\{\sup_{\\|\eta-\eta'\\|<\delta_n}\left\\| \frac{1}{n}\sum_{k=1}^n{\mbox{\rm grad}}_\eta\tau^l(\eta , x_k ) - { \mathbb e}\big[{\mbox{\rm grad}}_\eta\tau^l(\eta',x)\big]\right\\| \geq \epsilon \right\}\\ & \leq \frac{1}{\epsilon}\,{\mathbb e}\left[\sup_{\\|\eta-\eta'\\|<\delta_n}\left\\| \frac{1}{n}\sum_{k=1}^n{\mbox{\rm grad}}_\eta\tau^l(\eta , x_k ) - { \mathbb e}\big[{\mbox{\rm grad}}_\eta\tau^l(\eta',x)\big]\right\\| \right]~\to~0 \end{aligned}\ ] ] as @xmath585 . yielding the first assertion . the second assertion follows similarly . we now introduce notation we use for the central limit theorem . let @xmath586 . by construction , every measurable selection @xmath587 of bn sample means minimizes @xmath588 under the constraints that it minimizes each of @xmath589 similarly , the bn population mean @xmath590 minimizes @xmath591\end{aligned}\ ] ] under the constraints that it minimizes each of @xmath592 , \quad\ldots\quad , { \mathbb e}\left [ \rho_{p^{j+1}}(\pi_{f^{j+1}}\circ x , p^{j})^2\right]\,.\end{aligned}\ ] ] in consequence , due to differentiability guaranteed by assumption [ as : clt - uniform - derivatives ] , with the notation of @xmath593 there , suitable random lagrange multipliers @xmath594 and deterministic lagrange multipliers @xmath595 exist such that for @xmath596 and @xmath597 the following hold @xmath598 & & + \sum_{l = j+1}^m \lambda^l \,{\mathbb e}\big[\tau^l(\eta , x)\big ] \label{eq : population - lagrange}\end{aligned}\ ] ] [ lem : lambda - lem1 ] suppose that ( [ bp - sc : eq ] ) holds together with assumption [ as : clt - uniform - derivatives ] . then the random lagrange multipliers @xmath599 in ( [ eq : sample - lagrange ] ) and @xmath600 in ( [ eq : population - lagrange ] ) satisfy @xmath601 by hypothesis , the vector @xmath602 is a linear combination of the vectors @xmath603 conveyed by @xmath599 . similarly , @xmath604 $ ] is a linear combination of the vectors @xmath605 $ ] , @xmath606 conveyed by @xmath600 . set @xmath607 , @xmath608 and @xmath609 , @xmath610 . by assumption [ as : clt - uniform - derivatives ] we have @xmath611 . we set @xmath612 . since the determinant is continuous , by corollary [ cor : lambda - lem2 ] , @xmath613 now consider the function @xmath614 where @xmath615 denotes the moore - penrose pseudoinverse of @xmath341 . then @xmath616 and @xmath617 . in consequence , for arbitrary @xmath310 and @xmath585 we have that @xmath618 because , for any @xmath619 the first term is smaller than @xmath620 here , due to continuity established by @xcite , the first term vanishes for @xmath621 sufficiently small , and , due to corollary [ cor : lambda - lem2 ] , for any fixed @xmath619 , the second term tends to zero . this yields the assertion . [ clt : thm ] let @xmath622 and consider random data @xmath554 on a data space @xmath23 admitting bnfds from @xmath623 to @xmath624 , a unique bn population mean @xmath625 and bn sample means @xmath626 due to a measurable selection @xmath627 , @xmath628 , @xmath629 . * assuming that assumption [ as : clt - uniform - derivatives ] hold as well as ( [ bp - sc : eq ] ) for all @xmath630 , we have that @xmath631 with a chart @xmath99 as specified in assumption [ as : clt - uniform - derivatives ] as well as @xmath632 \mbox { and } \\ b_{\psi } & \:=\ : { \mbox{\rm cov}}\left [ { \mbox{\rm grad}}_{\eta}\tau^j(\eta',x ) + \sum_{l = j+1}^m \lambda^l\ , { \mbox{\rm grad}}_{\eta}\tau^l(\eta',x)\right ] \ , , \end{aligned}\ ] ] with the notation from assumption [ as : clt - uniform - derivatives ] where @xmath633 are suitable such that @xmath634 + \sum_{l = j+1}^m \lambda^l\,{\mbox{\rm grad}}_\eta \,{\mathbb e}\big[\tau^l(\eta , x)\big]\ , \end{aligned}\ ] ] vanishes at @xmath635 . * if additionally @xmath636 , then @xmath637 satisfies a gaussian @xmath551-clt @xmath638 * if additionally the chart @xmath99 factors as in definition [ def : factoring ] , then also @xmath639 satisfies a gaussian @xmath551-clt @xmath640 with the notation of definition [ def : factoring ] . by taylor expansion we have for @xmath641 defined in ( [ eq : sample - lagrange ] ) , @xmath642{\big)}}\nonumber\\ & = \sqrt{n}\,{\mbox{\rm grad}}_{\eta}g_n(\eta ' ) ~&&+~ { \mbox{\rm hess\,}}_{\eta}g_n(\eta ' ) \sqrt{n}\ , ( \eta_n - \eta ' ) \label{eq : clt - proof-1a}\\ & & & + ~ \big({\mbox{\rm hess\,}}_{\eta}g_n(\widetilde{\eta}_n ) - { \mbox{\rm hess\,}}_{\eta}g_n(\eta ' ) \big ) \sqrt{n}\ , ( \eta_n - \eta ' ) \nonumber \end{aligned}\ ] ] with some random @xmath643 between @xmath644 and @xmath547 . in consequence of corollary [ lem : lambda - lem1 ] we have with the usual clt for the first term in ( [ eq : clt - proof-1a ] ) that @xmath645 with a zero - mean gaussian vector @xmath646 of covariance @xmath647\ , . \end{aligned}\ ] ] similarly , we have for the first factor in the second term in ( [ eq : clt - proof-1a ] ) , @xmath648\ , . \end{aligned}\ ] ] finally , for the first factor in the last the term in ( [ eq : clt - proof-1a ] ) , invoking also corollary [ cor : lambda - lem2 ] , we obtain that @xmath649 this yields assertion ( i ) . if @xmath650 is invertible , as asserted in ( ii ) , joint normality follows at once for @xmath651 . ( iii ) : in case of factoring charts we can rewrite @xmath652 where @xmath653 with @xmath654 and @xmath655 , and @xmath656 is defined by @xmath657 . with @xmath658 in assertion ( ii ) we obtain thus @xmath659 since under projection to the first coordinates @xmath660 , asymptotic normality is preserved , assertion ( iii ) follows at once . suppose that we have two independent i.i.d . samples @xmath661 , @xmath662 in a data space @xmath23 admitting bnfds and we want to test @xmath663 using descriptors in @xmath664 . here , @xmath664 stands either for a single @xmath665 for which we have established factoring charts , or for a suitable sequence @xmath666 . we assume that the first sample gives rise to @xmath667 , the second to @xmath668 , and that these are unique . under the corresponding assumptions of theorem [ clt : thm ] , define a statistic @xmath669 under @xmath670 , up to a suitable factor , this is hotelling @xmath671 distributed if @xmath672 is the corresponding empirical covariance matrix . therefore , for @xmath672 we use the empirical covariance matrix from bootstrap samples . with this fixed @xmath171 , we simulate that statistic under @xmath670 by again bootstrapping @xmath52 times . namely from @xmath673 we sample @xmath674 and compute the corresponding @xmath675 ( @xmath676 ) from @xmath677 ( @xmath678 , @xmath58 ) . from these , for a given level @xmath679 we compute the empirical quantile @xmath680 such that @xmath681 arguing as in ( * ? ? ? * corollary 2.3 and remark 2.6 ) which extends at once to our setup , we assume that the corresponding population covariance matrix @xmath682 or @xmath683 , respectively , from theorem [ clt : thm ] is invertible . we have then under @xmath670 that @xmath680 gives an asymptotic coverage of @xmath684 for @xmath685 , i. e. @xmath686 as @xmath687 if @xmath688 with a fixed @xmath689 . to illustrate our clt for principle nested spheres ( pns ) and principle nested great spheres ( pngs ) , we simulate three data sets , each from two paired random variables @xmath464 and @xmath690 , displayed in figure [ simulated_data ] . a. data on an @xmath691 concentrate on the same proper small @xmath692 and there on segments of orthogonal great circles such that their nested means are antipodal . b. data on an @xmath691 concentrate on the same proper small @xmath692 and there on segments of orthogonal great circles such that their nested means coincide . c. data on an @xmath692 concentrate on segments of different small circles , have different nested means under pns , but , under pngs , coinciding principal geodesics and nested means . [ 0.3 ] concentrate on a common proper small @xmath692 , their projections to estimated small two - spheres is depicted . the simulated dataset iii ( right ) is on @xmath692.__,title="fig:",scaledwidth=30.0% ] [ 0.3 ] concentrate on a common proper small @xmath692 , their projections to estimated small two - spheres is depicted . the simulated dataset iii ( right ) is on @xmath692.__,title="fig:",scaledwidth=30.0% ] [ 0.3 ] concentrate on a common proper small @xmath692 , their projections to estimated small two - spheres is depicted . the simulated dataset iii ( right ) is on @xmath692.__,title="fig:",scaledwidth=30.0% ] we apply pns and pngs to the simulated data and perform the two - sample test for identical respective nested submanifolds ( means , small and great circles ) and for identical small and great two - spheres . the resulting p - values are displayed in table [ table ] . these values are in agreement with the intuition guiding the design of the data . ' '' '' data set & method & @xmath693d & @xmath694d & @xmath695d + ' '' '' i & pns & @xmath696 & @xmath696 & @xmath697 + ' '' '' & pngs & @xmath696 & @xmath696 & @xmath696 + ' '' '' ii & pns & @xmath697 & @xmath696 & @xmath698 + ' '' '' & pngs & @xmath699 & @xmath696 & @xmath696 + ' '' '' iii & pns & @xmath696 & @xmath696 & + ' '' '' & pngs & @xmath700 & @xmath701 & + [ table ] understanding differentiation of adult human stem cells with the perspective of clinical use ( see e.g. @xcite who emphasize their potential for cartilage and bone reconstruction ) is an ongoing fundamental challenge in current medical research , still with many open questions ( e.g. @xcite ) . to investigate mechanically guided differentiation , _ human mesenchymal stem cells _ ( hmscs , pluripotent adult stem cells taken from the bone marrow ) are placed on gels of varying elasticity , quantified by the young s modulus , to mimic different environments in the human body , e.g. @xcite . it is well known that within the first day the surrounding elasticity measured in kilopascal ( kpa ) induces differentiation through biomechanical cues , cf . @xcite , where the changes manifest in orientation and ordering of the _ actin - myosin filament skeleton_. in particular , in order to direct future , more focused research , it is of high interest to more precisely identify time intervals in which such changes of ordering occur and to separate changes due to differentiation from changes due to other causes . [ 0.48 ] [ 0.48 ] + [ 0.48 ] [ 0.48 ] * experimental setup . * we compare hmsc skeletons that have been cultured at the third institute of physics of the university of gttingen on gels with young s moduli of 1 kpa mimicking neural tissue , 10 kpa mimicking muscle tissue , and 30 kpa mimicking bone tissue . the cells have been fixed after multiples of 4 hours on the respective gel and have then been immuno - stained for nmm iia , the motor proteins making up small filaments that are responsible for cytoskeletal tension and imaged ( as described in @xcite ) table_samples ] shows their sample sizes and the data will be published and made available after completion of current research , cf . because earlier research ( @xcite ) suggests that during the first 24 hours , 10 kpa and 30 kpa hmscs develop rather similarly and quite differently from 1 kpa hmscs , for this investigation , we pool the former . ' '' '' time & 1 kpa & 10 kpa and 30 kpa + ' '' '' 4h & 159 & 321 + ' '' '' 8h & 163 & 317 + ' '' '' 12h & 176 & 344 + ' '' '' 16h & 135 & 274 + ' '' '' 20h & 138 & 253 + ' '' '' 24h & 166 & 304 + [ table_samples ] the actin - myosin filament structure has been automatically retrieved from the fluorescence images using the filament sensor from @xcite . since neighboring filaments share the same orientation , the 3d structure of the cellular skeleton can be retrieved by separating the filament structure into different orientation fields , cf . figure [ fig : filament - sensor ] . * orientation fields * for filament structures are determined via a relaxation labeling procedure , see @xcite . the source code of our implementation is available as supplementary material . a detailed description is deferred to a future publication . the algorithm results in a set of contiguous areas with slowly varying local orientation , and , corresponding to each of these areas , a set of filaments which closely follow the local orientation . also , these data will be published and made available after completion of current research , cf . @xcite . * data analysis . * for each single hmsc image , let @xmath426 be the number of pixels of all detected filaments , @xmath702 the number of all filament pixels of filaments of the largest orientation field and @xmath703 the number of all filament pixels of filaments of all smaller orientation fields . @xmath704 is then the number of pixels in all `` rogue '' filaments which are not associated to any field , because they are too inconsistent with neighboring filaments . define @xmath705 where the square roots ensure that @xmath519 does not describe relative areas but rather relative diameters of fields . this representation is confined to the @xmath706 part in the first octant and every sample shows a distinct accumulation of points in the @xmath707 plane , corresponding to cells with only one orientation field . as common with biological data , especially from primary cells , their variance is rather high . in consequence , great circle fits are more robust under bootstrapping than small circle fits and we use the nested two - sample tests for pngs with the following null hypothesis . @xmath670 : : : hmsc orientation and ordering measured by random loci on @xmath706 as above does not change between successive time points . ' '' '' time & & + ' '' '' gel & 1 kpa & 10 kpa and 30 kpa & 1 kpa & 10 kpa and 30 kpa + ' '' '' 4h vs. 8h & @xmath708 & @xmath696 & @xmath709 & @xmath696 + ' '' '' 8h vs. 12h & @xmath696 & @xmath696 & @xmath710 & @xmath696 + ' '' '' 12h vs. 16h & @xmath711 & @xmath696 & @xmath712 & @xmath696 + ' '' '' 16h vs. 20h & @xmath713 & @xmath714 & @xmath715 & @xmath716 + ' '' '' 20h vs. 24h & @xmath696 & @xmath696 & @xmath696 & @xmath717 + [ table_fields ] * results . * as visible in table [ table_fields ] , while for hmscs on harder gels ( 10 kpa and 30 kpa ) , nested means and the joint descriptor of nested mean and great circle change over each 4 hour interval until 16 hours for both the null hypothesis is rejected at the highest level possible similar changes are less clearly visible for hmscs on the soft gel ( 1 kpa ) between the intervals between 8 and 16 hours and not at all visible for the first time interval . strikingly , for hmscs on all gels , no changes seem to occur between 16 and 20 hours . in contrast , in the final interval between 20 and 24 hours , nested means and great circles clearly change for hmscs on the soft gel rejecting the null hypothesis at the highest level possible . this effect is also there for the nested mean of hmscs on the harder gels , but not as clearly visible for the joint descriptor including the circle . [ 0.4]_,title="fig:",scaledwidth=40.0% ] [ 0.4]_,title="fig:",scaledwidth=40.0% ] ._displaying p - values of two - sample tests for pngs of filament orientation field distribution data for all time points . we use @xmath718 bootstrap samples , thus the penultimate p - value is @xmath719 . _ [ cols="<,^,^",options="header " , ] [ 24hrs_table ] visualization in figure [ fig_fields ] reveals further details . as seen from the loci of the nested means , hmscs on the soft gel ( figure [ fields_1kpa ] ) tend to loose minor orientation field filaments with a nearly constant ratio of large orientation field filaments and rogue filaments until the _ critical slot _ , the time interval between 16 and 20 hours . their great circles , indicating the direction of largest spread , change at the beginning of the critical slot , suggesting that the major variation there occurs in the amount of rogue filaments . while , until the critical slot , the temporal motion of nested means for 1 kpa is mainly vertical , the corresponding motion for the hmscs on harder gels ( cf . figure [ fields_pooled ] ) is horizontal , indicating that the number of rogue filaments decreases in favor of the main orientation field . curiously for the nested means , there is also a sharp drop in height at the beginning of the critical slot as well as a backward horizontal motion . after the critical slot , hmscs seem to continue the direction of their previous journey , at a lower smaller fields level , though . in contrast , for the hmscs on the soft gel , the critical slot seems to represent a true change point since afterward , the nested mean travels not much longer towards reducing the smaller fields , but like hmscs on harder gels , mainly reduces the number of rogue filaments . indeed , taking into account the auxiliary mesh lines , it can be seen that descriptors are rather close at time 24 hours , cf . 24hrs_table ] , where , in contrast they are rather far away from each other for all other time points . * discussion . * we conclude that hmscs react clearly distinctly and differently on both gels already for short time intervals , where at the critical time slot some kind of reboot happens . a generic candidate for this effect is cell division . as all cells used in the experiments were thawed at the same time ( 72 hours before seeding ) and treated identically , cell division is expected to occur at similar ( at least for each environment ) time points . dividing cells completely reorganize their cell skeleton which would explain the change point found . in particular , it seems that due to cell division , the time point 24 hours ( as used in @xcite ) may not be ideal if differences in hmscs differentiation due to different young s moduli are to be detected . our results clearly warrant further analysis using higher time resolution , in particular time resolved in - vivo imaging , that among others , allow to register cell division times . we thank rabi bhattacharya and vic patrangenaru for their valuable comments on the bootstrap and our collaborators florian rehfeldt and carina wollnik for their stem cell data . the authors also gratefully acknowledge dfg hu 1575/4 , dfg crc 755 and the niedersachsen vorab of the volkswagen foundation . with @xmath724 which , in conjunction with assumption [ projection - close : as ] and the induction hypothesis @xmath725 a.s . can be made arbitrary small a.s . due to assumption [ uniform - link : as ] . in consequence of the usual strong law the first assertion follows . the second follows with the same argument . by separability of @xmath53 it follows at once from lemma [ basic - ziezold : lem ] that there is a measurable set @xmath727 with @xmath728 and a dense subset @xmath729 such that @xmath730\end{array}\right\ } \quad \forall s \in \{s_k : k\in \mathbb n\}\mbox { and } \omega\in \omega'\,.\end{aligned}\ ] ] in order to obtain ( [ dense - as - conv1:eq ] ) for all @xmath61 , consider @xmath731 and the following estimates . @xmath732 with @xmath733 now , w.l.o.g . , consider @xmath734 ( which implies that @xmath735 ) for which ( [ dense - as - conv1:eq ] ) is valid . using twice the first line in ( [ dense - as - conv2:ineq ] ) for @xmath736 and @xmath737 we obtain @xmath738 due to assumption [ uniform - link : as ] and the strong law ( [ dense - as - conv1:eq ] ) ( and the argument applied in the proof of lemma [ basic - ziezold : lem ] ) , for every @xmath739 there is @xmath740 such that for all @xmath741 we have @xmath742 + \epsilon\big)\epsilon & \leq\lim\inf_{n\to \infty } f_{n , f_n}({s}^{p_n})\\ & \leq \lim\sup_{n\to \infty } f_{n , f_n}({s}^{p_n})\\ & \leq f_f(s_k ) + 2\big(\mathbb e[\rho_p(\pi_f\circ x , s_k ) ] + \epsilon\big)\epsilon\end{aligned}\ ] ] for all @xmath743 . taking into account the continuity of @xmath744 , letting @xmath745 yields @xmath746 similarly we see that @xmath747\quad \forall s \in s_p\mbox { and } \omega\in \omega'\quad\mbox { for } n\to \infty\,.\end{aligned}\ ] ] next , we consider a sequence @xmath748 . note that in consequence of assumption [ almost - triangle : as ] we have that @xmath749 using the bottom line of ( [ dense - as - conv2:ineq ] ) yields that @xmath750 with the same @xmath751 for all @xmath61 due to ( [ as - conv4:eq ] ) . hence , in consequence of this and ( [ as - conv3:eq ] ) , for all @xmath748 we have that @xmath752 for all @xmath566 . finally let us show @xmath753 note that assertion ( [ ziezold - sc : eq ] ) is trivial in case of @xmath754 . otherwise , for ease of notation let @xmath755 , @xmath756 , @xmath757 . then @xmath758 for all @xmath759 . hence , there is a sequence @xmath760 , @xmath761 . moreover , there is a sequence @xmath762 such that @xmath763 for a suitable @xmath764 . then @xmath765 by ( [ as - conv5:eq ] ) a.s .. on the other hand , by lemma [ basic - ziezold : lem ] for arbitrary fixed @xmath61 , there is a sequence @xmath766 such that @xmath767 . first letting @xmath585 and then considering the infimum over @xmath61 yields @xmath768 in consequence @xmath769 in particular we have shown that @xmath770 which means that @xmath771 thus completing the proof of ( [ ziezold - sc : eq ] ) * proof of ( [ bp - sc : eq ] ) . * using the notation of the previous proof of ( [ ziezold - sc : eq ] ) , let @xmath772 and consider @xmath773 . if the assertion ( [ bp - sc : eq ] ) was false , there would be a measurable set @xmath774 with @xmath775 such that for every @xmath776 there is @xmath777 and @xmath778 . in consequence of the heine borel property we have thus @xmath785 for all @xmath786 . since @xmath787 < \infty$ ] for all @xmath61 , there is a @xmath494 such that @xmath788 hence , in consequence of assumption [ coercivity : as ] we have thus a subset @xmath789 with @xmath790 such that for all @xmath791 , due to the usual strong law , @xmath792 this is a contradiction to ( [ lw - lim - sup : ineq ] ) . this yields ( [ bp - sc : eq ] ) completing the proof of theorem [ sc : thm ] . bianco , p. , x. cao , p. s. frenette , j. j. mao , p. g. robey , p. j. simmons , and c .- y . wang ( 2013 ) . the meaning , the sense and the significance : translating the science of mesenchymal stem cells into medicine . _ 19_(1 ) , 3542 . huckemann , s. , k .- kim , a. munk , f. rehfeldt , m. sommerfeld , j. weickert , c. wollnik , et al . the circular sizer , inferred persistence of shape parameters and application to early stem cell differentiation . _ 22_(4 ) , 21132142 . huckemann , s. f. and b. eltzner ( 2015 ) . polysphere pca with applications . in _ proceedings of the 33th lasr workshop _ , pp . 5155 . leeds university press . http://www1.maths.leeds.ac.uk/statistics/workshop/lasr2015/proceedings15.pdf . pittenger , m. , a. mackay , s. beck , r. jaiswal , r. douglas , j. mosca , m. moorman , d. simonetti , s. craig , and d. marshak ( 1999 ) . multilineage potential of adult human mesenchymal stem cells . _ 284_(5411 ) , 143 . pizer , s. m. , s. jung , d. goswami , j. vicory , x. zhao , r. chaudhuri , j. n. damon , s. huckemann , and j. marron ( 2013 ) . nested sphere statistics of skeletal models . in _ innovations for shape analysis _ , pp . 93115 . springer .	for sequences of random backward nested subspaces as occur , say , in dimension reduction for manifold or stratified space valued data , asymptotic results are derived . in fact , we formulate our results more generally for backward nested families of descriptors ( bnfd ) . under rather general conditions , asymptotic strong consistency holds . under additional , still rather general hypotheses , among them existence of a.s . local twice differentiable charts , asymptotic joint normality of a bnfd can be shown . if charts factor suitably , this leads to individual asymptotic normality for the last element , a principal nested mean or a principal nested geodesic , say . it turns out that these results pertain to principal nested spheres ( pns ) and principal nested great subsphere ( pngs ) analysis by @xcite as well as to the intrinsic mean on a first geodesic principal component ( imo1gpc ) for manifolds and kendall s shape spaces . a nested bootstrap two - sample test is derived and illustrated with simulations . in a study on real data , pngs is applied to track early human mesenchymal stem cell differentiation over a coarse time grid and , among others , to locate a change point with direct consequences for the design of further studies . _ keywords : _ frchet means , dimension reduction on manifolds , principal nested spheres , asymptotic consistency and normality , geodesic principal component analysis , kendall s shape spaces , flags of subspaces _ ams subject classifications : _ primary 62g20 , 62g25 . secondary 62h11 , 58c06 , 60d05 .	26,363	463
there is almost irrefutible evidence for an increase in the star formation density with redshift , as demonstrated by emission line and continuum star formations tracers in wavebands from the ultraviolet to the submillimeter and radio wavebands . this evolution appears to be stronger for tracers which are less sensitive to dust obscuration ( e.g. ivison et al . 2006 ) , suggesting that an increasing proportion of the activity in more distant galaxies may be highly obscured ( e.g. blain et al . 1999 , 2002 ) . indeed , recent results on the mid- to far - infrared emission of luminous but dust obscured galaxies at high redshift ( @xmath103 ) suggests that the origin of their large infrared luminosities is a mix of dust obscured vigorous star formation and/or dust enshrouded active galactic nucleus ( agn ) ( yan et al . 2005 ; houck et al . 2005 ; lutz et al . 2005 ; desai et al.2006 ) . in many sources it is likely that both agn and star formation contribute to the emission as a result of the close link required between the growth of super - massive black holes and bulges in massive galaxies ( e.g. borys et al.2005 ) . one of the best - studied populations of high - redshift , far - infrared luminous galaxies is that identified in the submillimeter waveband using the scuba camera ( holland et al . 1999 ) on the james clerk maxwell telescope ( jcmt ) . although they span less than an order of magnitude in submillimeter flux , these galaxies are responsible for much of the energy density in the submillimeter background ( barger et al . 1998 ; hughes et al . 1998 ; smail et al . 2002 ; cowie , barger & kneib 2002 ; scott et al . the faintness of these obscured galaxies in the optical waveband has made it difficult to obtain precise redshifts ( e.g. simpson et al . 2004 ) , although some progress has been made using ultraviolet / blue spectrographs ( chapman et al . 2003a ; 2005 ) . the median redshift for submillimeter galaxies with 850@xmath11 m fluxes of @xmath12mjy , ( hereafter smgs ) is @xmath13 ( chapman et al . 2003a , 2005 ) . the submillimeter and radio fluxes of these systems indicate their bolometric luminosities are @xmath14l@xmath15 ( kovacs et al . 2006 ) , confirming that they are examples of high - redshift ultraluminous infrared galaxies ( ulirgs ) . this population provides critical constraints on models of galaxy formation and evolution . in particular , if the bolometric emission from smgs is powered solely by star formation , then these galaxies form about half of the stars seen in the local universe ( lilly et al.1999 ) . however , it appears likely that both agn and star formation activity contribute to the immense far - infrared luminosities of these systems , although it has been difficult to disentangle the precise balance between these two energy sources . recent sensitive x - ray analysis suggest that star formation is likely to be the dominant source of the bolometric luminosity in smgs ( alexander et al.2005a , b ) . further evidence suggest it is plausible to identify smgs as the progenitor of massive elliptical galaxies at the present - day , based on their large gas , stellar and dynamical masses ( neri et al . 2003 ; greve et al . 2005 ; tacconi et al . 2006 ; smail et al . 2004 ; borys et al . 2005 ; swinbank et al . 2004 , 2006 ) . furthermore , combining the x - ray constraints on the agn within this population with the typical mass estimates suggests that smgs are the sites of coeval growth of stellar bulges and central black holes ( borys et al . 2005 ) . rest - frame optical emission lines provide a powerful tool to investigate many fundamental properties of galaxies , such as star formation rates ( sfrs ) , power sources , internal extinction and metallicity . swinbank et al . ( 2004 ) conducted a systematic near - infrared spectroscopic survey of thirty smgs to investigate their sfrs and metallicities and the kinematics of the emission line gas . however , the wavelength coverage was limited to the region around h@xmath3 and so they did not include several emission lines at shorter wavelengths , such as h@xmath0 and [ oiii]@xmath1 , which are useful for evaluating internal extinction and metallicity or determining the power source . we present in this paper the results from a near - infrared spectroscopic survey of redshifted [ oiii]@xmath1 , h@xmath0 and [ oii]@xmath2 lines for a sample of far - infrared luminous galaxies . the sample is composed of smgs and optically faint radio galaxies ( ofrgs ) , at @xmath103.5 . chapman et al . ( 2004 ) and blain et al . ( 2004 ) claim that high - redshift ofrgs are ulirgs , with similar bolometric luminosities to smgs but warmer characteristic dust temperature , resulting in them being undetectable in the submillimeter waveband . we use h@xmath3/h@xmath0 emission line ratios to derive the dust extinction in these systems and then employ these estimates to derive extinction - corrected sfrs from the h@xmath3luminosities . in addition , we also use x - ray observations of these objects to compare the strength of the [ oiii]@xmath7 emission to their x - ray emission , and so investigate the power of the agn in these galaxies . we adopt cosmological parameters of h@xmath1672 km sec@xmath17 mpc@xmath17 , and @xmath18 and @xmath190.7 throughout . our sample was selected from the catalogs of smgs and ofrgs in chapman et al . ( 2005 , 2004 ) . we chose smgs / ofrgs in the redshift ranges @xmath202.56 and @xmath211.68 , where nebular emission lines such as [ oii ] , h@xmath0 , [ oiii ] and/or h@xmath3 are redshifted into clear parts of the @xmath22 , @xmath23 and @xmath24-bands respectively . in total 22 targets were observed using the ohs spectrograph on subaru , isaac on the vlt or nirspec on keck . the log of the observations is given in table 1 . the majority of our spectroscopic observations were taken with the oh suppression spectrograph ( ohs ; iwamuro et al . 2001 ) with the cooled infrared spectrograph and camera for ohs ( cisco ; motohara et al . 2002 ) attached to the nasmyth focus of subaru telescope ( iye et al . observations were obtained on the nights of 2004 april 6 , 7 , june 2425 , and 2005 feb 1416 . sky conditions were photometric on all these nights with typical seeing 0.50.7@xmath25 at 1.6@xmath11 m . we used a slit width of 0.95@xmath25 , which gives a resolution of @xmath26/@xmath27 ( @xmath28kmsec@xmath17 ) and used the `` sp4 '' dither pattern , which shifts the object along the slit to four positions in one sequence . after completing each observation , we observed bright a- or f - type stars with the same configuration as the science observation to calibrate the extinction and sensitivity variation with wavelength . during each night we observed at least two photometric standard stars selected from the ukirt faint standards catalog ( hawarden et al . we used fs27 and fs127 for the observations taken in 2004 april , fs23 and fs30 in 2004 june and fs133 and fs127 in 2005 february . the data reduction was performed in the standard manner using custom scripts in iraf and some c programs provided by the ohs / cisco instrument teams . first , we subtracted the sky background using the object frames at different dither positions . next we fitted the sky line residuals using two dimensional polynomials and subtracted these from the data . we then shift - and - added the images from the different dithering positions , using a median combine . as the instrument is stable , wavelength calibration was performed using the nominal conversion of pixel coordinates to wavelength . to confirm the stability of the wavelength solution , we analysed argon calibration lamp exposures taken during our runs and checked for systematic shifts in wavelength . we found typical systematic shifts of 79 ( @xmath29% ) which is ignorable in our analysis due to the low resolution of our spectra . extinction , sensitivity and photometric calibration were performed by dividing the calibrated spectra with those of the bright a- or f - type standard star observations after fitting the stellar spectra with models . we conducted observations of four smgs and one ofrg using the isaac spectrograph on the 8-m vlt on 2004 november 2223 ( table 1 ) . isaac was used in medium - resolution mode , which provides spectral resolution of 3000 ( @xmath30 kmsec@xmath17 ) . seeing was steady at @xmath31 over the course of the observations and the observations were taken with a standard 10@xmath25 abba chop . preliminary data reduction was performed using the eclipse pipeline , using flat - fields generated from night - calibrations taken after each observation , and wavelength calibration from a solution using the oh sky lines . the remaining flux - calibration was achieved in iraf , using corresponding hipparcos standard stars observed throughout the observing run and near - infrared fluxes derived from the 2mass catalog . the observations of smmj09431 + 4700 ( h6/h7 ) and smmj131201.17 + 424208.1 were taken on 2004 april 8 in photometric conditions and 0.8@xmath25 seeing using the nirspec spectrograph on keck . these observations employed the standard abba configuration to achieve sky subtraction . each exposure was 600s in length and the total integration time was 2400s . the data were reduced using the wmkonspec package in iraf . we remapped the two dimensional spectra using linear interpolation to rectify the spatial and spectral dimensions . after subtracting pairs of nod - positions ( the nod was 20@xmath25 along the slit ) , residual sky features were removed in idl using sky regions on either side of the object spectrum . for the wavelength calibration we used an argon arc lamp . the output pixel scale is 4.3 pix@xmath17 , and the instrumental profile has a fwhm of 15 ( measured from the widths of the sky - lines ) , which corresponds to @xmath32 kmsec@xmath17 . we used fs27 for photometric calibration . we show all of our spectra in figure 1 . we identified emission lines in 20 spectra out of 22 targets which were observed . most of the smgs show weak h@xmath0 emission , but many show strong ( and sometimes broad and distorted ) profiles in [ oiii]@xmath1 . some of our spectra show additional emission lines of [ neiii ] , [ nev]and [ oi]@xmath33 , which are common in agn . five of the smgs from our sample ( smmj09431 + 4700 ( h6 ) , smmj123549.44 + 621536.8 , smmj123716.01 + 620323.3 , smmj163639.01 + 405635.9 , and smmj163650.43 + 405734.5 ) display spatially extended structures ( @xmath341.0 ) in either [ oiii]@xmath7 and/or h@xmath3 emission line ( figure 2 , see smail et al . 2003 ; swinbank et al . 2005 for evidence of the spatial extension in smmj163650.43 + 405734.5 ) . several of our observations are particularly noteworthy and we discuss them here . this galaxy is identified as a submillimeter source associated with a spiral galaxy at @xmath36 , which shows features typical of a seyfert 1 ( smail et al . 1997 , 2002 ; soucail et al . the strong and featureless continuum , together with the spatially compact emission line flux indicates agn activity ; an interpretation which is further supported by the detection of this source in hard x - rays by bautz et al . ( 2000 ) . our spectrum shows at least two peaks in the h@xmath3 emission line with fwhm@xmath37400kmsec@xmath17 , consistent with these lines arising from independent components within the system . if we force fit a single gaussian profile to the h@xmath3 emission , we determine fwhm@xmath38kmsec@xmath17 , which if it arises from an agn is narrower than typical seyfert 1 galaxies , although broader than seyfert 2 galaxies ( @xmath39kmsec@xmath17 ) . this source is also detected by co observation by greve et al . ( 2005 ) with a double peaked profile with a fwhm of @xmath40kmsec@xmath17and a separation between the two peaks of @xmath41kmsec@xmath17 , consistent within the errors with our measurements from h@xmath3 . we therefore choose to interpret the double - peaked h@xmath3 line as evidence for a merger or interaction in this system , with any agn - produced broad component undetected in our spectrum . this source was discovered by cowie , barger & kneib ( 2002 ) , and has been identified with two distinct @xmath11jy radio counterparts : h6 and h7 ( ledlow et al . these are lensed sources , lying behind a massive cluster abell851 at @xmath42 although the amplification is modest : @xmath43 . the redshift for h6 was measured by ledlow et al . ( 2002 ) as @xmath44 from ly@xmath3 , h7 was not observed . the restframe ultraviolet properties of h6 suggest it hosts an agn with spectral features similar to a narrow - line seyfert 1 ( ledlow et al . we placed the nirspec slit across both radio components and detected [ oiii]@xmath7emission from both sources at redshifts of @xmath45 and @xmath46 for h6 and h7 , respectively . we also detected narrow ( fwhm@xmath47 kmsec@xmath17 ) h@xmath0 emission from h6 . the [ oiii]@xmath7 emission from h6 is spatially extended(@xmath48 or 14.5kpc ; figure 2 ) , but has no significant velocity gradient across @xmath49kpc in projection . no hard x - ray emission was detected with the upper limits on f@xmath50 as @xmath5110@xmath52 erg sec@xmath17 @xmath53 ( ledlow et al . co line emission is also detected by neri et al . ( 2003 ) and tacconi et al . ( 2006 ) based on our restframe optical redshift , originating from h7 at @xmath54 . millimeter continuum emission has been seen from h6 , but assuming the gas reservoir is at the redshift we find from [ oiii]@xmath7 , the gas mass of the agn - dominated component , h6 , is a factor of a few lower than that of h7 . this source has apparent double - peaked , narrow ( @xmath55kmsec@xmath17 ) emission lines in [ oii]@xmath2 and [ oiii]@xmath1 , with the two components spatially offset by @xmath56 . the one dimensional spectra also shows signs of broad h@xmath0emission at @xmath57 with a fwhm of @xmath58 kmsec@xmath17 . both the [ oiii]@xmath7 and the [ oii]@xmath2 emissions are spatially extended with faint wings on scales of approximately 1@xmath25 ( @xmath49kpc ) , see figure 2 . there may also be a very weak , broad multiplet of feii@xmath59 ( figure 1 ) , potentially indicating the presence of the narrow line seyfert 1 ( nls1 ) type agn component ( osterbrock & pogge 1985 ; goodrich 1989 ) . this is consistent with the results of alexander et al . ( 2005b ) , which indicated the presence of a heavily obscured agn with n@xmath60@xmath53 based on their x - ray spectral analysis . the spatial extension in the bright core of the [ oiii]@xmath7 likely indicates merging components or rotation along the slit , while the extended wings may reflect `` superwind '' activity . this source is very bright in the optical ( @xmath61 ) with a redshift of @xmath62 and it was classified as a qso by chapman et al . ( 2005 ) based on the broad rest uv emission lines and comparable luminosities in rest optical and far - infrared wavelength , which exceed 10@xmath63 erg sec@xmath17 . the source has also been detected in hard x - rays by alexander et al . ( 2005b ) . our spectrum shows several hydrogen balmer lines such as h@xmath0 , h@xmath64 and h@xmath65 with broad fwhm@xmath66 ( @xmath672700kmsec@xmath17 ) and the [ oiii]@xmath1 doublet with fwhm@xmath66 of @xmath67kmsec@xmath17 . we also detected the [ neiii ] and several feii lines at 34@xmath68 significance . the restframe optical spectrum is dominated by continuum emission without stellar absorption features , suggesting a large contribution from the agn component to the total rest - frame optical flux . the [ oiii]@xmath7 emission lines are wide fwhm@xmath69kmsec@xmath17 and spatially extended ( @xmath70 ; 12kpc ) indicating dynamically active gas motion ( figure 2 ) . the estimated hydrogen column density from the x - ray spectral analysis is relatively low ( n@xmath71@xmath53 ) , which implies the agn does not suffer from large extinction . it should be noted that the redshift based on the restframe - uv emission lines is @xmath72 , which is blueshifted by @xmath73kmsec@xmath17 from the redshift indicated by the restframe optical nebular emission line . this velocity offset may arise due to broad ly@xmath3 emission which may be affected by dust extinction and resonance scattering . this source is another example of a nls1 type agn . it lies at @xmath74 and our spectrum displays broad h@xmath0 emission , with fwhm@xmath75kmsec@xmath17 and a low [ oiii]@xmath7/h@xmath0 ratio ( @xmath76 ) . this source has ly@xmath3 [ civ ] and heii emission lines in the rest uv spectrum and was classified as a qso by chapman et al . the rest - frame optical emission is dominated by very strong continuum emission without stellar absorption lines , supporting the presence of a luminous agn component . unfortunately , there is no coverage of h@xmath3 emission for this object and so we could not constrain the internal extinction . the [ nev ] line ( which is a very clean indicator of agn activity ; osterbrock 1989 ) is detected . furthermore , this source was detected by the x - ray imaging by mushotzky et al . ( 2000 ) , confirming the presence of a luminous agn in the source . this source is a good example of a heavily extincted starburst in an smg and was recently discussed by swinbank et al . this @xmath77 galaxy has weak h@xmath0 emission line with h@xmath3/h@xmath0@xmath78 . the h@xmath3 and [ oiii]@xmath7 emission lines are spatially extended ( @xmath79 or 10 kpc)(figure 2 ) . there is only an upper limit on its x - ray emission , f@xmath80ergsec@xmath17@xmath53 from manners et al.(2003 ) , which does not strongly constrain the presence of a luminous agn given the possibility of substantial absorption ( e.g. alexander et al . the possible detection of [ oi]@xmath33 emission line may hint at the presence of an agn , although the line ratios of [ oi]@xmath33/h@xmath3@xmath81 and [ oiii]@xmath7/h@xmath0@xmath82 can be explained by a relatively highly ionized starburst nebulae ( osterbrock et al . 1989 ) . this heavily obscured agn at @xmath83 ) was found in the mambo survey of greve et al . ( 2004 ) ( and is also called n21200.18 ) and was detected in x - ray imaging with _ chandra _ ( manners et al . it has broad , fwhm@xmath842500kmsec@xmath17 , emission lines of ly@xmath3 , [ civ ] and h@xmath3 in the rest - frame uv and optical wavelengths , with a high [ oi]@xmath33/h@xmath3 ratio ( @xmath85 ) ( willott et al . 2003 ; swinbank et al . 2006 ) which is typical of agn ( osterbrock 1989 ) . our data also show asymmetric h@xmath0 and [ oiii]@xmath86 emission line profiles , which exhibit `` blue wings '' in their profiles . such profiles have been interpreted as evidence for wind activity from the agn , although contribution from other components is possible ( swinbank et al . 2006 ) . our isaac spectrum shows strong , narrow h@xmath3 h@xmath0 [ oiii]@xmath1 and [ nii ] emission lines at a redshift of @xmath87 ( fwhm@xmath88 of h@xmath0 is @xmath89kmsec@xmath17 ) . to investigate the restframe optical properties , we retrieved an archival @xmath90-band image taken with subaru telescope s prime focus camera ( suprime - cam ) using smoka . the image shows an elongated structure , @xmath91 , towards the north - west and the spectrum was taken with the slit aligned along the major axis of this source . we identify two separate h@xmath3 emission lines with a velocity offset of @xmath92kmsec@xmath17and a spatial offset @xmath93@xmath94 ( @xmath95kpc ) . these suggest the system is a merger . the h@xmath3 and h@xmath0 emission lines do not show asymmetric profiles or detectable broad line components . since many of our individual spectra have modest signal - to - noise , we have also constructed several composite spectra to investigate the general properties of subsets of the smg population . we create the composite spectra by deredshifting each spectrum based on redshifts measured from the [ oiii]@xmath7 lines , subtracting continuum emission using a first order spline fit and averaging all of the spectra with 3-@xmath68 clipping after normalizing by [ oiii]@xmath7flux . we smoothed the higher resolution spectra taken at keck and vlt to match the low resolution subaru spectra before stacking . either stacking the spectra with weights based on their individual signal - to - noise ratio or an unweighted stack does not alter any of the conclusions below . we derive a composite spectrum for those sources which show qso signatures ( `` qso '' ; i.e. , classified as qso ) and for those galaxies that individually show signs of an agn in their optical spectra ( `` opt - agn '' ; i.e. , those classified as agn in the column of `` class '' under `` opt '' category in table 2 ) . the former is made from only three individual spectra , while the latter comes from nine spectra . the resulting composite spectra are shown in figure 3 . we do not make a composite of starburst ( `` sb '' ) sources since there are only two sources in our sample classified as `` sb '' or intermediate ( `` int '' ) from their restframe optical spectra . the details of the classification will be discussed in 4.1 . the emission lines of h@xmath0 and [ oiii]@xmath1 lines are clearly seen in both the composite spectra . in addition in the `` qso '' spectrum , many strong lines are visible , including [ neiii]@xmath96 and several feii lines at @xmath97 , 5167 and 52005360 , although the [ oii]@xmath2 line is only marginally detected . by fitting a gaussian to the h@xmath0 and [ oiii]@xmath1 emission lines , we measure the fwhm@xmath88 of h@xmath0as @xmath98kmsec@xmath17 after correction for the instrumental resolution . this is @xmath99kmsec@xmath17 lower than the average fwhm of qsos at @xmath1002.1 ( jarvis & mclure 2006 ) . the [ oiii]@xmath7/h@xmath0 ratio is @xmath101 . all these spectral features are typical of type 1 agns studied locally . on the other hand in the composite `` opt - agn '' spectrum , a gaussian profile fit to the h@xmath0 emission line yields fwhm@xmath88 of @xmath102kmsec@xmath17 ( it should be noted that the h@xmath0 line fit is not improved by including a narrow line component due to the low spectral resolution of our spectra ) and [ oiii]@xmath7/h@xmath0ratio of @xmath103 , in addition the [ oii]@xmath2 line is well - detected . the h@xmath0 line , which is broader than typical type 2 agns , and relatively low [ oiii]@xmath7/h@xmath0 line ratio , is similar to that of local nls1 ( although by definition these should have [ oiii]@xmath7/h@xmath0/@xmath1043.0 ) . the feii emission lines , which are one of the characteristic features seen in local nls1 s , are marginally detected with @xmath105 features seen around 5200 in the spectrum , and we can see some marginal detections in individual spectra ( smmj123549.44 + 621536.8 , smmj123635.59 + 621424.1 , smmj163650.43 + 405734.5 , and smmj163706.51 + 405313.8 ) , all of which have broad h@xmath0 emission of fwhm@xmath106kmsec@xmath17(figure 1 ) . the resultant spectrum is consistent with a scenario where the restframe optical spectra classified as `` agn '' in the uv in reality comprise two types : one has relatively broad , @xmath107kmsec@xmath17 , fwhm for the h@xmath0 lines and the other has narrow h@xmath0 lines with a relatively high [ oiii]@xmath7/h@xmath0 ratio , typical of type 2 agns . there are clearly differences in the extinction of the circumnuclear region of these two types of objects implied by the difference in luminosity and spectroscopic properties of the restframe - uv emission , although there is no systematic difference in the h@xmath3/h@xmath0 ratio we measure for them . in figure 4 , we plot the observed [ oiii]@xmath7/h@xmath0 versus [ nii]/h@xmath3emission line ratios of the 13 galaxies in our sample for which we have secure h@xmath3 detections and some information about [ nii]/h@xmath3 . this diagnostic plot , termed the bpt diagram , can be used to identify the source of gas excitation ( baldwin et al . 1981 ) . based on this diagram we classify the spectra into three types , starburst ( sb ) , intermediate ( int ) or non - thermal ( agn ) , as listed in table 2 . we use the definitions from kauffman et al . ( 2003 ) which are derived for a large sample of local sdss galaxies . we classify the sources between the boundary of kauffman et al . and the classical definition of veilluex & osterbrock ( 1987 ) as `` int '' . we also classify galaxies as agn which have h@xmath3 and/or h@xmath0 fwhm@xmath88 greater than 1500kmsec@xmath17 , as it is difficult to understand the formation of such large line widths from gas motions in star - forming regions . this limit is also greater than the coarse spectral resolution of ohs ( @xmath108kmsec@xmath17 ) . for comparison we also plot the emission line flux ratios from local ulirgs ( veilleux , kim & sanders 1999 ) and note that the smgs in our sample occupy the same region of the diagnostic diagram as local ulirgs . the curves show various criteria for separating agns and the star - forming galaxies ( see figure 4 ) . it is clear that the majority , 8/13 , of sources in our sample ( including all but one , smmj163639.01 + 405635.9 , with all four emission lines detected ) are classified as agn based on these criteria . we reiterate that this subsample may be biased towards strong line emitters ( due to the requirement to have detected lines in our low - resolution spectra ) and so this is perhaps not a surprising result . moreover , we must interpret the bpt diagram with caution since `` superwind '' ejecta ( shock - driven line emitting gas ) can occupy a very similar region to agn ( dopita & sutherland 1995 ) . to illustrate this possibility in more detail , we plot the emission line ratio of the wind structure in m82 from a @xmath109kpc region ( shopbell & bland - hawthorn 1998 ) and in a @xmath110kpc@xmath111 area of ngc6240 ( schmitt et al . 1996 ) . the former is indicative of a wind which is dominated by photoionization , and the latter illustrates the line ratios expected from shocks in a very dense environment . some smgs show very similar emission line ratios to ngc6240 , although most of them have lower [ nii]/h@xmath3 and higher [ oiii]@xmath7/h@xmath0 ratios . further support for the wind scenario is that p - cygni features are seen in the rest - uv emission lines ( chapman et al . 2003a , 2005 ) of a significant fraction of the smg population , supporting the presence of `` winds '' arising from the vigorous starburst activity . indeed , the majority ( 6/8 ) of this subsample ( with four detected emission lines ) are classified as `` sb '' or `` int '' from their rest uv spectroscopic features ( chapman et al . 2005 ; table 2 ) . the power sources in smgs is discussed further in 4.2 . looking at the individual sources in figure 4 , we note that smmj123622.65 + 621629.7 has a very low [ nii]/h@xmath3(@xmath112 ) emission line ratio and no detection of [ oiii]@xmath1 and h@xmath0(see also figure 5 in smail et al . 2004 ) with h@xmath3/h@xmath0@xmath1134.25 and [ oiii]@xmath7/h@xmath3@xmath114 . this sources is an interacting system of a relatively blue(@xmath115 ) galaxy with extremely red ( @xmath116 ) companion , where the latter is a hard x - ray source . whilst the slit was aligned along the major axis of the red x - ray source , it is possible that it also passed through the blue component and the line emission may be contaminated . to avoid biasing our sample , we have therefore eliminated this source from our subsequent analysis and discussion . smgs are dusty systems with large dust masses , @xmath117m@xmath15 , and high bolometric luminosities ( @xmath118 ) . the presence of large quantities of dust and its associated reddening may also explain the large discrepancies between the sfrs derived for smgs from their far - infrared and h@xmath3luminosities ( swinbank et al . 2004 ) , which imply extinction in h@xmath3 of factors @xmath119100 . there are of course alternative explanations : that the bulk of the far - infrared emission originates from other sources which are too dusty to see even at restframe optical wavelengths , such as very highly obscured agn , or due to emission which falls outside of the slits used in the h@xmath3 measurements . although , the latter explanation is unlikely as these observations are based on radio - identified sources with precise positions ( @xmath120 ; chapman et al . 2005 ) , and so it is unlikely that a major source of bolometric emission has been missed by the observations . to investigate the internal reddening of smgs ( at least for those regions which are visible in the restframe optical ) we plot the h@xmath3/h@xmath0 ratios as a function of their far - infrared luminosities in figure 5 to calculate @xmath121 , we use the reddening curve from calzetti et al . ( 2000 ) , and assume an intrinsic h@xmath3/h@xmath0 ratio of 3.0 , which is between the values for typical seyfert 2 galaxies and/or liners ( 3.1 , halpern & steiner 1983 ; gaskel & ferland 1984 ) and star - forming galaxies , 2.85 ( veilleux & osterbrock 1987 ) . the observed h@xmath3/h@xmath0 ratio for the smgs is typically 520 and the derived extinction spans @xmath1224 with a median value of @xmath123 ( where the error comes from bootstrap resampling ) . this estimate is consistent with the results based on the spectral energy distribution ( sed ) fitting of optical to near - infrared photometric data ( smail et al . 2004 ) , and slightly higher than that derived from optical to mid - infrared seds ( @xmath124 , borys et al.2005 ) where the latter did not include any contribution from thermally pulsed - agb stars in the model seds , which might lead to an underestimation of the reddening ( maraston 206 ) . in figure 6 we compare the extinction corrected sfrs derived from the h@xmath3 and far - infrared luminosities . the far - infrared luminosities come from chapman et al . ( 2003b ; 2004 ) based on sed model fitting to the observed 850@xmath11 m and 1.4-ghz fluxes at their known redshifts , assuming the local far - infrared - radio correlation holds ( condon et al . 1991 ; garrett 2002 ) . we also include observations for local _ iras _ galaxies ( kewley et al . 2002 ) and _ iso _ galaxies ( flores et al . the typical @xmath121 in these samples are @xmath125 and @xmath126 , respectively . the extinction corrected h@xmath3 luminosities for the _ iras _ and the _ iso _ galaxies are all calibrated in the same manner as for our smg samples based on their h@xmath3/h@xmath0 ratio . the sfrs from the h@xmath3 and from the far - infrared luminosities are derived using the equations given in kennicutt ( 1998 ) . the correlation between the far - infrared and reddening - corrected h@xmath3 luminosities ( figure 6 ) appears to be relatively good with a linear relation extending over five orders of magnitude in sfr , although with some scatter , with the most luminous smgs in our sample having sfrs approximately an order of magnitude higher than those of the brightest _ iso _ galaxies . the good agreement between the two sfrs when using the reddening - corrected h@xmath3-estimate confirms that the discrepancies between the sfrs seen in swinbank et al . ( 2004 ) are in large part due to dust extinction and moreover that the bulk of the far - infrared luminosity in these galaxies is probably derived from star formation . we note that it is likely that slit - losses and placement contribute to the scatter in these measurements as we are combining observations of h@xmath0 and h@xmath3 from different telescopes and instruments . for example , our brightest far - infrared source , smmj163650.43 + 405734.5 ( n2850.4 ) , has a lower sfr measured from h@xmath3 than from the far - infrared . however , this galaxy is spatially extended and has a very complex structure in the restframe optical ( smail et al . 2003 ; swinbank et al . it is therefore likely that our slit covered only a part of the h@xmath3 emitting region . finally , any remaining systematic offset between the two sfr estimates may be caused by the fact that our a@xmath127 estimates only reflect the reddening to the optically detectable gas and thus are not necessarily a good indicator of the total column towards the bolometric sources in these objects . smgs are proposed to be the progenitors of present - day massive spheroidal galaxies , because of their high star - formation rates and their large stellar , gas and dynamical masses ( smail et al . 2004 ; neri et al . 2003 ; greve et al . 2005 ; borys et al . 2005 ; tacconi et al . 2006 ; swinbank et al . most massive galaxies in the local universe contain super - massive black holes ( smbhs ) ( e.g. , ferrarese & merritt 2000 ; gebhardt et al . 2000 ; marconi & hunt 2003 ; heckman et al.2004 ) . equally an agn appears to be almost universally present in smgs : based on the extremely sensitive x - ray observations of the _ chandra _ deep field north ( cdfn ) , alexander et al . ( 2003 ; 2005a , b ) found more than @xmath128% of smgs to be detected in hard x - rays , indicating they contain an accreting smbh . it is therefore interesting to estimate the mass of , and accretion rates onto , the central black holes of smgs to constrain the coevolution of the smbhs and the stellar masses of their surrounding bulge ( kawakatu et al . 2003 ; granato et al . 2004 ) . the three sources classified as `` qso '' in our sample have characteristics typical of local nls1 : low [ oiii]@xmath7/h@xmath0 ratios ( @xmath1291.8 ) , and detectable feii emission in their individual and also composite spectra ( figures 1 & 3 ) . nls1 are commonly interpreted as hosting rapidly growing smbhs ( collin & kawaguti 2004 ) , and hence the spectral similarities of these smg - qsos with local nls1s could imply comparable physical conditions in the accretion disk around the smbh in the smgs . however , the smgs have fwhm@xmath1302500kmsec@xmath17for their balmer emission lines , and so they are not formally nls1s because these lines width are higher than the definition used for nls1 ( fwhm@xmath88 of h@xmath0 of @xmath131kmsec@xmath17 ) . nevertheless , it is however worth noting that their h@xmath0 fwhm@xmath88 are close to the minimum for qsos at @xmath1322.1 ( jarvis & mclure 2006 ) , and narrower than the average of radio quiet / loud qsos(@xmath1336500kmsec@xmath17 ) . although , we caution that with the limited signal to noise in our spectra we may underestimate the line widths , missing weak and broader line components . for instance , in the composite spectrum of smg - qsos we estimate the fwhm@xmath88 of the h@xmath0 line as @xmath134kmsec@xmath17 using a single gaussian fit . this is 5001000kmsec@xmath17 broader than the mean of the individual spectra , suggesting there may be an undetected broad component present in them . more secure estimates of the line widths would either need observations of stronger emission lines such as h@xmath3 which are not available for these sources , or much deeper observations . the fwhm@xmath88 of the balmer emission lines in those smgs with agn - like features ( but omitting the three sources classified as `` qso '' ) , are 10003000 kmsec@xmath17 . they are at least 10002000 kmsec@xmath17 lower than the average fwhm of qsos at @xmath1322.1 measured from h@xmath0 and/or mgii lines ( jarvis & mclure 2006 ) suggesting that the smgs host lower mass smbhs . this would support the claims of alexander et al,(2005a , b ; see also borys et al . 2005 ) based on eddington - limited assumptions . the similarities of the rest - frame optical spectral features of some smgs to nls1s implies rapid growth of the smbh in smg s nuclei . a total of 5/9 of the smgs classified as `` agn '' in our sample have relatively narrow fwhm@xmath66 ( up to @xmath1353700kmsec@xmath17 ) for their h@xmath3 or h@xmath0 emission lines , and 3/5 show marginal feii emission . therefore , the eddington - limited accretion determined for local nls1 galaxies may also be appropriate for smgs . assuming this , the measured line - widths are then consistent with the estimate of the central bh masses derived from their x - ray luminosities under the assumption of eddington - limited accretion ( @xmath136 , alexander et al . 2005a ) . however , this conclusion appears to be undermined by the fact that three of these nls1-like smgs display high ( @xmath137 ) [ oiii]@xmath7/h@xmath0 ratios which far exceed the nls1 definition of [ oiii]@xmath7/h@xmath0 @xmath138 , and thus these comparisons may not be appropriate . to further test the claim that smgs have small smbh masses we compare the [ oiii]@xmath7 and hard x - ray luminosities . there is a well - studied correlation between the hard x - ray and the optical [ oiii]@xmath7 emission line luminosities in local agn ( e.g. mulchaey et al . this correlation can be used to gauge the black hole masses and the accretion rates of agns within our sample . in figure 7 , we show the hard x - ray versus [ oiii]@xmath7 luminosities of the smgs ( uncorrected for any extinction / absorption ) . all 22 smgs in our sample have hard x - ray coverage , but of varying depth : cdfn : alexander et al . ( 2003 ) , cfrs03hr : waskett et al . ( 2004 ) , ssa13 : mushotzky et al . ( 2000 ) , ssa22 : basu - zych & scharf ( 2005 ) , and elais n2 : manners et al . we adopt the hard x - ray fluxes from these observations , although 9/22 of them yield only the upper limits . for comparison , we also plot observations of local ulirgs ( ptak et al . 2003 ; franceschini et al . 2003 ) , as well as seyfert 1 and seyfert 2 galaxies and the pg qsos , representative of more luminous type 1 agns ( alonso - herrero et al . 1997 ; mulchaey et al . all of these comparison samples are the _ observed _ luminosities : there are no extinction corrections applied to either the x - ray or [ oiii]@xmath7 measurements . figure 7 also shows the relation for seyfert 2 galaxies suggested by mulchaey et al . compared to the qsos and seyfert 1 galaxies , which are selected to represent unabsorbed hard x - ray sources , the majority of our smgs are typically an order of magnitude brighter in [ oiii]@xmath7 for a given hard x - ray luminosity . we note that a similar excess of [ oiii]@xmath7 emission is also seen in local ulirgs . could this apparent excess be due to absorption / extinction ? the typical hydrogen column densities to the agn in smgs have been determined by alexander et al . ( 2005b ) , yielding n@xmath139@xmath53 , with corrections to their hard x - ray luminosities of 2.520@xmath9 . equally , the typical [ oiii]@xmath7 luminosity correction , adopting the extinction estimated from the balmer decrement , @xmath140 , is also approximately a factor ten : @xmath141{\lambda5007}}_{corrected } } = { \rm f}_{{[oiii]{\lambda5007}}_{obs}}\cdot((h\alpha / h\beta)/(h\alpha_{0}/h\beta_{0}))^{2.94},$ ] where h@xmath142/h@xmath143 is assumed to be 3.0 ( see bassani et al . 1999 ) . unfortunately we have only one source ( smmj123549.44 + 621536.8 ) with reliable estimates of the hi column density and reddening correction which has @xmath144 and @xmath145 corrections to the hard x - ray and [ oiii]@xmath7 luminosities respectively and with yields corrected luminosities of @xmath146 ergsec@xmath17 and @xmath147 ergsec@xmath17 respectively ( figure 7 ) . as the extinction corrections for [ oiii]@xmath7 and hard x - ray luminosities run parallel to the trend in figure 7 , the [ oiii]@xmath7 excess can not be explained by a simple reddening effect . we also caution that the reddening corrections applied to the [ oiii]@xmath7 fluxes are uncertain since [ oiii]@xmath7 may arise in external shocks which suffer much less extinction than the h@xmath3/h@xmath0 ratio suggests . we also note that the apparent [ oiii]@xmath7 excess could arise simply due to the relatively shallow x - ray coverage in several of our fields where the sources only have upper - limits on their hard x - ray fluxes . however the fact that 3/4 sources with [ oiii]@xmath7 and hard x - ray detections from the cdfn , which has by - far the best x - ray data , show the excess provides good evidence for the reality of this feature . while some of the [ oiii]@xmath7 flux we see arises from the obscured agn , we suggest that the excess [ oiii]@xmath7 flux arises , at least in part , from shock - induced ( `` superwind '' ) activity . there are some cases of plausible `` superwind '' driven [ oiii]@xmath7 excesses seen in our smg sample as seen by the structured [ oiii]@xmath7 line profiles ( asymmetric / broad / multi - peaked ) and the spatially extended emission ( figures 1 & 2 ; see also smail et al . 2003 ) . in order to examine the possibility that shock - induced gas causes the excess [ oiii]@xmath7 emission , we first search for the signature of shock - excited nebular emission using the simple criterion of [ nii]@xmath148/h@xmath3@xmath149 and [ sii]@xmath150/h@xmath3@xmath151 . using the line ratios from the stacked spectrum of smgs in swinbank et al . ( 2004 ) we find that they lie outside of this shock induced criterion . however , this criterion is only valid for the shocks with large outflow velocities and a relatively weak starburst radiation field ( veilleux et al . 2005 ) and therefore may not be applicable to the smgs . another test is to use the line ratios of [ oiii]@xmath1/[oiii]@xmath152 and/or [ nii]@xmath153/[nii]@xmath154 , which can be used to estimate the temperature of the nebular gas . these ratios will provide robust estimates of electron temperature of the emission nebulae , yielding high ( @xmath155k ) temperatures if the gas is ionized mainly by shocks ( as in the cygnus loop ) and lower temperature ( @xmath156k ) for photoionization - dominated clouds seen in star - forming regions ( osterbrock 1989 ) . as these methods rely on measurements of relatively weak emission lines [ oiii]@xmath152 and [ nii]@xmath154 , only our composite spectra have sufficient signal to noise to be useful . from the composite spectrum in figure 3 and also from the total smg composite spectrum in swinbank et al . ( 2004 ) , we derive an upper limit on [ oiii]@xmath1/[oiii]@xmath152@xmath157 , and [ nii]@xmath153/[nii]@xmath158 . both ratios imply an upper limit to the electron temperature of less than 20,000k . this is consistent with the expected temperature in photoionization - dominated clouds ( with an electron density of @xmath159@xmath53 ) . if the electron density is higher than this , collisional de - excitation begins to play a role and the estimated temperature is reduced . these results would appear to rule out the dominance of shock excitation similar to that seen in galactic supernova remnants . as described in dopita & sutherland ( 1995 ) , the optical line ratios of seyfert 2s can also be explained by fast ( 300500kmsec@xmath17 ) shocks , if the precursor hii regions in front of the shock absorb most of the uv photons generated by the shocks . the calculated electron temperature is @xmath160k for this `` shock + precursor '' model from dopita & sutherland ( 1995 ) , which is consistent with the limit on the electron temperatures in smgs estimated from our [ nii ] and [ oiii ] emission line ratios . thus there is a plausible origin for the [ oiii]@xmath7 excess we see compared to typical agn : shocks associated with supernova explosions in relatively dense gas environments , where the precursor hii clouds are still present . thus we suggests that those sources with high ( @xmath16110 ) [ oiii]@xmath7/h@xmath0 ratios and broad ( @xmath1622000 kmsec@xmath17 ) fwhm of h@xmath0 lines can be explained by a combination of a nls1-type agn residing in an environment of shocks associated with supernova explosions in relatively dense gas . this would explain all their observable properties , including the high [ oiii]@xmath7/h@xmath0 ratios ( dopita & sutherland 1995 ) . using near - infrared spectroscopy we have observed the redshifted h@xmath0 the [ oiii]@xmath1 and [ oii]@xmath2 emission lines in a sample of 22 ultra - luminous infrared galaxies at high redshifts . twenty of the sources in our sample are submillimeter galaxies at @xmath1633.5 . combining our observations with previous studies of the h@xmath3 and the [ nii ] emission from these galaxies and also with observations of their hard x - ray and far - infrared emission , we have placed constraints on the physical properties of this population . we conclude the following : 1 . a majority of our sample ( 14/22 ) have spectra which are classified as `` agn '' or `` qso '' based on several restframe optical spectroscopic diagnostics . specifically , for those sources with detections of the four emission lines necessary to construct a bpt diagram , 8/9 are classified as `` agn '' . it should be noted that there is no confirmed pure starburst galaxy in our sample , although several sources show intermediate spectral properties . this is likely to be caused by our sample selection , which is biased towards galaxies with bright near - infrared magnitudes and also to those exhibiting strong line emission . thus we caution that our results should not be taken as representative of the whole smg population . 2 . using the h@xmath3/h@xmath0 flux ratio we are able to estimate the internal extinction in our smgs . we measure a median extinction of @xmath164 , which is similar to the extinction measured in local ulirgs . this value is also consistent with the estimates from the sed fitting in the restframe uv / optical which are derived under the assumption of a dominant dust - reddened young starburst ( smail et al . we compare the sfrs derived from the dust - extinction - corrected h@xmath3 luminosities with those derived from the far - infrared luminosities , and find reasonable consistency between these for most of the smgs in our sample . the fact that the corrected h@xmath3-derived sfrs correspond closely to those estimated from the far - infrared suggests that star - formation is the major contributor to the far - infrared luminosities in smgs . at least 11/19 of the smgs in our sample show a clear excess in the ratio of their [ oiii]@xmath7 to x - ray luminosities relative to values for local agns . the five sources with the highest [ oiii]@xmath7/h@xmath0 ratios ( @xmath137 ) , which are classified as `` agn '' from our spectral diagnostics , show this [ oiii]@xmath7 excess . one possible explanation for the [ oiii]@xmath7 excess is that it is produced by `` compton - thick '' agns . however , this is inconsistent with the column density measurements ( n@xmath165 ) from fitting of the x - ray spectra for the sources in cdfn and we argue that this is unlikely in most smgs . instead , we suggest that the most plausible cause of the [ oiii]@xmath7 excess is shock - induced emission arising from vigorous star formation ( `` super - wind '' activity ) . this scenario is supported in several galaxies by spatially extended and/or distorted / multiple [ oiii]@xmath7 emission line profiles . furthermore , using limits on the electron temperatures from [ oiii ] and [ nii ] emission line ratios , we can explain the excess [ oiii]@xmath7 emission as arising from shocks in dense regions within these systems . the balmer line widths in 9/22 sample galaxies exhibit broad emission components with relatively small fwhms ( @xmath1663700kmsec@xmath17 ) . three of them are classified as `` qso '' , but have smaller h@xmath0 fwhm ( 21002600kmsec@xmath17 ) than are typical for qsos . they also have lower [ oiii]@xmath7/h@xmath0 ratios and relatively strong feii emission , both of which are characteristics of local narrow line seyfert 1s . among the other six sources , only one shows a low [ oiii]@xmath7/h@xmath0 ratio , and four show high [ oiii]@xmath7/h@xmath0 ratios ( larger than seen in nls1 s ) . however , the high [ oiii]@xmath7/h@xmath0 ratios may arise from [ oiii]@xmath7 excesses due to shock excitation and hence removing this contribution would yield lower ratios more consistent with nls1 classification . several of these sources also have tentative evidence for feii emission , again characteristic of nls1s . thus , once account is taken of the potential contribution from shocks to the excess [ oiii]@xmath7 emission , there appears to be close similarities between smgs and nls1s . the spectral classification of smgs as nls1s may then indicate ( as has been claimed for local nls1s ) that smgs have small mass black holes which are rapidly growing at high accretion rates ( alexander et al . 2005ab ; borys et al . deeper spectroscopic observations are essential to search for any obscured broad balmer lines which might indicate larger smbh masses and confirm the presence of feii lines which are common in the nls1s . summarising our results : we conclude that our sample of smgs contains a population of vigorously star - forming galaxies with high sfrs and strong extinction . the activity in these systems is driving shocks through the dense gas reservoirs they contain and some of this material is being expelled from the galaxies . in addition , many of our sources show evidence for low - mass , but rapidly growing , super - massive black holes . these results confirm the critical place of the submillimeter - bright phase in defining the properties of massive galaxies forming at high redshifts . we are grateful to michael balogh , bob nichol , chris miller and dave alexander for their providing invaluable information and discussions . tt and ks are also thank to all staffs of subaru telescope , especially to dr . kentaro aoki and dr . takuya fujiyoshi for the supports on our subaru / ohs observation . we also thank to the anonymous referee for the various comments and suggestions to improve our manuscript . irs acknowledges support from the royal society . jeg acknowledges support from a pparc postgraduate studentship . ams acknowledges a pparc fellowship . alexander , d.m . 2003 , 126 , 539 alexander , d. , smail , i. , bauer , f. , chapman , s.c . , blain , a.w . , ivison , r. 2005a , 434 , 738 . alexander , d. , smail , i. , bauer , f. , chapman , s.c . , blain , a.w . , ivison , r. 2005b , 632 , 736 alonso - 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hawthorn , j. 2005 , 43 , 769 wandel , a. 2002 , 565 , 762 waskett , t.j.,eales , s.a . , gear , w.k . , mccracken , h.j . , brodwin , m. , nandra , k. , laird , e.s . , & lilly , s. 2004 , 350 , 785 willott , c. et al . 2003 , 339 , 397 yan , l. , chary , r. , armus , l. , teplitz , h. , helou , g. , frayer , d. , fadda , d. , surace , j. & choi , p. 2005 , 628 , 604 and 2@xmath167 significance respectively . the upper axis gives the restframe wavelength scale at the source redshift . the shaded regions are areas effected by strong sky emission or absorption . [ fig1 ] ] lrrrrrl & & & & & & + + & & & & & & + smmj123549.44 + 621536.8 & 7200 & apr 6 2004 & j&h & 6.8@xmath168 & 12.0@xmath1691.4 & + smmj123606.85 + 621021.4 & 7200 & feb 15 2005 & j&h & 8.7@xmath170 & 7.4@xmath1691.4 & + smmj123622.65 + 621629.7 & 7200 & apr 7 2004 & j&h & 9.0@xmath171 & 11.0@xmath1691.4 & + smmj123635.59 + 621424.1 & 7200 & feb 16 2005 & j&h & 7.5@xmath172 & 25.0@xmath1691.4 & + smmj123716.01 + 620323.3 & 4000 & feb 16 2005 & j&h & 6.3@xmath173 & 74.0@xmath1691.4 & + smmj123721.87 + 621035.3 & 7200 & jun 24 2004 & j&h & 0.43@xmath174 & 21.0@xmath1691.4 & + smmj131215.27 + 423900.9 & 600 & feb 16 2005 & j&h & 13.9@xmath175 & 190.0@xmath16923.0 & + smmj131222.35 + 423814.1 & 2000 & jun 25 2004 & j&h & 12.7@xmath176 & 120.0@xmath16920.0 & + smmj163639.01 + 405635.9 & 7200 & jun 24 2004 & j&h & 5.5@xmath177 & @xmath10422.0 & + smmj163650.43 + 405734.5 & 7000 & may 18 2003 & j&h & 50.5@xmath178 & @xmath10422.0 & smail et al . ( 2003 ) + mmj163655 + 4059 & 3600 & apr 7 2004 & j&h & 10.9@xmath179 & 150.0@xmath16921.0 & + smmj163706.51 + 405313.8 & 7200 & apr 6 2004 & j&h & 7.2@xmath180 & @xmath10422.0 & + smmj221733.02 + 000906.0 & 3000 & jun 24 2004 & j&h & 1.9@xmath181 & @xmath10428.0 & + smmj221733.79 + 001402.1 & 7200 & jun 25 2004 & j&h & 4.9@xmath182 & @xmath10428.0 & + & & & & & & + + & & & & & & + smmj02399 - 0134 & 4500 & nov 24,25 2004 & j & 6.5@xmath183 & 32.0@xmath1695.0 & + & 2400 & nov 25 2004 & z & & & + smmj030227.73 + 000653.5 & 6000 & nov 23 2004 & j & 5.8@xmath184 & 60.0@xmath16910.0 & + rgj030257.94 + 001016.3 & 4500 & nov 23 2004 & h & 7.7@xmath185 & 60.0@xmath16910.0 & + smmj105702.50 - 033602.6 & 4500 & nov 23 2004 & h & 5.0@xmath173 & @xmath10410.0 & + smmj221737.39 + 001025.1 & 9000 & nov 23,24 2004 & k & 21.0@xmath186 & @xmath10428.0 & + & 6000 & nov 25 2004 & h & & & + & & & & & & + + & & & & & & + smmj09431 + 4700(h6 ) & 2400 & apr 8 2004 & k & 15.0@xmath187 & @xmath10413.0 & + smmj09431 + 4700(h7 ) & 2400 & apr 8 2004 & k & 15.0@xmath187 & @xmath10413.0 & + smmj131201.17 + 424208.1 & 2400 & apr 8 2004 & k & 20.2@xmath188 & 52.8@xmath16940.0 & + & & & & & & + + & & & & & & + smmj04431 + 0210 & & & & 3.5@xmath172 & & frayer et al . ( 2003 ) + smmj14011 + 0252(j1 ) & & & & 6.8@xmath176 & @xmath10444.0 & motohara et al.(2005 ) + smmj163658.19 + 410523.8 & & & & 10.9@xmath179 & @xmath10422.0 & simpson et al.(2004 ) + lrrrrrrcccrl object & @xmath189 & h@xmath3 flux & h@xmath0 flux & [ o iii]@xmath190 flux & [ o iii]@xmath191 flux & [ o ii]@xmath192 flux & & rest fwhm@xmath193 & comment + & & & uv & h@xmath194 & opt & km@xmath195sec@xmath17 & + smmj02399 - 0134 & 1.061 & 75.8@xmath169 15.0 & @xmath10411.0 & @xmath10411.0 & @xmath10411.0 & & agn & agn & & 1530@xmath169500 + smmj030227.73 + 000653.5 & 1.408 & 15.2@xmath169 2.0 & 1.9@xmath169 1.1 & 10.7@xmath169 3.2 & 7.2@xmath169 2.6 & & sb & agn & agn & @xmath104100 & + rgj030258.94 + 001016.3 & 2.239 & 1.8@xmath169 0.5 & @xmath1040.3 & 9.2@xmath169 2.1 & 0.85@xmath169 0.30 & & int & agn & agn & & + smmj09431 + 4700(h6 ) & 3.350 & & 0.3@xmath169 0.2 & 1.7@xmath169 0.3 & 0.7@xmath169 0.4 & & sb & & sb ? & 350@xmath16950 & extended [ oiii]@xmath7 + smmj09431 + 4700(h7 ) & 3.347 & & @xmath1040.15 & 0.5@xmath169 0.1 & @xmath1040.15 & & & & & & + smmj105702.50 - 033602.6 & 2.423 & 0.6@xmath169 0.2 & @xmath1040.19 & @xmath1040.19 & @xmath1040.19 & & & sb & & & van dokkum et al . ( 2004 ) ; miss the slit ? + smmj123549.44 + 621536.8 & 2.195 & 15.0@xmath169 1.0 & 1.6@xmath169 1.0 & 6.4@xmath169 1.3 & 1.9@xmath169 0.8 & 2.9@xmath169 0.8 & sb & int & agn & 2150@xmath169500 & [ oiii]@xmath7 double peaks and extended + smmj123606.85 + 621021.4 & 2.505 & 2.0@xmath169 0.3 & @xmath1040.043 & @xmath1040.043 & @xmath1040.043 & @xmath1040.043 & sb & int & & & slit on companion object ? + smmj123622.65 + 621629.7 & 2.462 & 3.4@xmath169 0.6 & @xmath1040.8 & @xmath1040.8 & @xmath1040.8 & 2.1@xmath169 0.5 & sb & sb & & & + smmj123635.59 + 621424.1 & 2.005 & 11.1@xmath169 1.2 & & 2.0@xmath169 0.6 & 2.4@xmath169 0.5 & & agn & agn & agn ? & & + smmj123716.01 + 620323.3 & 2.053 & & 28.1@xmath169 3.2 & 19.5@xmath169 2.4 & 9.7@xmath169 2.2 & & qso & & agn & 2130@xmath169500 & multiple peaks and extended [ oiii]@xmath7 + smmj123721.87 + 621035.3 & 0.979 & 6.2@xmath169 1.1 & & & & & & sb & & & + smmj131201.17 + 424208.1 & 3.408 & & 0.18@xmath1690.15 & 0.79@xmath1690.19 & 0.27@xmath1690.19 & & agn & & agn ? & & + smmj131215.27 + 423900.9 & 2.555 & 11.8@xmath169 1.0 & 3.1@xmath169 2.6 & 5.6@xmath169 2.2 & 2.2@xmath169 1.6 & @xmath1040.23 & qso & & agn & 2540@xmath169500 & + smmj131222.35 + 423814.1 & 2.560 & & 12.1@xmath169 4.5 & 5.5@xmath169 2.5 & 2.6@xmath169 2.5 & 2.0@xmath169 1.5 & qso & & agn & 2580@xmath1691000 & + smmj163639.01 + 405635.9 & 1.485 & 7.3@xmath169 0.7 & 0.7@xmath169 0.5 & 2.5@xmath169 0.5 & 1.1@xmath169 0.4 & & sb & sb & int & @xmath1041400 & extended h@xmath3 + smmj163650.43 + 405734.5 & 2.380 & 14.2@xmath169 1.5 & 2.3@xmath169 0.6 & 27.0@xmath169 2.1 & 6.8@xmath169 1.3 & 14.0@xmath169 0.8 & int & agn & agn & 3720@xmath169500 & reevaluated after smail et al . ( 2003 ) + mmj163655 + 4059 & 2.605 & 18.4@xmath169 2.4 & 2.7@xmath169 1.0 & 47.1@xmath169 1.6 & 15.7@xmath169 1.6 & 3.6@xmath169 0.9 & agn & agn & agn & 2410@xmath169600 & blue wings in h@xmath0 and [ oiii]@xmath86 + smmj163706.51 + 405313.8 & 2.373 & 7.4@xmath169 1.4 & 0.6@xmath169 0.3 & 5.7@xmath169 0.6 & 1.9@xmath169 0.6 & 1.0@xmath169 0.3 & agn & agn & agn & 1590@xmath169500 & + smmj221733.02 + 000906.0 & 0.926 & 9.6@xmath169 1.2 & & & & & & sb & & & + smmj221733.79 + 001402.1 & 2.551 & 8.5@xmath169 3.5 & 1.2@xmath169 0.6 & 1.2@xmath169 0.2 & @xmath1040.5 & 0.7@xmath169 0.3 & sb & sb & agn ? & 1860@xmath169600 & + smmj221737.39 + 001025.1 & 2.610 & 20.7@xmath169 6.0 & 0.6 @xmath1690.4 & 6.0@xmath169 0.4 & 1.2@xmath169 0.4 & & sb & agn & agn & 290@xmath16950 & + & & & & & & & & & + smmj04431 + 0210 & 2.510 & 1.6@xmath169 0.1 & @xmath1040.3 & 0.4@xmath169 0.1 & & & & agn ? & agn & & frayer et al . ( 2003 ) + smmj14011 + 0252(j1 ) & 2.565 & 1.3@xmath169 0.4 & 0.3@xmath169 0.1 & @xmath1040.2 & @xmath1040.2&0.4@xmath1690.1 & sb & sb & sb & & motohara et al . ( 2005 ) + smmj163658.19 + 410523.8 & 2.448 & 1.9@xmath169 0.4 & 0.2@xmath169 0.1 & 0.4@xmath169 0.1 & 0.3@xmath169 0.1 & 0.4@xmath169 0.2 & sb & sb & agn & & simpson et al . ( 2004 ) + & & & & & & & + composite(all ) & & & 1.0 & 0.7@xmath196 & 0.4@xmath196 & 0.3@xmath197 & & & & 3100@xmath169500 & + composite(qso ) & & & 1.0 & 0.36@xmath198 & 0.19@xmath199 & @xmath1040.12 & & & & 3200@xmath1691000 & + composite(opt - agn ) & & & 1.0 & 3.2@xmath200 & 1.2@xmath201 & 1.2@xmath202 & & & & 1730@xmath169500 & +	we report the results of a systematic near - infrared spectroscopic survey using the subaru , vlt and keck telescopes of a sample of high redshift ultra - luminous infrared galaxies ( ulirgs ) mainly composed of submillimeter - selected galaxies . our observations span the restframe optical range containing nebular emission lines such as h@xmath0 , [ oiii]@xmath1 , and [ oii]@xmath2 , which are essential for making robust diagnostics of the physical properties of these ulirgs . using the h@xmath3/h@xmath0 emission line ratios , we derive internal extinction estimates for these galaxies similar to those of local ulirgs : @xmath4 . correcting the h@xmath3 estimates of the star formation rate for dust extinction using the balmer decrement , results in rates which are consistent with those estimated from the far - infrared luminosity . the majority ( @xmath5% ) of our sample show spectral features characteristic of agn ( although we note this partially reflects an observational bias in our sample ) , with @xmath6% exhibiting broad balmer emission lines . a proportion of these sources show relatively low [ oiii]@xmath7/h@xmath0 line ratios , which are similar to those of narrow line seyfert 1 galaxies suggesting small mass black holes which are rapidly growing . in the subsample of our survey with both [ oiii]@xmath7 and hard x - ray coverage , at least @xmath8% show an excess of [ oiii]@xmath7 emission , by a factor of 510@xmath9 , relative to the hard x - ray luminosity compared to the correlation between these two properties seen in seyferts and qsos locally . from our spectral diagnostics , we propose that the strong [ oiii]@xmath7 emission in these galaxies arises from shocks in dense gaseous regions in this vigorously star - forming population . we caution that due to sensitivity and resolution limits , our sample is biased to strong line emitters and hence our results do not yet provide a complete view of the physical properties of the whole high - redshift ulirg population .	25,122	609
_ steganography _ is a scientific discipline within the field known as _ data hiding _ , concerned with hiding information into a commonly used media , in such a way that no one apart from the sender and the intended recipient can detect the presence of embedded data . a comprehensive overview of the core principles and the mathematical methods that can be used for data hiding can be found in @xcite . an interesting steganographic method is known as _ matrix encoding _ , introduced by crandall @xcite and analyzed by bierbrauer et al . matrix encoding requires the sender and the recipient to agree in advance on a parity check matrix @xmath2 , and the secret message is then extracted by the recipient as the syndrome ( with respect to @xmath2 ) of the received cover object . this method was made popular by westfeld @xcite , who incorporated a specific implementation using hamming codes in his f5 algorithm , which can embed @xmath3 bits of message in @xmath4 cover symbols by changing , at most , one of them . there are two parameters which help to evaluate the performance of a steganographic method over a cover message of @xmath5 symbols : the _ average distortion _ @xmath6 , where @xmath7 is the expected number of changes over uniformly distributed messages ; and the _ embedding rate _ @xmath8 , which is the amount of bits that can be hidden in a cover message . in general , for the same embedding rate a method is better when the average distortion is smaller . following the terminology used by fridrich et al . @xcite , the pair @xmath9 will be called _ @xmath10-rate_. furthermore , as willems et al . in @xcite , we will also assume that a discrete source produces a sequence @xmath11 , where @xmath5 is the block length , each @xmath12 , and @xmath13 depends on whether the source is a grayscale digital image , or a cd audio , etc . the message @xmath14 we want to hide into a host sequence @xmath15 produces a composite sequence @xmath16 , where @xmath17 and each @xmath18 . the composite sequence @xmath19 is obtained from distorting @xmath15 , and the distortion will be assumed to be a squared - error distortion ( see @xcite ) . in these conditions , if information is only carried by the least significant bit ( lsb ) of each @xmath20 , the appropriate solution comes from using binary hamming codes @xcite , improved using product hamming codes @xcite . for larger magnitude of changes , but limited to @xmath21 , that is , @xmath22 , where @xmath23 , the situation is called @xmath1-steganography " , and the information is carried by the two least significant bits . it is known that the embedding becomes statistically detectable rather quickly with the increasing amplitude of embedding changes . therefore , our interest goes to avoid changes of amplitude greater than one . with this assumption , our steganographic scheme will be compared with the upper bound from @xcite for the embedding rate in @xmath1-steganography " , given by @xmath24 , where @xmath25 is the binary entropy function @xmath26 and @xmath27 is the average distortion . a main purpose of steganography is designing schemes in order to approach this upper bound . in most of the previous papers , @xmath1-steganography " has involved a ternary coding problem . willems et al . @xcite proposed a schemed based on ternary hamming and golay codes , which were proved to be optimal . fridrich and lisonk @xcite proposed a method based on rainbow colouring graphs which , for some values , outperformed the scheme obtained by direct sum of ternary hamming codes with the same average distortion . however , both methods from @xcite and @xcite show a problem when dealing with extreme grayscale values , since they suggest making a change of magnitude greater than one in order to avoid having to apply the change @xmath28 and @xmath29 to a host sequence of value @xmath30 and @xmath31 , respectively . note that the kind of change they propose would obviously introduce larger distortion and therefore make the embedding more statistically detectable . in this paper we also consider the @xmath1-steganography . our new method is based on perfect @xmath0-linear codes which , although they are not linear , they have a representation using a parity check matrix that makes them as efficient as the hamming codes . as we will show , this new method not only performs better than the one obtained by direct sum of ternary hamming codes from @xcite , but it also deals better with the extreme grayscale values , because the magnitude of embedding changes is under no circumstances greater than one . to make this paper self - contained , we review in section [ sec : additivecodes ] a few elementary concepts on perfect @xmath0-linear codes , relevant for our study . the new steganographic method is presented in section [ sec : stegoz2z4 ] , whereas an improvement to better deal with the extreme grayscale values problem is given in section [ sec : anomalies ] . finally , the paper is concluded in section [ sec : conclusions ] . in general , any non - empty subgroup @xmath32 of @xmath33 is a _ @xmath0-additive code _ , where @xmath34 denotes the set of all binary vectors of length @xmath35 and @xmath36 is the set of all @xmath37-tuples in @xmath38 . let @xmath39 , where @xmath40 is given by the map @xmath41 where @xmath42 , @xmath43 , @xmath44 , and @xmath45 is the usual gray map from @xmath38 onto @xmath46 . a @xmath0-additive code @xmath32 is also isomorphic to an abelian structure like @xmath47 . therefore , @xmath32 has @xmath48 codewords , where @xmath49 of them are of order two . we call such code @xmath32 a _ @xmath0-additive code of type @xmath50 _ and its binary image @xmath51 is a _ @xmath0-linear code of type @xmath50_. note that the lee distance of a @xmath0-additive code @xmath32 coincides with the hamming distance of the @xmath0-linear code @xmath39 , and that the binary code @xmath51 does not have to be linear . the _ @xmath52-additive dual code _ of @xmath32 , denoted by @xmath53 , is defined as the set of vectors in @xmath33 that are orthogonal to every codeword in @xmath32 , being the definition of inner product in @xmath54 the following : @xmath55 where @xmath56 and computations are made considering the zeros and ones in the @xmath35 binary coordinates as quaternary zeros and ones , respectively . the binary code @xmath57 , of length @xmath58 , is called the _ @xmath52-dual code _ of @xmath51 . a @xmath0-additive code @xmath32 is said to be _ perfect _ if code @xmath39 is a perfect @xmath0-linear code , that is all vectors in @xmath59 are within distance one from a codeword and the distance between two codewords is , at least , three . for any @xmath60 and each @xmath61 @xmath62 @xmath63 there exists a perfect @xmath0-linear code @xmath51 of binary length @xmath64 , such that its @xmath0-dual code is of type @xmath50 , where @xmath65 , @xmath66 and @xmath67 ( note that the binary length can be computed as @xmath58 ) . the above result is due to @xcite and it allows us to write the parity check matrix @xmath2 of any @xmath0-additive perfect code for a given value of @xmath61 . matrix @xmath2 can be represented taking all possible vectors in @xmath68 , up to sign changes , as columns . in this representation , there are @xmath35 columns which correspond to the binary part of vectors in @xmath32 , and @xmath37 columns of order four which correspond to the quaternary part . we agree on a representation of the @xmath35 binary coordinates as coordinates in @xmath69 . take a perfect @xmath0-linear code and consider its @xmath0-dual , which is of type @xmath50 . as stated in the previous section , this gives us a parity check matrix @xmath2 which has @xmath70 rows of order two and @xmath61 rows of order four . for instance , for @xmath71 and according to @xcite , there are three different @xmath0-additive perfect codes of binary length @xmath72 which correspond to the possible values of @xmath73 . for @xmath74 , the corresponding @xmath0-additive perfect code is the usual binary hamming code , while for @xmath75 the @xmath0-additive perfect code has parameters @xmath76 , @xmath77 , @xmath78 , @xmath75 and the following parity check matrix : @xmath79 let @xmath80 , for @xmath81 , denote the @xmath82-th column vector of @xmath2 . note that the all twos vector @xmath83 is always one of the columns in @xmath2 and , for the sake of simplicity , it will be written as the column @xmath84 . we group the remaining first @xmath35 columns in @xmath2 in such a way that , for any @xmath85 , the column vector @xmath86 is paired up with its complementary column vector @xmath87 , where @xmath88 . to use these perfect @xmath0-additive codes in steganography take @xmath89 and let @xmath90 be an @xmath5-length source of grayscale symbols such that @xmath91 , where , for instance , @xmath92 for grayscale images . we assume that a grayscale symbol @xmath20 is represented as a binary vector @xmath93 such that @xmath94 where @xmath95 is the inverse of gray map . we will use the two least significant bits ( lsbs ) , @xmath96 , of every grayscale symbol @xmath20 in the source , for @xmath97 , as well as the least significant bit @xmath98 of symbol @xmath99 to embed the secret message . each symbol @xmath20 will be associated with one or more column vectors @xmath80 in @xmath2 , depending on the grayscale symbol : 1 . grayscale symbol @xmath99 is associated with column vector @xmath84 by taking the least significant bit @xmath98 of @xmath99 . grayscale symbol @xmath20 , for @xmath85 , is associated with the two column vectors @xmath80 and @xmath100 , by taking , respectively , the two least significant bits , @xmath101 , of @xmath20 . grayscale symbol @xmath102 , for @xmath103 , is associated with column vector @xmath104 by taking its two least significant bits @xmath105 and interpreting them as an integer number @xmath106 in @xmath38 . in this way , the given @xmath5-length packet @xmath107 of symbols is translated into a vector @xmath108 of @xmath35 binary coordinates and @xmath37 quaternary coordinates . the embedding process we are proposing is based on the matrix encoding method @xcite . the secret message can be any vector @xmath109 @xmath62 @xmath47 . vector @xmath110 indicates the changes needed to embed @xmath109 within @xmath107 ; that is @xmath111 , where @xmath112 is an integer whose value will be described bellow , @xmath113 is the syndrome vector of @xmath108 and @xmath80 is a column vector in @xmath2 . the following situations can occur , depending on which column @xmath80 needs to be modified : 1 . if @xmath114 , then the embedder is required to change the least significant bit of @xmath99 by adding or substracting one unit to / from @xmath99 , depending on which operation will flip its least significant bit , @xmath98 . if @xmath80 is among the first @xmath35 column vectors in @xmath2 and @xmath115 , then @xmath112 can only be @xmath116 . in this case , since @xmath80 was paired up with its complementary column vector @xmath100 , then this situation is equivalent to make @xmath117 or @xmath118 , where @xmath119 and @xmath120 are the least significant bits of the symbol @xmath20 which had been associated with those two column vectors . hence , after the inverse of gray map , by changing one or another least significant bit we are actually adding or subtracting one unit to / from @xmath20 . note that a problem may crop up at this point when we need to add @xmath21 to a symbol @xmath20 of value @xmath121 or , likewise , when @xmath20 has value @xmath122 and we need to subtract @xmath21 from it . 3 . if @xmath80 is one of the last @xmath37 columns in @xmath2 we can see that this situation corresponds to add @xmath123 to @xmath124 . note that because we are using a @xmath0-additive perfect code , @xmath112 will never be @xmath125 . hence , the embedder should add ( @xmath116 ) or subtract ( @xmath126 ) one unit to / from symbol @xmath124 . once again , a problem may arise with the extreme grayscale values . [ ex : noanomalies ] let @xmath127 be an @xmath5-length source of grayscale symbols , where @xmath128 and @xmath129 , and let @xmath2 be the matrix in ( [ pcm ] ) . the source @xmath107 is then translated into the vector @xmath130 in the way specified above . let @xmath131 be the vector representing the secret message we want to embed in @xmath107 . we then compute @xmath132 and see , by the matrix encoding method , that @xmath126 and @xmath133 . according to the method just described , we should apply the change @xmath134 . in this way , @xmath135 becomes @xmath136 , and then @xmath137 , which has the expected syndrome @xmath138 . as already mentioned at the beginning of this paper , the problematic cases related to the extreme grayscale values are also present in the methods from @xcite and @xcite , but their authors assume that the probability of gray value saturation is not too large . we argue that , though rare , this gray saturation can still occur . however , in order to compare our proposal with these others we will not consider these problems either until the next section . therefore , we proceed to compute the values of the average distortion @xmath139 and the embedding rate @xmath140 . our method is able to hide any secret vector @xmath109 @xmath62 @xmath47 into the given @xmath5 symbols . hence , the embedding rate is @xmath141 bits per @xmath5 symbols , @xmath142 . concerning the average distortion @xmath139 , we are using a perfect code of binary length @xmath143 , which corresponds to @xmath144 grayscale symbols . there are @xmath145 symbols @xmath20 , for @xmath146 , with a probability @xmath147 of being subjected to a change ; a symbol @xmath99 with a probability @xmath148 of being the one changed ; and , finally , there is a probability of @xmath148 that neither of the symbols will need to be changed to embed @xmath109 . hence , @xmath149 . the described method has a @xmath10-rate @xmath150 , where @xmath144 and @xmath151 is any integer @xmath60 . we are able to generate a specific embedding scheme for any value of @xmath151 but not for any @xmath10-rate . with the aim of improving this situation , convex combinations of @xmath10-rates of two codes related to their direct sum are extensively treated in @xcite . actually , it is possible to choose the @xmath139 coordinate and cover more @xmath10-rates by taking convex combinations . therefore , if @xmath139 is a non - allowable parameter for the average distortion we can still take @xmath152 , where @xmath153 are two contiguous allowable parameters , and by means of the direct sum of the two codes with embedding rate @xmath154 and @xmath155 , respectively , we can obtain a new @xmath10-rate @xmath9 , with @xmath156 and @xmath157 . from a graphic point of view , this is equivalent to draw a line between two contiguous points @xmath158 and @xmath159 , as it is shown in [ fig : graphicwoanom ] . in the following theorem we claim that the @xmath10-rate of our method improves the one given by direct sum of ternary hamming codes from @xcite . for @xmath160 , the @xmath10-rate given by the method based on @xmath0-additive perfect codes improves the @xmath10-rate obtained by direct sum of ternary hamming codes with the same average distortion . optimal embedding ( of course , in the allowable values of @xmath139 ) can be obtained by using ternary codes , as it is shown in @xcite . the @xmath10-rate of these codes is @xmath161 for any integer @xmath162 . our method , based on @xmath0-additive perfect codes , has @xmath10-rate @xmath150 , for any integer @xmath60 and @xmath144 . take , for any @xmath60 , two contiguous values for @xmath162 such that @xmath163 and write @xmath164 , where @xmath165 . we want to prove that , for @xmath160 , we have @xmath166 , which is straightforward . however , since it is neither short nor contributes to the well understanding of the method , we do not include all computations here . the graphic bellow compares the @xmath10-rate of the method based on ternary hamming codes with that one based on @xmath0-additive perfect codes . as one may see in this graphic , for some values of the average distortion @xmath139 , the scheme based on @xmath0-additive perfect codes has greater embedding rate @xmath140 than the one based on ternary hamming codes . * remark : * the same argumentation can be used and the same conclusion can be reached taking @xmath167 instead of @xmath168 and comparing our method with the method described in @xcite . -rate @xmath9 , for @xmath92 , of steganographic methods based on ternary hamming codes and on @xmath0-additive perfect codes . ] in section [ sec : stegoz2z4 ] we described a problem which may raise when , according to our method , the embedder is required to add one unit to a source symbol @xmath20 containing the maximum allowed value ( @xmath121 ) , or to substract one unit from a symbol @xmath20 containing the minimum allowed value , @xmath122 . to face this problem , we will use the complementary column vector @xmath100 of columns @xmath80 in matrix @xmath2 , where @xmath169 and @xmath80 is among the last @xmath37 columns in @xmath2 . note that @xmath80 and @xmath100 can coincide . the first @xmath35 column vectors in @xmath2 will be paired up as before , and the association between each @xmath20 and each column vector @xmath80 in @xmath2 will be also the same as in section [ sec : stegoz2z4 ] . however , given an @xmath5-length source of grayscale symbols @xmath90 , a secret message @xmath170 and the vector @xmath110 , such that @xmath171 , indicating the changes needed to embed @xmath109 within @xmath107 , we can now make some variations on the kinds of changes to be done for the specific problematic cases : * if @xmath80 is among the first @xmath35 columns in @xmath2 , for @xmath115 , and the embedder is required to add @xmath21 to a symbol @xmath31 , then the embedder should instead substract @xmath21 from @xmath20 as well as perform the appropiate operation ( @xmath172 or @xmath173 ) over @xmath99 to have @xmath98 flipped . likewise , if the embedder is required to substract @xmath21 from a symbol @xmath30 , then ( s)he should instead add @xmath21 to @xmath20 and also change @xmath99 to flip @xmath98 . * if @xmath80 is one of the last @xmath37 columns in @xmath2 , and the embedder has to add @xmath21 to a symbol @xmath31 , ( s)he should instead substract @xmath21 from the grayscale symbol associated to @xmath100 and also change @xmath99 to flip @xmath98 . if the method requires substracting @xmath21 from @xmath30 , then we should instead add @xmath21 to the symbol associated to @xmath100 and , again , change @xmath99 to flip @xmath98 . [ ex : anomalies ] let @xmath174 be an @xmath5-length source of grayscale symbols , where @xmath128 and @xmath129 , and let @xmath2 be the matrix ( [ pcm ] ) . as in example [ ex : noanomalies ] , the packet @xmath107 is translated into vector @xmath130 , and @xmath131 . however , note that now we are not able to make @xmath134 because @xmath175 . instead of this , we will add one unit to @xmath176 , which is the symbol associated with @xmath177 , and substract one unit from @xmath99 so as to have its least significant bit flipped . therefore , we obtain @xmath178 and then @xmath179 , which has the desired syndrome . the method above described has the same embedding rate @xmath180 as the one from section [ sec : stegoz2z4 ] but a slightly worse average distortion . we will take into account the squared - error distortion defined in @xcite for our reasoning . as before , among the total number of grayscale symbols @xmath144 , there are @xmath145 symbols @xmath20 , for @xmath146 , with a probability @xmath147 of being changed ; a symbol @xmath99 with a probability @xmath148 of being the one changed ; and , finally , there is a probability of @xmath148 that neither of the symbols will need to be changed . as one may have noted in this scheme , performing a certain change to a symbol @xmath20 , associated with a column @xmath80 in @xmath2 , has the same effect as performing the opposite change to the grayscale symbol associated with @xmath100 and also changing the least significant bit @xmath98 of @xmath99 . this means that with probability @xmath181 we will change a symbol @xmath20 , for @xmath146 , a magnitude of @xmath21 ; and with probability @xmath182 we will change two other symbols also a magnitude of @xmath21 . therefore , @xmath183 and the average distortion is thus @xmath184 . hence , the described method has @xmath10-rate @xmath185 . as we have already mentioned , the problem of grayscale symbols with @xmath122 and @xmath121 values was previously detected in both @xcite and @xcite . with the aim of providing a possible solution to this problem , the authors suggested to perform a change of a magnitude greater than @xmath21 . the effects of doing this were are out of the scope of @xmath1-steganography . in the remainder of this section we proceed to compare the @xmath10-rate of our method with the @xmath10-rate that those methods would have if their proposed solution was implemented . the scheme presented by willems et al . @xcite is based on ternary hamming codes , which are known to have length @xmath186 , where @xmath162 denotes the number of parity check equations . let us assume that whenever the embedder is required to perform a change ( @xmath172 or @xmath173 ) that would lead the corresponding symbol @xmath20 to a non - allowed value , then a change of magnitude @xmath125 ( @xmath187 or @xmath188 ) is made instead . while the embedding rate @xmath140 of this scheme would still be @xmath189 , the average distortion @xmath139 would no longer be @xmath190 . the actual expected number of changes @xmath7 is computed by noting that a symbol will be changed with probability @xmath191 , and will not with probability @xmath192 . among the cases in which a symbol would need to be changed , there is a probability of @xmath181 that a symbol will be changed a magnitude of @xmath21 , and a probability of @xmath182 that it will be changed a magnitude of @xmath125 . by the squared - error distortion , @xmath193 and therefore @xmath194 . fridrich and lisonk propose in their paper to pool the grayscale symbols source @xmath107 into cells of size @xmath195 , then rainbow colour these cells and apply a @xmath167-ary hamming code , where @xmath196 is a prime power . they measure the distortion by counting the maximum number of embedding changes , thus just considering the covering radius of the @xmath167-ary hamming codes . however , we will now consider the average number of embedding changes ( see @xcite ) . as willems et at . , the authors from @xcite also suggest to perform a change of magnitude @xmath197 to solve the extreme grayscale values problem . if this is done , the embedding rate would still be the same , @xmath198 , but the average distortion would now be @xmath199 . one can see in [ fig : graphicwanom ] how our steganographic method for @xmath0-additive perfect codes deals with the extreme grayscale values problem , for some values of @xmath139 , better than those using ternary hamming codes ( @xmath200 ) from @xcite and @xcite . -rates @xmath9 , for @xmath92 , of steganographic methods based on ternary hamming codes and on @xmath0-additive perfect codes , when they are dealing with the extreme grayscale values problem described in section [ sec : anomalies ] . ] in this paper , we have presented a new method for @xmath1-steganography , based on perfect @xmath0-linear codes . these codes are non - linear but still there exists a parity check matrix representation that makes them efficient to work with . as we have shown in sections [ sec : stegoz2z4 ] and [ sec : anomalies ] , this new scheme outperforms the one obtained by direct sum of ternary hamming codes ( see @xcite ) as well as the one obtained after rainbow colouring graphs by using @xmath167-ary hamming codes for @xmath200 . if we consider the special cases in which the technique might require to substract one unit from a grayscale symbol containing the minimum allowed value , or to add one unit to a symbol containing the maximum allowed value , our method performs even better than those aforementioned schemes . this is so because unlike them , our method never applies any change of magnitude greater than @xmath21 , but two changes of magnitude @xmath21 instead , which is better in terms of distortion . therefore , our method makes the embedding less statistically detectable . as for further research , since the approach based on product hamming codes in @xcite improved the performance of basic lsb steganography and the basic @xmath201 algorithm , we would also expect a considerable improvement of the @xmath10-rate by using product @xmath0-additive codes or subspaces of product @xmath0-additive codes in @xmath1-steganography . j. bierbrauer and j. fridrich , constructing good covering codes for applications in steganography " , in transactions on data hiding and multimedia security iii , vol . 4920 of lecture notes in computer science , pp . 1 - 22 , springer , berlin , germany , 2008 .	steganography is an information hiding application which aims to hide secret data imperceptibly into a commonly used media . unfortunately , the theoretical hiding asymptotical capacity of steganographic systems is not attained by algorithms developed so far . in this paper , we describe a novel coding method based on @xmath0-linear codes that conforms to @xmath1-steganography , that is secret data is embedded into a cover message by distorting each symbol by one unit at most . this method solves some problems encountered by the most efficient methods known today , based on ternary hamming codes . finally , the performance of this new technique is compared with that of the mentioned methods and with the well - known theoretical upper bound .	7,758	181
we present new measurements of the @xmath4 and @xmath5 absolute branching fractions , their ratio , and measurements of the semileptonic form factors controlling these decays.@xmath11 modes as @xmath12 decays , and use the clebsch - gordan factor 1.5 to correct for @xmath13 decays , which we do not detect . ] exclusive charm semileptonic decays provide particularly simple tests of over decay dynamics since long distance effects only enter through the hadronic form factors @xcite . a wide variety of theoretical methods have been brought to bear on the calculation of these form factors including quark models @xcite , qcd sum rules @xcite , lattice qcd @xcite , analyticity @xcite , and others @xcite . using a technique developed by focus @xcite , we present non - parametric measurements of the @xmath14 dependence of the helicity basis form factors that give an amplitude for the @xmath15 system to be in any one of its possible angular momentum states where @xmath14 is the invariant mass squared of the lepton pair in the decay . the ultimate goal of this study is to obtain a better understanding of the semileptonic decay intensity . cleo - c produces @xmath16 mesons at the @xmath17(3770 ) , which ensures a pure @xmath18 final state with no additional final state hadrons . in events where the @xmath12 is produced against a fully reconstructed @xmath19 the missing neutrino can be reconstructed with unparalleled precision using energy - momentum balance . hence , cleo - c data offer unparalleled @xmath14 and decay angle resolution allowing one to resolve fine details in the structure of these form factors without the complications of a deconvolution procedure . the various helicity basis form factors are distinguished based on their contributions to the decay angular distribution . , @xmath20 , and @xmath21 angles.,width=307 ] the amplitude @xmath22 for the semileptonic decay @xmath12 is described by five kinematic quantities : @xmath14 ; the kaon - pion mass ( @xmath23 ) ; the kaon helicity angle ( @xmath24 ) , which is computed as the angle between the @xmath25 and the @xmath16 direction in the @xmath15 rest frame ; the lepton helicity angle ( @xmath26 ) , which is computed as the angle between the @xmath27 and the @xmath16 direction in the @xmath28 rest frame ; and the acoplanarity angle between the two decay planes ( @xmath21 ) . the decay angles are illustrated in fig . [ fig : kinem - definition ] . the amplitude @xmath22 can be expressed in terms of four helicity amplitudes representing the transition to the vector @xmath10 : @xmath29 , @xmath30 , @xmath31 , @xmath32 and a fifth form factor , @xmath33 describing a non - resonant , @xmath7-wave @xmath12 contribution . the diferential decay width for the 4-body semileptonic process is @xmath34 where @xmath35 is the decay intensity , @xmath36 is the @xmath15 momentum in the @xmath37 rest frame , @xmath38 is the momentum of the kaon in the @xmath15 rest frame , and @xmath39 is the momentum of the @xmath40 in the @xmath41 rest frame . upon integration over @xmath21 , the differential decay width is proportional to : @xmath42 the @xmath32 form factor , which appears in the second term of eq . ( [ amp1 ] ) , is helicity suppressed by a factor of @xmath43 . the mass - suppressed terms are negligible for @xmath1 but can be measured in @xmath2 . the @xmath32 form factor can only be effectively measured in @xmath2 decays at low @xmath14 where the mass suppression effects are least severe . the semimuonic to semielectric branching ratio is sensitive to the magnitude of the @xmath32 form factor . we study the form factor of the non - resonant , spin zero , @xmath7-wave component to @xmath5 first described in ref . @xcite . according to the model of ref . @xcite , 2.4% of the decays in the mass range @xmath44 are due to this @xmath7-wave component@xcite , where @xmath23 is the @xmath15 mass . the underlined term in eq . ( [ amp1 ] ) represents the interference between the @xmath7-wave , @xmath15 amplitude and the @xmath10 amplitude , represented as a simplified , breit - wigner function of the form : @xmath45 where @xmath38 is the kaon momentum in the @xmath15 rest frame , and @xmath46 is the value of @xmath38 when the @xmath15 mass is equal to the @xmath10 mass - wave breit - wigner form with a width proportional to the cube of the kaon momentum in the kaon - pion rest frame . our breit - wigner intensity is proportional to @xmath47 as expected for a @xmath48-wave breit - wigner resonance . two powers of @xmath38 come explicitly from the @xmath38 in the numerator of the amplitude and one power arises from the 4-body phase space as shown in eq . ( [ gamma ] ) . we are not including additional , small corrections such as the blatt - weisskopf barrier penetration factor . ] . the @xmath7-wave form factor is denoted as @xmath49 in the underlined piece of eq . ( [ amp1 ] ) . following ref . @xcite we model the @xmath7-wave contribution as an amplitude with a phase ( @xmath50 ) and modulus ( @xmath51 ) that are independent of @xmath23 . we have dropped the second - order , @xmath7-wave intensity contribution ( @xmath52 ) in eq . ( [ amp1 ] ) since @xmath53 . the @xmath21 integration significantly simplifies the intensity by eliminating all interference terms between different helicity states of the virtual @xmath54 with relatively little loss in form factor information . the four helicity basis form factors for the @xmath5 component are generally written @xcite as linear combinations of a vector ( @xmath55 ) and three axial - vector ( @xmath56 ) form factors according to @xmath57 \,,\label{helicity } \\ h_t({\ensuremath{q^2 } } ) & = & { m_d k\over m_{k\pi}\sqrt{{\ensuremath{q^2 } } } } \left [ ( m_d+m_{k\pi})a_1({\ensuremath{q^2 } } ) -{(m^2_d -m^2_{k\pi}+{\ensuremath{q^2 } } ) \over m_d+m_{k\pi}}a_2({\ensuremath{q^2 } } ) \right . \nonumber\\ & & \mbox { } \hspace{2 cm } \left . + { 2{\ensuremath{q^2}}\over m_d+m_{k\pi}}a_3({\ensuremath{q^2 } } ) \right ] \,,\nonumber \end{aligned}\ ] ] where @xmath58 is the mass of the @xmath37 and @xmath36 is the momentum of the @xmath15 system in the rest frame of the @xmath37 . in the spectroscopic pole dominance ( spd ) model @xcite , these axial and vector form factors are given by @xmath59 where @xmath60 and @xmath61 . the spd model allows one to parameterize the @xmath30 , @xmath29 , @xmath31 , and @xmath32 form factors using just three parameters , which are ratios of form factors taken at @xmath62 : @xmath63 and @xmath64 . there are accurate measurements @xcite of @xmath65 and @xmath66 , but very little is known about @xmath67 , which is an important motivation for this work . in this paper , we use a _ projective weighting _ technique @xcite to disentangle and directly measure the @xmath14 dependence of these helicity basis form factors free from parameterization . we provide information on the six form factor products @xmath68 , @xmath69 , @xmath33 , @xmath70 and @xmath71 in bins of @xmath14 by projecting out the associated angular factors given by eq . ( [ amp1 ] ) . we next describe some of the experimental and analysis details used for these measurements . the cleo - c detector @xcite consists of a six - layer inner stereo - wire drift chamber , a 47-layer central drift chamber , a ring - imaging cerenkov detector ( rich ) , and a cesium iodide electromagnetic calorimeter inside a superconducting solenoidal magnet providing a 1.0 t magnetic field . the tracking chambers and the electromagnetic calorimeter cover 93% of the full solid angle . the solid angle coverage for the rich detector is 80% of @xmath72 . identification of the charged pions and kaons is based on measurements of specific ionization ( @xmath73 ) in the main drift chamber and rich information . electrons are identified using the ratio of the energy deposited in the electromagnetic calorimeter to the measured track momentum ( @xmath74 ) as well as @xmath73 and rich information . although there is a muon detector in cleo , it was optimized for b - meson semileptonic decay , and is ineffectual for charm semileptonic decay since a muon from charm particle decay will typically range out in the first layer of iron in the muon shield . in this paper , we use 818 pb@xmath75 of data taken at the @xmath3 center - of - mass energy with the cleo - c detector at the cornell electron storage ring ( cesr ) @xmath76 collider , which corresponds to a ( produced ) sample of 1.8 million @xmath77 pair events @xcite . we select the events containing a @xmath19 decaying into one of the following six decay modes : @xmath78 , @xmath79 , @xmath80 , @xmath81 , @xmath82 , and @xmath83 along with a 4-body semileptonic candidate . to avoid complications due to having two or more @xmath12 decay candidates in the event , we select the decay candidate with the smallest @xmath84 value where @xmath85 is the beam - constrained mass . the beam - constrained mass @xmath85 is defined as @xmath86 where @xmath87 is the beam energy and @xmath88 is the d - tag momentum . more details on selecting the tagging @xmath19 candidates as well as identifying @xmath89 and @xmath90 candidates are described in ref . @xcite . we used extensive monte carlo ( mc ) studies to design efficient , background - suppressing selections . the @xmath12 reconstruction starts by requiring three well - measured tracks not associated with the tagging @xmath19 decay . in order to select semileptonic decays , we require a minimal missing momentum and energy of 50 mev/@xmath91 and 50 mev , respectively . both the minimal missing momentum and energy are calculated using the center - of - mass momentum and energy . in order to reduce backgrounds from charm decays with missing @xmath89 s , we require an unassociated shower energy of less than 250 mev . the unassociated shower energy refers to electromagnetic showers , which are statistically separated from all measured , charged tracks . charged kaons and pions are required to have momenta of at least 50 mev/@xmath91 and are identified using @xmath73 and rich information . we require that the pion deposits a shower energy , which is inconsistent with the electron hypothesis . electron candidates are required to have momenta of at least 200 mev/@xmath91 , lie in the good shower containment region ( @xmath92 ) , and pass a requirement on a likelihood variable that combines @xmath74 , @xmath73 , and rich information . our simulations indicate that contamination of our kaon sample due to pions is less than 0.06% using this likelihood variable . the only final state particle not detected is the neutrino in the semileptonic decay . the neutrino four - momentum vector can be reconstructed from the missing energy and momentum in the event . the @xmath14 resolution , predicted by our monte carlo simulation , is roughly gaussian with an r.m.s . width of 0.02 @xmath93 , which is negligible on the scale that we will bin our data . for @xmath2 candidates , it is difficult to distinguish the @xmath94 track from the @xmath95 track . because @xmath2 decay is strongly dominated by @xmath96 , which is a relatively narrow resonance , we select the positive track with the smallest @xmath97 as the pion and the other track as the muon . our monte carlo studies concluded that this @xmath10 arbitration approach was correct 84% of the time and works better than pion - muon discrimination based on the electromagnetic calorimeter response . we apply a variety of additional requirements to suppress backgrounds in @xmath2 candidates . we require that the muon is inconsistent with the electron hypothesis according to the electron likelihood variable . we require that missing momentum ( @xmath98 ) lies within 20 mev of the missing energy ( @xmath99 ) . for @xmath2 candidates , we also require @xmath100 . the @xmath101 distributions for muon and @xmath100 . the @xmath101 distributions for muon and electron candidates are illustrated in fig . [ mmsq ] . in order to suppress cross - feed from @xmath1 decay to our @xmath2 sample , we construct the squared invariant mass of the lepton candidate , @xmath102 , where @xmath103 is the reconstructed energy of the @xmath19 produced against the @xmath104 candidate and @xmath105 , @xmath106 , @xmath107 are the reconstructed kaon energy , pion energy , and lepton momentum . we require @xmath108 to eliminate both @xmath1 cross - feed and @xmath109 decays . in order to suppress backgrounds to @xmath2 from @xmath110 decays with an accompanying bremsstrahlung photon , we require that cosine of the minimum angle between three charged tracks and the missing momentum direction be less than 0.90 . this requirement is illustrated in fig . [ cone ] . we obtain 11801 @xmath12 candidates . the @xmath23 distribution for these @xmath12 candidates is shown in fig . [ signal ] . finally , we require @xmath111 and select 10865 events . distributions for events satisfying our _ nominal _ @xmath12 selection requirements apart from the @xmath101 requirement . ( a ) shows the @xmath101 distribution for @xmath2 candidates , while ( b ) shows the @xmath101 distribution for @xmath1 candidates . for @xmath2 candidates , we require that @xmath101 lies between the vertical lines . this cut is placed asymmetrically on our semimuonic sample to suppress cross - feed from @xmath1 . in each plot , the solid histogram shows the signal plus background distribution predicted by our monte carlo simulation , while the dashed histogram shows the predicted background component . [ mmsq],width=268 ] two types of monte carlo simulations are used throughout this analysis . the _ generic _ monte carlo simulation is a large charm monte carlo sample consisting of generic @xmath112 decays , which is primarily used in this analysis to simulate the properties of backgrounds to our @xmath12 signal states . the _ generic _ monte carlo events are generated by evtgen @xcite and the detector is simulated using a geant - based @xcite program . in much of the form - factor work , we use an spd monte carlo simulation based on the spd model described in sec . [ intro ] and summarized by eqs . ( [ amp1][spdpole ] ) . we use the spd parameters of ref . @xcite , @xmath65 = 1.504 , @xmath66 = 0.875 , and we set @xmath67=0 . the background shapes in fig . [ signal ] are obtained using _ monte carlo simulations . our simulation predicts a 6.5 % background for our @xmath2 sample with 4% due to misidentified @xmath1 cross - feed events and the rest due to various charm decays . the simulation also predicts a 1% background to our @xmath1 sample with 0.03 % due to @xmath2 cross - feed . distribution for @xmath2 candidates , while ( b ) shows the @xmath113 distribution for @xmath1 candidates . we remove all combinations to the right of the vertical line , which removes the major part of remaining @xmath114 background for the semimuonic sample . in each plot , the solid histogram shows the signal plus background distribution predicted by our monte carlo simulation , while the dashed histogram shows the predicted background component . [ cone],width=268 ] distributions for events satisfying our nominal @xmath12 selection requirements . ( a ) shows the @xmath23 distribution for @xmath2 candidates , while ( b ) shows the @xmath23 distribution for @xmath1 candidates . over the full displayed mass range , there are 11801 ( 6227 semielectric and 5574 semimuonic ) events satisfying our nominal selection . for this analysis , we use a restricted mass range from 0.8 1.0 @xmath115 , which is the region between the vertical lines . in each plot , the solid histogram shows the signal plus background distribution predicted by our monte carlo simulation , while the dashed histogram shows the predicted background component . in this restricted region , there are 10865 ( 5658 semielectric and 5207 semimuonic ) events . the inserted figures are on a finer scale to better show the estimated background contributions . [ signal],width=268 ] we have measured both the semimuonic to semielectric relative branching ratio and the @xmath4 and @xmath5 absolute branching fractions , which we will denote as @xmath116 and @xmath117 , respectively . the @xmath118 relative branching ratio is expected to be less than 1 due to the reduced phase space available to the semimuonic decay relative to the semielectric decay . the mass - suppressed terms in eq . ( [ amp1 ] ) will change the relative branching ratio compared to the phase space ratio . in the context of the spd model , eq . ( [ spdpole ] ) , the relative branching fraction will depend on @xmath119 , which controls the strength of the @xmath32 form factor and is essentially unknown . it is expected that @xmath118 will increase with increasing values of @xmath67 . in order to obtain the semimuonic to semielectric branching ratio , we write the observed @xmath120 and @xmath121 yields as @xmath122 where @xmath123 are the observed yields , @xmath124 are non - semileptonic backgrounds , and @xmath125 give the number of produced semileptonic decays in our data sample . the cross - feed matrix , which multiplies the @xmath126 and @xmath127 signal vector , is constructed from @xmath128 , which are the @xmath129 and @xmath130 detection efficiencies , and @xmath131 , which are the cross - feed efficiencies . for example , @xmath132 is the efficiency for reconstructing a @xmath129 event as a @xmath130 candidate . the @xmath123 yields are obtained by counting the number of semimuonic and semielectric events in our mass range @xmath133 . the relative branching ratio is given by @xmath134 . the vector @xmath135 represents parameters that the efficiencies and cross - feeds can depend on such as the spd parameters : @xmath65 , @xmath66 , and @xmath67 and the @xmath7-wave amplitude and phase . the detection efficiencies , @xmath136 , and the cross - feed efficiencies , @xmath137 , were obtained using our monte carlo simulations . we will refer to the use of eq . ( [ eq : brratio - yield ] ) to obtain the relative branching ratio , @xmath118 , as the _ cross - feed method_. we used the double - tag technique , described in ref . @xcite , to measure the @xmath138 and @xmath139 absolute semileptonic branching fractions ( @xmath140 ) . we define single tag ( st ) events as events where the @xmath141 was fully reconstructed against one of our six tag modes without any requirement on the recoil @xmath142 . we estimate the number of st events by fitting the @xmath143 distributions , shown in fig . [ fig : abs - st - data ] , using a binned maximum likelihood fit . here @xmath144 , where @xmath145 is the energy of the d - tag candidate . the total number of reconstructed @xmath141 st events is then @xmath146 where @xmath147 is the number of st reconstructed events in the @xmath148-th mode , @xmath149 is the number of produced @xmath150 events in our data sample , @xmath151 is the st detection efficiency , and @xmath152 is the tag mode branching fraction . for double tag ( dt ) events , we reconstruct @xmath153 into one of our six tagging modes , and require the presence of either a @xmath120 or @xmath121 candidate . the dt yields are then @xmath154 and @xmath155 respectively . the yields @xmath156 and @xmath157 represent the number of reconstructed dt events in semielectric and semimuonic decay modes after the background subtraction . the efficiencies @xmath158 and @xmath159 are the dt event detection efficiencies for the semielectric and semimuonic decay modes . the cross - feed efficiency @xmath160 describes how often a semimuonic decay is reconstructed as a semielectric candidate , while the cross - feed efficiency @xmath161 describes how often a semielectric decay is reconstructed as semimuonic candidate . the variables @xmath162 , @xmath163 are the respective @xmath139 and @xmath138 branching fractions , which we wish to measure . dividing eq . ( [ eq : dt - el ] ) and eq . ( [ eq : dt - mu ] ) by eq . ( [ eq : st ] ) , we have : @xmath164 equation ( [ eq : brratio - yield - abs ] ) shows how the branching fractions of @xmath165 and @xmath166 semileptonic modes depend on the ratio of the dt and the st yields , the detection efficiencies , and the cross - feed efficiencies . figures [ fig : abs - dt - data - el ] and [ fig : abs - dt - data - mu ] shows the @xmath143 distributions for our double tag sample . for both semileptonic decay modes , about half of our sample comes from the @xmath167 d - tag mode . the st yields for this mode are nearly background free . the cross - feed fraction for the @xmath168 semileptonic mode is less than 0.02% , while , for the @xmath169 semileptonic mode , the cross - feed fraction is 3.7% . the background level is about 2.5 times smaller for the @xmath168 mode than for the @xmath169 mode . the semielectric mode is nearly background free because of the effectiveness of the electromagnetic calorimeter , while our semimuonic mode uses a variety of less effective kinematic cuts to suppress background and cross - feed . our absolute branching fraction results are summarized by tables [ tab - abs - br - cond ] and [ tab - abs - br - comparison ] . table [ tab - abs - br - cond ] gives a `` conditional '' absolute branching fraction based only on @xmath12 decays into the mass range @xmath170 . this mass range is required for events entering into figs . [ fig : abs - dt - data - el ] and [ fig : abs - dt - data - mu ] . we find that the total systematic error for the semielectric and semimuonic absolute branching fractions , presented in table i , are comparable . the dominant systematic error for the semielectric decay is due to the 1% uncertainty in the efficiency our electron identification requirements , while the dominant systematic error for the semimuonic branching fraction is due to the 0.8% uncertainty in the background subtraction . the remaining systematic error , which is 1.2% for both the semielectric and semimuonic branching fractions , includes uncertainties in the final state radiation corrections , as well as uncertainties in the tracking and particle identification efficiencies for the kaon and pion tracks . table [ tab - abs - br - comparison ] , on the other hand , relies on models for the @xmath10 line - shape to extrapolate outside of the 200 @xmath171 wide mass region where our measurements are made in order to report the conventional @xmath6 absolute branching ratios , which includes events over the entire @xmath23 spectrum . we include an additional , clebsch - gordan factor of 1.5 in order to correct for the undetected @xmath13 decay mode assume that all of the signal events in the @xmath172 mass region , where our @xmath143 measurements made , are due to @xmath6 decay . ] . finally , we have included an additional @xmath173 contribution to the quoted systematic error in table [ tab - abs - br - comparison ] based on the difference between the @xmath10 extrapolations made using our generic and spd models . this @xmath173 systematic error contribution includes both distortions to the @xmath10 line shape as well as uncertainties in level of non - resonant contributions due to the @xmath7-wave amplitude . .conditional absolute branching fractions . these branching fractions only represent the @xmath174 spectrum from @xmath175 . [ cols="^,^",options="header " , ] [ summary ] figure [ finite ] illustrates our sensitivity to the pole masses in eq . ( [ helicity ] ) by comparing measurements of the @xmath176 form factor product to a model with spectroscopic axial and vector pole masses versus a model with infinite pole masses , implying _ constant _ axial and vector form factors . our data favor the spectroscopic pole masses given in eq . ( [ spdpole ] ) , for the high @xmath14 bins of the @xmath177 form factor product . the other five form factor products are consistent with either pole mass choice . form factor shown in fig . [ sixnormz ] overlayed with two models . ( a ) uses the same spd model shown in fig . [ sixnormz ] while ( b ) overlays the data with a spd model where the axial and vector poles [ @xmath178 and @xmath179 in eq . ( [ amp1 ] ) ] are set to infinity . we show the data with @xmath2 and @xmath1 combined . the slight scale difference between the data points in the two plots is an artifact of our @xmath180 as @xmath181 normalization scheme , which is based on the two different pole mass predictions for the @xmath69 form factor product . [ finite],height=192 ] it is of interest to search for the possible existence of additional non - resonant amplitudes of higher angular momentum . it is fairly simple to extend eq . ( [ amp1 ] ) to account for potential @xmath8-wave or @xmath9-wave interference with the @xmath10breit - wigner amplitude . we search specifically for a possible zero helicity @xmath8-wave or @xmath9-wave piece that interferes with the zero helicity @xmath10 contribution . one expects that such potential @xmath182 and @xmath183 form factors would peak as @xmath184 near @xmath181 as is the case for the other zero helicity contributions @xmath31 and @xmath49 . if so , the zero helicity contributions should be much larger than potential @xmath8- or @xmath9-wave @xmath185 helicity contributions . the @xmath8-wave projectors are based on an additional interference term of the form @xmath186 to search for the presence of zero helicity @xmath8-wave amplitude we use the technique of ref . @xcite to construct a projector which is orthogonal to the projectors for each of the six terms in eq . ( [ amp1 ] ) . the @xmath9-wave weights are based on an additional interference term of the form @xmath187 averaging over the breit - wigner intensity , the interference should be proportional to @xmath188 and will disappear when the non - resonant amplitude is orthogonal to the average , accepted @xmath10 amplitude . [ bestfd ] shows the @xmath189 form factor products obtained in the data using projective weights generated assuming a phase of zero . the projective weights are normalized so that @xmath190 in the limit @xmath191 if the putative @xmath8,@xmath9 -wave amplitude had the same strength as the @xmath7-wave amplitude relative to the @xmath10 breit - wigner amplitude . -wave form factor product ( a ) and @xmath9-wave form factor product ( b ) for an assumed phase of 0 radians relative to the @xmath10 breit - wigner amplitude . [ bestfd],height=192 ] there is no evidence for either a @xmath8-wave or @xmath9-wave component with this phase . and @xmath192 represent the phase of possible d and f -wave contributions relative to the phase of the @xmath10breit - wigner amplitude . they are measured in radians.,height=211 ] figure [ wavelimits ] shows our amplitude and limits for sixteen phase assumptions . as illustrated by fig . [ phase ] , our ability to measure a non - resonant amplitude can depend critically on its phase relative to the average , accepted @xmath10 phase . in order to maximize our sensitivity to the non - resonant amplitude , for each phase assumption and @xmath14 bin we made our measurement based on three @xmath23 mass regions : @xmath193 , @xmath194 , and @xmath195 , which puts the average @xmath10 reference phase at roughly @xmath196 , @xmath197 , and @xmath198 for these three mass regions , respectively . we chose the mass region with the smallest expected error according to the monte carlo simulation . under the assumption @xmath199 , used in ref . @xcite , we performed a @xmath200 fit of fig . [ bestfd ] to the form @xmath201 over the region @xmath202 to find the amplitude and limits shown in fig . [ wavelimits ] . figure [ wavelimits ] shows that this `` mass selection '' method produced non - amplitude limits , which are reasonably independent of assumed phase . if , on the other hand , one used the full @xmath203 mass range for all sixteen phase assumptions , one would get dramatically poorer limits for phase choices orthogonal to the breit - wigner amplitude phase . it is apparent from fig . [ wavelimits ] that we have no compelling evidence for either a @xmath8-wave , or an @xmath9-wave component . we present a branching fraction and form factor analysis of the @xmath12 decay based on a sample of approximately 11800 @xmath1 and @xmath2 decays collected by the detector running at the @xmath3 . we find @xmath204 and @xmath205 . our direct measurement of the relative semimuonic to semielectric branching ratio using eq . ( [ eq : brratio - yield ] ) is @xmath206 . we also present a non - parametric analysis of the helicity basis form factors that control the kinematics of the @xmath12 decays . we used a projective weighting technique that allows one to determine the helicity form factor products by weighted histograms rather than likelihood based fits . we find consistency with the spectroscopic pole dominance model for the dominant @xmath207 , @xmath177 and @xmath69 form factors . our measurement on the @xmath33 form factor product suggests that the @xmath208 form factor falls faster than @xmath209 with increasing @xmath14 . the form factors determined using @xmath2 decays are consistent with those determined using @xmath1 decays and are consistent with our earlier study @xcite of @xmath1 . our measured @xmath177 form factor data are more consistent with axial and vector form factors with the expected spectroscopic pole dominance @xmath14 dependence than with constant axial and vector form factors . our measurements of the @xmath70 and @xmath71 form factor suggests a much smaller @xmath32 form factor than expected in lattice gauge theory models @xcite . within the context of the spectroscopic pole dominance model eq . ( [ spdpole ] ) , our @xmath71 measurements are most consistent with a small @xmath32 form factor contribution implying a very negative value for @xmath119 , such as @xmath210 , which would place the predicted @xmath118 relative branching ratio close to the phase space estimate of 91% . finally , we have searched for possible @xmath8-wave or @xmath9-wave non - resonant interference contributions to @xmath12 . we have no statistically significant evidence for @xmath8-wave or @xmath9-wave interference , but are only able to limit these terms to roughly less than 1.0 and 1.5 times the observed @xmath7-wave interference for @xmath8-wave and @xmath9-wave respectively . we gratefully acknowledge the effort of the cesr staff in providing us with excellent luminosity and running conditions . d. cronin - hennessy thanks the a.p . sloan foundation . this work was supported by the national science foundation , the u.s . department of energy , the natural sciences and engineering research council of canada , and the u.k . science and technology facilities council . m. bauer , b. stech , and m. wirbel , z. phys . c * 29 * , 637 ( 1985 ) ; m. bauer and m. wirbel , z. phys . c * 42 * , 671 ( 1989 ) ; j.g . korner and g.a . schuler , z. phys . c * 46 * , 93 ( 1990 ) ; f.j . gilman and r.l . singleton , phys . d * 41 * , 142 ( 1990 ) ; d. scora and n. isgur , phys . d * 52 * , 2783 ( 1995 ) ; b. stech , z. phys . c * 75 * , 245 ( 1997 ) ; d. melikhov and b. stech , phys . d * 62 * , 014006 ( 2000 ) . bernard , a.x . el - khadra , and a. soni , phys . d * 45 * , 869 ( 1992 ) ; v. lubicz , g. martinelli , m.s . mccarthy , and c.t . sachrajda , phys . b * 274 * , 415 ( 1992 ) ; a. abada et al . b * 416 * , 675 ( 1994 ) ; k.c . bowler et al . 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( cleo collaboration ) , phys . d * 74 * , ( 2006 ) 052001 .	using a large sample ( @xmath0 11800 events ) of @xmath1 and @xmath2 decays collected by the cleo - c detector running at the @xmath3 , we measure the helicity basis form factors free from the assumptions of spectroscopic pole dominance and provide new , accurate measurements of the absolute branching fractions for @xmath4 and @xmath5 decays . we find branching fractions which are consistent with previous world averages . our measured helicity basis form factors are consistent with the spectroscopic pole dominance predictions for the three main helicity basis form factors describing @xmath6 decay . the ability to analyze @xmath2 allows us to make the first non - parametric measurements of the mass - suppressed form factor . our result is inconsistent with existing lattice qcd calculations . finally , we measure the form factor that controls non - resonant @xmath7-wave interference with the @xmath6 amplitude and search for evidence of possible additional non - resonant @xmath8- or @xmath9-wave interference with the @xmath10 .	10,974	282
the theoretical treatment of the longstanding problem of turbulent flows @xcite has to relate dynamical systems theory with non - equilibrium statistical physics @xcite . the central notion of physical turbulence theory is the concept of the energy cascade , highlighting the fact that turbulent flows are essentially transport processes of quantities like energy or enstrophy in scale . although well - established theories due to richardson , kolmogorov , onsager , heisenberg and others ( for reviews we refer the reader to @xcite ) can capture gross features of the cascade process in a phenomenological way , the dynamical aspects are by far less understood , and usually are investigated by direct numerical simulations of the navier - stokes equations . an exception , in some sense , are inviscid fluid flows in two dimensions . based on the work of helmholtz @xcite , it was kirchhoff @xcite who pointed out that the partial differential equation can be reduced to a hamiltonian system for the locations of point vortices , provided one considers initial conditions where the vorticity is a superposition of delta - distributions ( we refer the reader to the works of aref @xcite as well as the monographs @xcite ) . due to onsager @xcite ( for a discussion we refer the reader to @xcite ) a statistical treatment of point vortex dynamics is possible for equilibrium situations because of the hamiltonian character of the dynamics , provided the ergodic hypothesis holds . extensions to non - equilibrium situations based on kinetic equations have been pursued , e.g. , by joyce and montgomery @xcite , lundgren and pointin @xcite , as well as more recently by chavanis @xcite . the purpose of the present article is to generalize kirchhoff s point vortex model to a rotor model that exhibits the formation of large - scale vortical structures due to the formation of rotor clusters . the existence of such a process in two - dimensional flows where a large - scale vorticity field spontaneously emerges from an initially random distribution of vortices was first predicted by kraichnan @xcite and is termed an inverse cascade . thereby , the energy that is injected into the small scales is transfered to larger scales , whereas the enstrophy follows a direct cascade from large to small scales . it was also kraichnan @xcite , who gave an intuitive explanation of the possible mechanism of the cascade : he considered a small - scale axisymmetric vortical structure that is exposed to a large - scale strain field . eventually , the vortex is elongated along the stretching direction of the strain , i.e. to a first approximation drawn out into an elliptical structure . this thinning mechanism induces relative motions between vortices that have been deformed under their mutual strain , which leads to a decrease of the kinetic energy of the small - scale motion and consequently to an energy transfer upscale . more recently , it has been pointed out numerically and experimentally by chen et al . @xcite that the effect of vortex thinning is indeed an important feature of the inverse cascade . an appropriate vortex model for the inverse cascade therefore has to provide a mechanism similar to that identified in @xcite . although , several point vortex models have been known for a long time to form large - scale vortical structures from an initially random distribution of point vortices due to the events of vortex merging @xcite or special forcing mechanisms @xcite , an explicit inclusion of the concept of vortex thinning never has been taken into account . in our vortex model , the small - scale vortical structure is represented by a rotor consisting of two point vortices with equal circulation that are glued together by a nonelastic bond . the main observation now is that the two co - rotating point vortices mimic a far - field that is similar to an elliptical vortex , which makes the rotor sensitive to a large - scale strain . the model is motivated by a representation of the vorticity field as a superposition of vortices with elliptical gaussian shapes along the lines of melander , styczek and zabusky @xcite . the nonelastic bond in a rotor can be considered as an over - damped spring which models the influence of forcing and viscous damping . however , the main renewal in this model is not the mechanism of how the energy is injected into the system , but how the energy is transfered upscale due to the strain - induced relative motions between the rotors in the sense of vortex thinning . the efficiency of the cascade in the rotor model is supported by the relatively fast demixing of the system as well as a kolmogorov constant of @xmath0 that is within the range of accepted values @xcite . this paper is organized as follows : first of all , we consider a decomposition of the vorticity field into localized vortices with different shapes in section [ dec ] . in section [ ans ] , we make an ansatz for the shapes which corresponds to an elliptical distribution of the vorticity and discuss the interaction of two vortices with like - signed circulation within the point vortex model , the gaussian vortex model and the elliptical model . it will explicitly be shown that the former two models do not lead to a relative motion between the vortices , and that the thinning mechanism is only taken into account by the elliptical model . a suitable forcing mechanism for the vorticity equation is introduced in section [ forcing ] and then used within our generalized vortex model , presented in section [ modelsection ] . as it is known from basic fluid dynamics , the vorticity @xmath1 only possesses one component in two - dimensional flows and obeys the evolution equation @xmath2 here , the advecting velocity field is determined by biot - savart s law according to @xmath3 we consider the two - dimensional vorticity equation in fourier space derived from equation ( [ omega ] ) in the appendix [ fourier - vorticity ] according to @xmath4 with @xmath5 $ ] . + in the following the vorticity is decomposed into vortices @xmath6 with the circulation @xmath7 that are centered at @xmath8 and that possess the shapes @xmath9 , namely @xmath10 our ansatz thus reads @xmath11 for @xmath12 , we recover the vorticity field @xmath13 of point vortices @xmath14 that are located at the positions @xmath8 and that are a solution of the ideal vorticity equation ( @xmath15 ) , which conserves the vorticity along a lagrangian trajectory . inserting the vorticity field from ( [ point ] ) into biot - savart s law ( [ biot ] ) immediately yields the evolution equation for the point vortices @xmath16 we now insert our ansatz ( [ ansatz ] ) into the vorticity equation and obtain @xmath17\\ \nonumber & = & i{\bf k } \cdot \sum_{j , l}\gamma_j \gamma_l \int \textrm{d}{\bf k } ' { \bf u}({\bf k } ' ) e^{i({\bf k}-{\bf k}')\cdot { \bf x_j}+i{\bf k}'\cdot { \bf x}_l } \\ & ~&\times e^{w _ j({\bf k}-{\bf k}',t)+w_l({\bf k}',t)}\end{aligned}\ ] ] the left - hand side of this equation contains the sweeping dynamic of the vortices encoded in the temporal change of @xmath8 as well as the temporal change of the shapes @xmath9 due to shearing and vorticity . in the inviscid case , the entire dynamic of the @xmath18-th vortex is determined by the nonlinearity on the right hand side of equation ( [ evom ] ) which couples the different fourier modes of the vortices @xmath19 as well as the self - interaction term from @xmath20 in a rather complicated manner . nevertheless , a separation of the effects becomes possible under the assumption that the overlap of the different vortex structures is negligible , which is valid for widely separated vortices . to this end , we single out the terms in the summations over @xmath18 and get @xmath21 the sweeping dynamic can now be defined via the terms in the evolution equation ( [ evom_neu ] ) which are proportional to @xmath22 . this immediately yields the evolution equations for the center of the vortices @xmath23 where we have defined the velocity kernels @xmath24 inserting the evolution equation of the vortex centers back into yields the evolution equations for the shapes @xmath25 } e^{w_i(-{\bf k}',t)+w_l({\bf k}',t ) } \nonumber \\ & & \times \left [ e^{w_i({\bf k}-{\bf k}',t)-w_i(-{\bf k}',t)-w_i({\bf k},t)}-1\right]\end{aligned}\ ] ] here the sum includes also the self - interaction term with @xmath20 . the system of equations ( [ x_j ] ) and ( [ shape ] ) is the extension of the set of evolution equations for the @xmath26-point vortices ( [ point ] ) and takes into account possible changes of the shapes @xmath9 of each vortex . it is important to stress that up to now we did not impose any restrictions on the shapes @xmath27 . the vorticity of an elliptical vortex with major and minor semi - axes @xmath28 and @xmath29 can be written according to @xmath30 where @xmath31 is the symmetric matrix of the dyadic products of the semi axes . a rotation of the coordinate system then turns ( [ elli ] ) into @xmath32 where @xmath33 is the is the ratio of the major to the minor semi axes . the vorticity in fourier space thus reads @xmath34 which again corresponds to an elliptical distribution of the vorticity . an elliptical representation of the shapes can thus be obtained via the approximation @xmath35 with the symmetric matrix @xmath36 . in approximating the last term on the right - hand side of equation ( [ shape ] ) by @xmath37\end{aligned}\ ] ] we are able to derive an evolution equation for the matrix @xmath36 , namely @xmath38\\ \nonumber & ~&+ \sum_{l \ne j } \gamma_l [ s_{jl}({\bf x}_j-{\bf x}_l ) c_j+c_j s_{jl}({\bf x}_j-{\bf x}_l)^t ] % \sum_l \gamma_l [ \nabla { \bf u}_{il}({\bf x}_i-{\bf x}_l ) % c_i+c_i % \nabla { \bf u}_{il}({\bf x}_i-{\bf x}_l)]\end{aligned}\ ] ] here , we explicitly have introduced the matrix @xmath39 $ ] and have singled out the term with @xmath20 . the velocity field is now determined from eq . ( [ u ] ) up to the first order in @xmath40 valid for widely separated vortices @xmath41 \frac{{\bf r}}{2\pi\|{\bf r}\|^2}\end{aligned}\ ] ] the evolution equation for the vortex centers then reads @xmath42\nabla_{{\bf x}_j } { \bf e}_z \times \frac{{\bf x}_j-{\bf x}_l } { 4\pi\|{\bf x}_j-{\bf x}_l\|^2}\end{aligned}\ ] ] a similar system of equations ( [ evolc ] ) and ( [ x_i ] ) has been obtained by melander et al . @xcite by means of a truncation of the stream function within their second - order moment model for the euler equations . it is illustrative to consider the interaction of two vortices @xmath43 and @xmath44 at the positions @xmath45 and @xmath46 that possess equal circulation @xmath47 in the realm of the different vortex models considered above , namely the point vortex model , the gaussian shape model , and the elliptical gaussian shape model . + + _ i. ) point vortex model : _ + for the case where @xmath48 we recover the evolution equations of two point vortices @xmath49 where we made use of @xmath50 with the inverse laplacian @xmath51 . + the evolution equation for the relative coordinate @xmath52 then reads @xmath53 or @xmath54 which is a circular motion of the point vortices around their center @xmath55 with the angular velocity @xmath56 . gaussian shapes : _ + let us consider the case of gaussian shapes @xmath57 and @xmath58 . the symmetry of the problem imposes that @xmath59 , and we arrive at the following evolution equations for the centers @xmath60 in making use of @xmath61 which is the velocity profile of a lamb - oseen vortex , the evolution equation for the relative coordinate reads @xmath62 the evolution equation for the shapes @xmath63 has to be evaluated in a similar fashion from eq . ( [ evolc ] ) , but for now we invoke the approximation @xmath64 where we have neglected the interaction - terms in ( [ evolc ] . this yields the evolution equations for two lamb - oseen vortices @xmath65 in comparison to the angular velocity of the point vortex pair from above , the angular velocity of the gaussian vortex pair is thus slowed down by viscosity . however , if we observe such two vortices in real flows , we would see a deformation of the two vortices due to their mutual strain . this deformation in turn , leads to an attractive motion of the vortex centers and ultimately to a merging process of the two vortices . at this point , it is important to notice that a direct consequence of an axisymmetric vorticity profile is that @xmath66 , which means that no relative motion is induced . furthermore , in this context we want to mention that a recent investigation of the two - point vorticity statistic in two - dimensional turbulence within a gaussian approximation revealed the absence of an energy flux from smaller scales to larger scales @xcite . the emergence of deformable structures that induce such relative motions in the context of vortex thinning can thus be considered as an important feature of the inverse cascade . + + _ iii . ) elliptical shapes : _ + as we have discussed in _ ii . ) _ , the mutual interaction of gaussian vortices in real flows leads to deformations and subsequently attractive motions of the vortex centers . such deformations can be considered in a first approximation as elliptical deformations . + therefore , the interaction of two elliptical vortices should for the first time lead to non - vanishing relative motions . + the evolution equation for two elliptically shaped vortices read @xmath67 { \bf k}'}\\ \dot { \bf x}_2&= & \gamma \int \textrm{d}{\bf k } ' { \bf u}({\bf k } ' ) e^{-i { \bf k } ' \cdot ( { \bf x}_2-{\bf x}_1 ) } e^{-\frac{1}{2 } { \bf k } ' [ c_1+c_2 ] { \bf k}'}\end{aligned}\ ] ] for widely separated vortices the evolution equation for the relative coordinate thus reads @xmath68\nabla_{{\bf r } } \right ) { \bf e}_z \times \frac{{\bf r } } { r^2}\end{aligned}\ ] ] which can lead to contributions to the relative motion @xmath69 , provided that the matrices @xmath70 and @xmath71 do not reduce to diagonal matrices as in the case of gaussian shapes . whether the motion is attractive or repulsive , is to a far extend determined by the alignment angle @xmath72 between @xmath73 and the major semi axis @xmath28 of the vortices , which is explicitly derived for the interaction of two rotors in section [ inter ] , for instance in eq . ( [ r_i ] ) . as it can be seen from eq . ( [ shape ] ) , the viscous contributions causes the broadening of the shape of a vortex . since this effect is more pronounced for smaller vortex structures , thus larger values of @xmath74 in ( [ evolc ] ) , an appropriate forcing mechanism has to counteract this effect and provide an energy input at small scales . the forcing mechanism we want to introduce , consists in forcing the semi axes of each elliptical vortex and thus the whole shape of this vortex back to a fixed shape @xmath75 . it will be seen in section [ modelsection ] that the influence of this kind of forcing makes the two like - signed point vortices of our rotor model to behave as if they were connected by an over - damped spring . the described forcing mechanism can now be introduced in the following way : @xmath76 \nonumber \\ \label{modelg3 } & + & \sum_l \gamma_l [ s_{il}({\bf x}_i-{\bf x}_l ) c_i+c_i s_{il}({\bf x}_i-{\bf x}_l)^t]\end{aligned}\ ] ] such type of forcing may be obtained from the vorticity equation ( [ vorticity ] ) by just adding a linear damping term , @xmath77 as well as the forcing term @xmath78 , @xmath79 \nonumber \end{aligned}\ ] ] where the centers @xmath80 as well as the shapes @xmath81 are close to the centers and the shapes of the elliptical vortices . the first contribution in eq . ( [ force ] ) leads to a modulation of the circulation , the second term describes a shift of the rotor center and the third one corresponds to a modification of the width of the gaussian vortex shape that forces the elliptical vortex back to a certain shape @xmath75 . the stretching of the semi axes of the elliptical vortex due to viscous broadening represented by the first term on the right - hand side in eq . [ modelg3 ] is thus counteracted by the second term trying to contract the shape of the vortex back to @xmath75 . a striking analogy to this forcing mechanism can be found in the explanation of the magneto - rotational instability @xcite . thereby , two elements of an electrically conduct- ing fluid that undergo a rotation around a fixed center are supposed to be connected by an elastic spring repre- senting the magnetic field . as a consequence the angular momentum of the system is not a conserved quantity anymore and the fluid motion becomes unstable . although , the introduced forcing mechanism is an ad - hoc forcing , it emerges in a physically plausible way from the basic equations of the elliptical model [ evolc ] and [ x_i ] . furthermore , it should be mentioned that the system of equations ( [ modelg1 ] ) can be obtained from the instanton equations of two - dimensional turbulence by means of a variational ansatz with gaussian elliptical vortices @xcite . as we have seen in section [ models ] about the interaction between two point vortices with equal circulation compared to the interaction between two elliptical vortices with equal circulation , the former model fails to describe a relative motion @xmath82 in the direction of @xmath73 . the thinning mechanism mentioned in @xcite is thus clearly neither captured by onsager s point vortex model nor by a gaussian distribution of the vorticity , in analogy to @xcite . our vortex model is based on the observation that the point vortex couple considered in section [ models ] under _ i. ) _ generates a far field that is similar to that of one elliptical vortex with circulation @xmath83 . we therefore consider point vortex couples with equal circulation @xmath84 at the positions @xmath85 and @xmath86 as indicated in fig . [ vector ] . the center of this object that we want to term a rotor is then given by @xmath87 . in order to model a forcing and viscous damping mechanism similar to that mentioned in section [ forcing ] , the two point vortices in a rotor are supposed to be glued together by an inelastic spring , such that each rotor possesses an additional degree of freedom and that the size of a single rotor relaxes with relaxation time @xmath88 to @xmath89 . our model then reads @xmath90 \nonumber \\ \dot { \bf y}_i & = & - \frac{\gamma}{2 } ( d_0-\|{\bf y}_i-{\bf x}_i\| ) { \bf e}_i + \gamma_i { \bf u}({\bf y}_i-{\bf x}_i ) \nonumber \\ & + & \sum_j \gamma_j [ { \bf u}({\bf y}_i-{\bf y}_j)+ { \bf u}({\bf y}_i-{\bf x}_j)]\end{aligned}\ ] ] where we have defined the unit vector @xmath91 and the velocity field @xmath92 is the velocity field of a point vortex centered at the origin , @xmath93 . the first two terms on the right - hand side of equation ( [ model ] ) describe the interaction within one rotor , whereas the last two terms describe the interaction with the other rotors . for vortices moving inside a closed regime , the velocity field has to be changed based on the introduction of mirror vortices @xcite . it is important to stress that the above system is not a hamiltonian system anymore due to the inelastic coupling which mimics an energy input to the system on a scale @xmath89 . furthermore , by the additional degree of freedom the rotor is sensitive with respect to a shear velocity field which can be seen from the multipole expansion of the relative coordinate @xmath94 with respect to the leading terms in @xmath95 , derived in the appendix [ app ] @xmath96 the influence of the forcing can be seen from the first term : if a rotor is subjected to shear , the spring between the point vortices in a rotor pulls back and the rotor relaxes to the size @xmath89 . the shear velocity in the last term is thereby generated by the other rotors . + in a similar way , the multipole expansion of the center coordinate of the rotor in appendix [ app ] leads to the evolution equation @xmath97 { \bf u}({\bf r}_{ij})\ ] ] the evolution equation is identical to equation ( [ x_i ] ) , provided that the matrix @xmath98 can be written as @xmath99 , which corresponds to an infinitely thin elliptical vortex oriented in @xmath100-direction . the relative distance @xmath100 can thus be considered as an elliptical deformation of the velocity field that depends on the shear velocity field induced by the remaining vortices and the effect of the overdamped spring . furthermore , we again want to emphasize that the last term in eq . ( [ locr ] ) induces relative motions between the rotors as we have seen in section [ models ] . the usual point vortex dynamics solely represented by the first term on the right hand side of equation ( [ locr ] ) is thus extended to a dynamical system that is sensitive to the effect of vortex thinning . we have numerically solved the dynamical system ( [ model ] ) in a square periodic domain @xmath101 . the temporal evolution of 200 rotors with an equal number of positive and negative circulations starting from a random initial condition exhibits the formation of a large scale vortical structure via the formation of rotor - clusters . a typical time series is exhibited in fig . [ unequal ] , for the parameter values ( @xmath102 , @xmath103 , @xmath104 , @xmath105 , @xmath101 ) . the boxes have been continued periodically , with up to 5 layers of neighboring boxes , which guarantees a sufficient degree of homogeneity . the temporal evolution of the system can be quantified by the introduction of a characteristic time scale of the system which is given as the period that a rotor possesses at a fixed distance @xmath89 and is in the following termed as one rotor turnover time @xmath106 , which follows from equation ( [ phi ] ) . 1= as it can be seen from fig . [ unequal ] , the clustering of like - signed rotors already occurs within the first 100 rotor turnover times , which means that the separation of the rotors takes place on a relatively short time - scale . the temporal evolution of 200 rotors with identical circulations starting from random initial positions of the rotors , is exhibited in fig . [ lattice ] . a fluctuating lattice of rotor clusters appears and after approximately 1500 rotor turnover times , the system forms a monopole which attracts the remaining rotors . we have calculated the kinetic energy spectra of the rotor system with @xmath107 at different times in fig . [ spec1 ] . starting from 20 different initial configurations of the rotors , we let the systems evolve in time and performed the ensemble average at a specific time @xmath108 . thereby , the spectrum is calculated from the velocity field in eq . ( [ biot ] ) that has been interpolated on a grid and then transformed into fourier space . initially , the rotors possess a clear point vortex spectrum following a power law @xmath109 . only at high values of @xmath110 deviations due to the singular structure of the vorticity and corresponding discontinuities in the velocity field manifest themselves in an increase of @xmath111 . this effect can be observed in the following spectra , too . however , after a few ( @xmath112 ) rotor turnover times , as the rotor clustering sets in , a more universal energy spectrum can be observed . due to an energy flux from smaller to larger scales , the spectra begin to steepen for smaller @xmath110-values , revealing a spectrum that is close to the predicted @xmath113 . as it can be seen from the compensated spectra in fig . [ spec1 ] , this slope remains constant for nearly @xmath114 and an energy flux into the large scales takes place . this is also in agreement with the time - averaged spectral energy flux @xmath115 , depicted in fig . [ flux ] . the inlet plot in fig . [ flux ] corresponds to the kinetic energy transfer rate @xmath116 , which is related to @xmath115 according to @xcite @xmath117 it is obvious that energy accumulates at small @xmath110-values . this is not surprising , since the rotor model only provides an energy input on small scales and it will be a task for the future to extend the model in order to achieve a damping at small values of @xmath110 and thus to extract energy at the integral scale . we now turn to the determination of the kolmogorov constant of the energy spectrum from the binary rotor system ( @xmath107 ) . the spectrum as it was predicted by kraichnan @xcite reads @xmath118 where @xmath119 is the energy dissipation rate . in the following , @xmath119 is determined from the time - dependence of the total kinetic energy that shows up to be linear in time within @xmath120% . for the sake of completeness , we have provided the corresponding plot in fig . [ energy ] in the appendix . the slope of the fitted line can thus be interpreted as the rate of energy input into the system and we obtain a value of @xmath121 . in order to make an estimate for @xmath122 , we take an average of the compensated spectra in fig . [ spec1 ] of times between @xmath123 and @xmath124 which yields @xmath125 . the kolmogorov constant @xmath126 of the rotor system for times t between @xmath123 and @xmath127 thus lies in the range @xmath0 . the high inaccuracy of our estimate is due to the estimation of @xmath122 . reported values from direct numerical simulations @xcite and experiments @xcite lie within the range from 5.8 to 7.0 . the kolmogorov constant of the rotor system thus lies on the lower end of that range . in comparison to the point vortex model of siggia and aref @xcite , who report a kolmogorov constant of @xmath128 which is twice the accepted value , the rotor model thus seems to provide an efficient mechanism for the energy transfer upscale due to the effect of vortex thinning . another important way to determine the distribution and the occuring structures in the rotor model will be discussed in the following . in order to quantify the emergence of the rotor clusters in fig . [ unequal ] and [ lattice ] , we make use of the radial distribution function @xmath129 which can be considered as the probability of finding a like - signed rotor at a distance @xmath73 away from a reference - rotor ( for further references see for instance @xcite ) . the radial distribution function is therefore given as @xmath130 where @xmath131 and the prime indicates that summation over @xmath132 is left out . the averaging is performed in such a way that the number of like - signed rotors populating a concentric segment of radius @xmath133 at a given radius r is divided by its area . in the following the radial distribution function is assumed to be isotropic , so that @xmath134 . for a disordered state one expects the radial distribution function to be equal to 1 for every @xmath135 . as the formation of the rotor clusters sets in , one should observe an increase of @xmath136 for small @xmath137 , since the probability of finding a like - signed rotor in the neighborhood of a reference - rotor increases . the radial distribution functions for the two time series are plotted in fig . [ radial_dis ] and one clearly observes an increase of @xmath136 at small @xmath137 . in order to get smooth curves , @xmath136 was calculated in such a way that it shows no discontinuities for @xmath138 due to a minimum distance between neighboring rotors . the radial distribution function can thus be used as a qualitative measure for the formation of the clusters and their typical sizes . furthermore , the radial distribution function is related to the structure factor @xmath139 in a way that @xmath140 where @xmath141 is the bessel function of order zero . the structure factor @xmath142 can thus be calculated via the hankel transform of @xmath143 , provided that the radial distribution function is isotropic . the structure factors for the two system are plotted in fig . [ structure ] . for the case of the mixed system of fig . [ unequal ] , one observes an increase of @xmath142 over time . whether this increase is governed by a power law for intermediate @xmath33 has to be evaluated within further simulations of the model equations ( [ model ] ) . furthermore , eq . ( [ structurefactor ] ) is of great importance for the investigation of the rotor model , since it relates macroscopic quantities on the left - hand side to microscopic quantites such as the radial distribution function . it is thus a good starting point for the interpretation of the fluctuations of the rotor clusters in the realm of phase transitions . the growth rate of the rotor clusters can be determined from the time dependence of the structure factor . the growth of the largest structures of the system is given by @xmath144 . in fig . [ temp ] , the temporal evolution of @xmath144 is plotted for the two systems . the fluctuating rotor lattice below exhibits a pronounced growth rate after @xmath145 , whereas the growth rate of the mixed system above already increases for @xmath146 . for comparison , two power laws @xmath147 and @xmath148 were plotted in the figures . the growth rate of our rotor clusters can thus be considered as relatively strong compared to typical growth rates from pattern formation , for instance compared to the growth rate of droplets in the cahn - hilliard equation where @xmath149 according to slyozov - lifshitz theory @xcite . the fact that the rotor vortex system exhibits a pronounced inverse cascade already for moderate numbers of rotors ( 200 rotors have been used for the figures ) on a small time - scale allows us to investigate the inverse cascade using methods of nonlinear dynamics . although , usual point vortex models such as @xcite , have been known for a long time to possess inverse energy cascades the present model incorporates the aspect of vortex thinning , due to a possible change of the ellipticity of the rotor in much the same way as identified in the experiments of chen et al @xcite . hence , it is a minimal dynamical model containing the mechanisms of the inverse cascade . in the following we shall discuss the origin of the formation of clusters of rotors with like - signed circulations . in the following , we consider the configuration of two rotors with circulations @xmath84 and @xmath7 , depicted in fig . [ vector ] , which can be considered as the interaction of two infinitely - thin elliptical vortices in the same manner as _ iii . ) _ from section [ models ] . it is straightforward to show that the center of vorticity @xmath150 is a conserved quantity . the distance vector @xmath151 between the two rotors obeys the evolution equation @xmath152 \\ \nonumber & ~&+ \frac{1}{8}\left . [ -2 \frac{\bf r}{\|{\bf r}\|^4 } { \bf r}_j^2- 4 \frac{{\bf r}_j}{\|{\bf r}\|^4}{\bf r}_j\cdot { \bf r}+8 \frac{{\bf r}}{\|{\bf r}\|^6 } ( { \bf r}_j\cdot { \bf r})^2 ] \right]\end{aligned}\ ] ] which follows from equation ( [ locr ] ) in calculating the corresponding velocity field gradients , described in the appendix [ app ] . + for the following it is convenient to represent the unit vectors according to @xmath153 , as well as @xmath154 , which yields @xmath155({\bf e}_i\cdot { \bf e}_r)=\frac{1}{2 } \sin(2 ( \varphi_i -\varphi_r))\ ] ] we obtain the equation for the relative distance @xmath156\end{aligned}\ ] ] the evolution equation for the relative coordinate of a rotor reads @xmath157\end{aligned}\ ] ] which follows from equation ( [ dipol1 ] ) and the calculation of the velocity field gradients , performed in the appendix [ app ] . we have to determine the quantities @xmath158 , @xmath159 , which are determined by the evolution equations @xmath160 we can solve iteratively for small deviations of @xmath161 from @xmath89 : @xmath162 a similar treatment applies to @xmath163 . splitting the rotation into its fast ( @xmath164 ) and slow varying parts @xmath165 , i.e. @xmath166 we obtain after a partial integration @xmath167^{-1}\\ \nonumber & ~&\times e^{2i(\tilde \varphi_i(t')-\tilde \varphi_r(t'))}(\dot{\tilde { \varphi}}_i(t')-\dot { \tilde { \varphi}}_r(t'))\\ ~\end{aligned}\ ] ] in order to proceed with the adiabatic approximation , we neglect the second term in eq . ( [ adiabatic ] ) since it contains time derivatives of the slowly varying parts of the rotations . assuming that the damping constant @xmath168 is large compared to the rotation frequency of the rotor , we obtain @xmath169 to lowest order in @xmath170 we thus obtain @xmath171 \nonumber \\ r_j^2 & = & d_0 ^ 2\left[1 + \frac{\gamma_i}{\pi \gamma r^2 } \sin(2 ( \varphi_j-\varphi_r))\right]\end{aligned}\ ] ] here , the last terms on the right - hand side arise due to the change of the size of the rotors , connected with a change of the far field , induced by the mutually generated shear . it thus mimics the mechanism of vortex thinning , identified in @xcite . + the relative motion of the rotors obeys the evolution equation @xmath172 \end{aligned}\ ] ] we now average the evolution equation with respect to the rotations of the vectors @xmath173 and @xmath174 taking into account that the averages @xmath175 vanish . furthermore , the averages @xmath176 are positive . as a consequence , the relative distance behaves according to @xmath177 two rotors approach each other , except for @xmath178 . it is important to stress that this attractive relative motion arises only if we include the irreversible effect of the strain induced stretching of the rotors . furthermore , the symmetry breaking of @xmath179 in equation ( [ r ] ) can be considered as an important feature of the rotor model in comparison to the point vortex model , which conserves this symmetry . we have presented a generalized point vortex model , a rotor model , exhibiting an inverse cascade based on clustering of rotors . we have discussed how this rotor model can be derived from the vorticity equation by an expansion of the vorticity field into a set of elliptical vortices at locations @xmath180 and shapes @xmath181 . an important point has been the inclusion of a forcing term , which prevents the elliptical far field of the rotors from diffusing away . the added forcing term breaks the symmetry @xmath182 , @xmath183 . this symmetry breaking lies at the origin of cluster formation and the inverse cascade , as can be seen from the two - rotor interaction inducing in average a relative motion proportional to @xmath184 . the numerical simulations of the model equations [ model ] reveal the formation of rotor clusters on a short time scale . in addition , the calculated energy spectra and energy fluxes give strong evidence for the important role of vortex thinning during the cascade process in two - dimensional turbulence . the presented rotor model can be investigated by applying methods from dynamical systems theory like the evaluation of finite time ljapunov exponents and ljapunov vectors . these and further dynamical aspects are the basis for future work and will be covered in a following paper . the model system ( [ modelg1 ] ) may also be studied as a stochastic system by considering the velocity @xmath185 to be a white noise force . the corresponding fokker - planck equation allows one to draw analogies with quantum mechanical many body problems . furthermore , we emphasize that a continuum version of the model equations ( [ modelg1 ] ) leads to a subgrid model exhibiting analogies with the work of eyink @xcite . it will be a task for the future to investigate the cluster formation from a statistical point of view , based on the formulation of kinetic equations , along the lines as has been performed for fully developed turbulence @xcite , and rayleigh - bnard convection @xcite . in this respect we hope to find a relation to the kinetic equation for the two - point vorticity statistics recently derived on the basis of the monin - lundgren - novikov hierarchy , taking conditional averages from direct numerical simulations @xcite . is very grateful for discussions with michael wilczek and frank jenko about the organization of this paper . sadly , rudolf friedrich ( 16th august 2012 ) unexpectedly passed away during this work . he was as much an inspiring physicist as well as a caring father . for the decomposition of the vorticity field into a discrete set of vortices with arbitrary shapes in section [ dec ] , we have made use of the vorticity equation in fourier space ( [ vorticity ] ) . this equation can be derived from the vorticity equation in real space ( [ omega ] ) in defining the vorticity and the velocity field in fourier space according to @xmath186 and @xmath187 the nonlinearity in eq . ( [ omega ] ) can thus be expressed in terms of the convolution between @xmath188 and @xmath189 which yields @xmath190 furthermore , the velocity field in fourier space can be calculated from biot - savart s law in eq . ( [ biot ] ) according to @xmath191 a substitution @xmath192 in the last integral yields @xmath193 where we have defined the inverse laplacian in @xmath22-space as @xmath194 . inserting the fourier space representation of @xmath195 into ( [ om ] ) yields the evolution equation ( [ vorticity ] ) used for the decomposition of the vorticity field in section [ dec ] . in this part we calculate the multipole expansion of a rotor , defined by @xmath85 and @xmath86 in fig . [ vector ] . to this end , we introduce relative and center coordinates , according to @xmath196 as well as the vector @xmath197 . + in using equation ( [ model ] ) , we obtain the evolution equation for the relative coordinate @xmath198 a taylor expansion of the curled bracket yields @xmath199 where we have only retained the leading terms in @xmath95 . the evolution equation for the center coordinate reads @xmath200 again , a taylor expansion yields @xmath201 ^ 2 + [ ( { \bf r}_i + { \bf r}_j ) \cdot \nabla_{{\bf r}_{ij } } ] ^2 \right \ } { \bf u } ( { \bf r}_{ij } ) \bigg ] \\ \nonumber & = & 2 \sum_j \gamma_j { \bf u } ( { \bf r}_{ij})\\ & ~&+ \frac{1}{4 } \sum_j \gamma_j [ ( { \bf r}_i \cdot \nabla_{{\bf r}_{ij}})^2 + ( { \bf r}_j \cdot \nabla_{{\bf r}_{ij}})^2 ] { \bf u } ( { \bf r}_{ij})\end{aligned}\ ] ] the gradients of the velocity fields are now calculated according to @xmath202 which is needed in equation ( [ dipol1 ] ) , and @xmath203 now , this is the counterpart of equation ( [ locr ] ) + +	we generalize kirchhoff s point vortex model of two - dimensional fluid motion to a rotor model which exhibits an inverse cascade by the formation of rotor clusters . a rotor is composed of two vortices with like - signed circulations glued together by an overdamped spring . the model is motivated by a treatment of the vorticity equation representing the vorticity field as a superposition of vortices with elliptic gaussian shapes of variable widths , augmented by a suitable forcing mechanism . the rotor model opens up the way to discuss the energy transport in the inverse cascade on the basis of dynamical systems theory .	11,991	165
the `` metallic - lined '' or am stars are a - type stars which have strong absorption lines of some metals such as zn , sr , zr and ba and weaker lines of other metals such as ca and/or sc relative to their spectral type as determined by the strength of the hydrogen lines @xcite . the strong metallic lines are more typical of an f star rather than an a star . the work of @xcite established radiative diffusion in a strong magnetic field as the likely cause of the chemical peculiarities in ap stars . when the magnetic field is absent , diffusion leads to the am / fm stars @xcite . the presence of magnetic fields in am stars has been investigated , but with negative results , ( e.g. @xcite ) . a peculiarity of am stars is that their projected rotational velocities are generally much smaller than normal a stars and they are nearly always members of close binary systems . rotational braking by tidal friction in a binary system is regarded as a possible explanation for the low rotational velocities in am stars . slow rotation further assists the segregation of elements by diffusion . the abundance anomalies predicted by the diffusion hypothesis are usually much larger than observed . @xcite developed detailed models of the structure and evolution of am / fm stars using opal opacities , taking into account atomic diffusion and the effect of radiative acceleration . these models develop a convective zone due to ionization of iron - group elements at a temperature of approximately 200,000 k. in addition to this convective zone , these stars also have a thin superficial convective zone in which h and hei are partially ionized . by assuming sufficient overshoot due to turbulence , these separate convective zones become one large convective zone . the resulting mixing dilutes the large abundance anomalies predicted by previous model , leading to abundances which closely resemble those observed in am / fm stars . a detailed abundance analysis of eight am stars belonging to the praesepe cluster @xcite show good agreement with the predictions of @xcite for almost all the common elements except for na and possibly s. the models of @xcite assume a certain ad - hoc parametrization of turbulent transport coefficients which are adjusted to reproduce observations . other parameterizations of turbulence have been proposed for other types of stars . @xcite have investigated to what extent these are consistent with the anomalies observed on am / fm stars . they find that the precision of current abundances is insufficient to distinguish between models . more recently , @xcite have studied the abundance anomalies of the mild am star sirius a. they find that except for b , n and na , there is good agreement with the predicted anomalies but turbulent mixing or mass loss is required . it is not clear whether it is turbulence or mass loss which competes with diffusion to lower the abundance anomalies . for example , @xcite find that diffusion in the presence of weak mass loss can explain the observed abundance anomalies of pre - main - sequence stars . this is in contrast to turbulence models which do not allow for abundance anomalies to develop on the pre - main - sequence . most of the pulsational driving in @xmath0 scuti stars is caused by the @xmath1 mechanism operating in the heii ionization zone . diffusion tends to drain he from this zone and therefore pulsational driving may be expected to be weaker or absent in am / fm stars @xcite . in fact , for many years it was thought that classical am / fm stars did not pulsate , though claims were made for some stars @xcite . recently , intensive ground - based observations by super - wasp @xcite , and also from the _ kepler _ mission @xcite have shown that many am / fm stars do pulsate . @xcite , for example , found that about 200 am / fm stars out of a total of 1600 ( 12.5 percent ) show @xmath0 sct pulsations , but with generally lower amplitudes . they found that the pulsating am / fm stars are confined between the red and blue radial fundamental edges , in agreement with @xcite . while there are many @xmath0 sct stars hotter than the fundamental blue edge , this does not seem to be the case for pulsating am / fm stars . the significance of this result remains to be evaluated . the effect of draining of he from the heii ionization zone is to reduce the width of the instability strip , the blue edge moving towards the red edge , eventually leading to the disappearance of the instability strip when he is sufficiently depleted @xcite . @xcite has discussed the effect of diffusion on pulsations in am / fm stars using the models by @xcite . one significant difference with earlier models is that a substantial amount of he remains in the heii ionization zone . the blue edge of the instability strip for am / fm stars is sensitive to the magnitude of the abundance variations and is thus indicative of the depth of mixing by turbulence . @xcite predict that pulsating am / fm stars should lie in a confined region of the hr diagram close to the red edge of the @xmath0 sct instability strip . however , @xcite show that there is no relationship between the predicted am / fm instability strip and the actual location of these stars in the hr diagram . a particularly interesting result of the pulsation analysis of @xcite is the prediction of long - period g modes in a - type stars . as the star evolves , the driving regions shift deeper into the star and the g modes become gradually more and more excited . whereas p modes are stabilized through diffusion , g modes tend to be excited as a result of that process . it appears that diffusion may act to enhance driving of long - period g modes due to a significant increase in opacity due to iron - group elements . this may have a bearing on the fact that nearly all a - type stars observed by _ kepler _ have unexplained low - frequencies @xcite . when dealing with objects with non - standard chemical composition , such as am stars , it is crucial that the opacities are correctly calculated . this question has been investigated by several authors in recent years . these studies show that a non - standard chemical composition of the stellar atmosphere alters the flux distribution of the star or modifies the profiles of the balmer lines ( @xcite , @xcite ) . therefore a determination of t@xmath2 and @xmath3 based on a comparison between observed and computed balmer - line profiles will not be correct unless one takes into account the metallicity of the star . thus , even estimates based on standard analysis of the spectra may be in error when applied to am / fm stars . in this paper , we investigate the determination of effective temperature and surface gravity of the am star hd27411 ( hr 1353 , a3 m ) using spectra in the eso archives . the purpose is to determine whether the stellar parameters of this star agree with those obtained from strmgren photometry and hence to test the reliability of the effective temperature calibration applied to am / fm stars . the star was used by @xcite as a comparison in their study on the calcium stratification in ap stars . hd27411 is not known to pulsate . however , as we know from _ kepler _ observations , pulsations in a and f stars with amplitudes too low to be detected from the ground are common . atmospheric models obtained with atlas9 @xcite use precomputed line opacities in the form of opacity distribution functions ( odfs ) . these are tabulated for multiples of the solar metallicity and for various microturbulent velocities . this approach allows very fast computation of model atmospheres , but with very little flexibility in choice of chemical profile and microturbulent velocity . while this is satisfactory for most applications , it fails for chemically peculiar stars where a non - standard chemical composition profile is required . this can be done with atlas12 @xcite , which is essentially identical to atlas9 , but uses the opacity sampling ( os ) method to evaluate line opacities . in this study we compare the abundances of hd27411 obtained with both codes to determine if the use of atlas12 is essential . the result that am stars are not confined to particular region of the @xmath0 sct instability strip depends , to a large extent , on effective temperatures and luminosities estimated from strmgren photometry @xcite . it is not clear whether the calibration , derived from normal af stars , can be applied to am / fm stars . in this paper we use synthetic strmgren photometry applied to models of am / fm stars to investigate the reliability of fundamental parameters estimated from the photometry . finally , we discuss the relative numbers of pulsating and non - pulsating am stars and compare these to the relative numbers of @xmath0 scuti and constant stars in the instability strip . from this comparison , one can deduce the effectiveness of pulsational driving in the heii ionization zone and compare the he abundance to that expected from diffusion calculations . .ions used to determine the microturbulent velocity in hd27411 . the number of spectral lines used , the microturbulent velocity , @xmath4 , the derived abundance and the radial velocity , rv , are listed . [ cols="<,^,^,^,^ " , ] in fig . [ a12_a9 ] we compare the lte abundances derived from model atmospheres computed using atlas9 and atlas12 . it is evident that there is good agreement in the abundances derived with the two different codes . there are some very small differences for na , v , zr and nd , but these are only 0.1 dex or less . thus we may use the faster atlas9 code with confidence . in fig . [ pattern ] we show the abundances relative to solar standard abundances @xcite . the chemical pattern displayed here is typical of that observed in am stars , i.e. an underabundance of c , n , o , ca and sc and a general increasing overabundance for heavy elements . the atmospheric abundance of li is interesting in the context of diffusion . @xcite and @xcite find that , in general , the li abundance in am stars is close to the cosmic value ( @xmath5 dex ) , although some am / fm stars appear to have an underabundance of li . normal a - type stars in the range @xmath6 k appear to have a higher li abundance , i.e. @xmath7 dex , @xcite . to determine the li abundance , we used the lii @xmath86707 line , taking into account the hyperfine structure @xcite . the abundance that gives the best fit is @xmath9=@xmath108.42@xmath110.10 , which is closer to the abundance in normal a stars and agrees with the average li abundance of three cluster am / fm stars observed by @xcite . @xcite predict abundances as a function of stellar age and effective temperature using their models of diffusion . fig . 14 of their paper allows us to estimate the predicted abundances for a star with a given t@xmath2 up to a maximum age of @xmath12670 myrs . we find that age of hd27411 to be about @xmath13 myr , which is considerably older than the maximum age of models in @xcite , but we will assume that models of 670 myr still give a fair approximation of the abundances . for t@xmath2=7400 k , which is our best estimate for hd27411 , the models by @xcite predicts underabundances ranging from -0.3 dex to -0.1 dex for c , n , o , na , mg , k , and ca . for si and s the abundances are normal , while overabundances of about 0.10.8 dex are found for for li , al , ti , cr , mn , fe , and ni . inspection of their figure reveals that for na , mg , al , si , s , ca , ti , cr , mn , and fe , the abundance anomaly is approximately constant with age and depends only on the turbulence . the abundances of li , c , n , and o vary with age . [ pattern ] shows the abundances of the elements at 670 myr predicted by @xcite compared with our abundances . there is indeed good agreement for li , c , n , o , mg , k , ca , ti , mn , fe , and ni , but abundances of na , al , si , s , and cr are somewhat discrepant . similar discrepancies for na and s were found by @xcite for the am star hd73730 . from the nlte analysis of the s abundance by @xcite , @xcite concluded that nlte effects should be taken into account to determine whether this resolves the discrepancy with diffusion predictions . following this idea , we performed a nlte analysis on na , al , si , s , and cr , to derive their abundances . we used the same technique of matching predicted and observed line profiles , but in this case the nlte line profiles were computed with version 43 of synspec @xcite . this code reads the same input model atmosphere previously computed using atlas12 and solves the radiative transfer equation , wavelength by wavelength in a specified spectral range . synspec also reads the same kurucz list of lines that we used for determining metal abundances . synspec allows one to compute the line profiles by using an approximate nlte treatment , even for lte models . this is done by means of second - order escape probability theory ( for details see @xcite ) . the results of these calculations are shown in table [ abund ] . all the nlte abundances are lower than the lte abundances by factors ranging from 0.23 dex ( s ) to 0.39 dex ( al ) . as can be seen from fig . [ pattern ] , the nlte calculations bring the observations closer to the diffusion predictions by @xcite . in fact , there is no longer any discrepancy in abundances within the observational errors . in order to investigate the effect of line blanketing on the strmgren colour indices , we used the method of synthetic photometry . for this purpose we computed the spectrum of a star at different effective temperatures with abundances shown in table[abund ] . we used the abundances given by the atlas12 models modified by nlte where necessary . the spectra were calculated using spectrum , version 2.76e @xcite . synthetic spectra with normal and peculiar abundances were calculated for @xmath14 k and @xmath15 . in all cases the microturbulence velocity was set to @xmath16 km s@xmath17 . these spectra were convolved with standard @xmath18 transmission functions to calculate synthetic strmgren indices . it should be noted that not all am / fm stars will have the same abundance anomalies as hd27411 . hence the results described here are only indicative of what might be typical in am / fm stars . individual am / fm stars will have different abundances and different line blanketing . in computing these synthetic strmgren indices , it is necessary to identify a particular model with a real star in order to determine the zero points . we chose a model of vega ( @xmath19 ) for this purpose . comparison of the synthetic @xmath20 as a function of @xmath21 with the standard relations of @xcite shows that the @xmath21 zero point required a further correction of -0.04 . the synthetic colours for normal and am / fm stars are listed in table[strom ] . comparison of indices for models with standard solar abundance and the abundances of table[abund ] are shown in fig.[colcol ] . rrrrr @xmath22 & @xmath20 & @xmath23 & @xmath24 & @xmath21 + + normal : + 6000 & 0.365 & 0.251 & 0.372 & 2.630 + 6250 & 0.323 & 0.210 & 0.425 & 2.645 + 6500 & 0.296 & 0.183 & 0.482 & 2.663 + 6750 & 0.264 & 0.168 & 0.544 & 2.684 + 7000 & 0.232 & 0.162 & 0.608 & 2.707 + 7250 & 0.200 & 0.163 & 0.677 & 2.733 + 7500 & 0.150 & 0.186 & 0.846 & 2.792 + 7750 & 0.121 & 0.191 & 0.914 & 2.819 + 8000 & 0.092 & 0.196 & 0.976 & 2.841 + 8250 & 0.063 & 0.199 & 1.030 & 2.858 + 8500 & 0.041 & 0.196 & 1.062 & 2.870 + 8750 & 0.026 & 0.189 & 1.076 & 2.875 + 9000 & 0.013 & 0.180 & 1.077 & 2.875 + 9250 & 0.003 & 0.172 & 1.071 & 2.872 + 9500 & -0.005 & 0.163 & 1.058 & 2.866 + 9750 & -0.012 & 0.156 & 1.038 & 2.857 + + am / fm : + 6000 & 0.434 & 0.351 & 0.202 & 2.637 + 6250 & 0.392 & 0.315 & 0.251 & 2.649 + 6500 & 0.352 & 0.288 & 0.312 & 2.665 + 6750 & 0.312 & 0.268 & 0.382 & 2.683 + 7000 & 0.274 & 0.255 & 0.459 & 2.705 + 7250 & 0.236 & 0.247 & 0.542 & 2.730 + 7500 & 0.183 & 0.267 & 0.705 & 2.786 + 7750 & 0.148 & 0.263 & 0.790 & 2.812 + 8000 & 0.114 & 0.258 & 0.870 & 2.835 + 8250 & 0.079 & 0.251 & 0.944 & 2.852 + 8500 & 0.053 & 0.239 & 0.995 & 2.865 + 8750 & 0.034 & 0.223 & 1.027 & 2.871 + 9000 & 0.019 & 0.207 & 1.042 & 2.872 + 9250 & 0.007 & 0.193 & 1.046 & 2.869 + 9500 & -0.002 & 0.180 & 1.042 & 2.863 + 9750 & -0.010 & 0.168 & 1.027 & 2.855 + [ strom ] various colour colour diagrams derived from the synthetic colours are shown in fig.[red ] together with reddening lines . the reddening lines are from @xcite : + + @xmath25 , @xmath26 , @xmath27 + + also shown are the locations of hd27411 . it is clear that the star lies nicely on the synthetic relations given by the enhanced abundances and that the star is unreddened or only very slightly reddened . it is also clear that in determining the reddenings of am / fm stars it is wise to avoid using @xmath23 and @xmath28 . the @xmath29 relations for am / fm abundances is almost the same as for normal abundances and it is preferable to use this diagram to deduce the reddening correction . assuming that hd27411 is unreddened and matching the observed indices with those for modified abundances in table[strom ] gives @xmath30 k. the @xmath21 index is correlated well with effective temperature for @xmath31 k and is practically unaffected by line blanketing . it is weakly sensitive to surface gravity , particularly for the hotter stars . @xmath20 is correlated with effective temperature but is affected by blanketing for the cooler stars . as can be seen from fig[colcol ] , the @xmath23 index is severely affected by blanketing . the @xmath24 index for am / fm stars is nearly always smaller than for normal stars . this index measures the strength of the balmer discontinuity ( and hence the surface gravity ) , but it is clearly not entirely free of blanketing effects . in estimating the absolute magnitudes , @xmath32 , of f stars , @xcite defines , first of all , a relationship for stars on the zams as a function of @xmath20 , @xmath33 the absolute magnitude for a star above the zams is calculated from the value of @xmath34 , i.e. the difference between the measured @xmath24 and the value of @xmath24 on the zams at the given @xmath20 . since there are significant line blanketing effects on @xmath24 for am / fm stars , their absolute magnitudes derived in this way are probably not free of systematic errors . we derived @xmath32 using the data of table[strom ] for stars with solar abundance and with the am abundance using the calibration of @xcite . we find that on average the am stars are estimated to be about 1.2 magnitudes fainter than normal stars of the same effective temperature and gravity . this is due to the systematically lower values of @xmath24 in the am star models . table[groundspec ] lists am stars which have good parallaxes . the general trend of lower luminosities derived from strmgren photometry is apparent . from this exercise we conclude that although the effective temperatures of am / fm stars derived from strmgren photometry are probably reliable , the absolute magnitudes may be systematically too faint . * for example , if we apply the @xcite calibration to hd27411 , assuming no reddening , we obtain @xmath35 or @xmath36 ( using bc = 0.035 derived from @xcite ) , whereas the most reliable estimate ( hipparcos parallax ) gives @xmath37 . * this effect can be seen in fig[hr ] where many of the am stars are below the line defining the zams . the luminosities of these stars were derived using the standard calibration and hence have been under - estimated . ( filled circles ) . the asterisk shows the observed location of hd27411 and the arrows are the reddening lines from @xcite.,width=321 ] rl@ rr@ r@ r & & & & parallax & strmgren + hd & class & @xmath38 & @xmath3 & @xmath39 & @xmath39 + 27411 & a3 m & 3.869 & 4.12 & 1.35 & 0.96 + 27628 & ka5hf0mf2 & 3.864 & 4.00 & 1.04 & 0.82 + 71297 & a5iii - iv & 3.900 & 4.19 & 1.14 & 0.82 + 104513 & a7 m & 3.879 & 4.24 & 0.88 & 0.87 + 204188 & ka6ha9mf0 & 3.898 & 4.34 & 0.89 & 0.74 + the diffusion models of @xcite are the best models presently available for am / fm stars . the models seem to be able to predict the abundances in these stars rather well , but we need to bear in mind that this is achieved because of adjustable free parameters to describe the turbulence . we still do not know if the mechanism competing with diffusion is turbulence , mass loss or some other factor . what we do know is that the current description of am stars is in trouble because it fails to account for the wide distribution of pulsating am stars in the @xmath0 sct instability strip @xcite . one question that is of interest is the fraction of am stars that pulsate . to answer this question we have to define what we mean by `` non - pulsating '' . clearly , a star could be pulsating but with amplitudes too small to be visible from the ground . @xcite discussed this issue in the context of _ kepler _ observations which , of course , allow pulsations to be detected at the micromagnitude level . they deduced that the fraction of pulsating stars in the @xmath0 sct instability strip is surprisingly low . there is clearly some damping mechanism which is currently not understood . the fraction of @xmath0 sct stars in the instability strip varies with effective temperature , but does not exceed about 50 percent . we can answer this question for am stars only in part because we do not have a sufficient number of am stars observed at the micromagnitude level . in order to compare these ground - based observations with the extensive _ kepler _ observations of @xmath0 sct stars , we need to degrade the _ data by considering as pulsating only those @xmath0 sct stars with amplitudes over 1.5 mmag . we chose this minimum amplitude as roughly representative of the detection limit in the catalogue of pulsating am stars in @xcite . the percentage of _ kepler _ @xmath0 sct stars with this minimum amplitude relative to all stars in a particular temperature range is shown in table[dsct ] . lrrr @xmath22 & @xmath40(all ) & @xmath40(@xmath0 sct ) & percent + : + 5500 - 6500 & 1509 & 33 & 2.19 + 6500 - 7000 & 3842 & 174 & 4.53 + 7000 - 7500 & 1412 & 263 & 18.63 + 7500 - 8000 & 811 & 172 & 21.21 + 8000 - 8500 & 512 & 51 & 9.96 + 8500 - 9000 & 297 & 11 & 3.70 + 9000 - 10000 & 263 & 1 & 0.38 + + am stars : + 5500 - 6500 & 16 & 2 & 12.50 + 6500 - 7000 & 75 & 21 & 28.00 + 7000 - 7500 & 307 & 52 & 16.94 + 7500 - 8000 & 489 & 9 & 1.84 + 8000 - 8500 & 185 & 1 & 0.54 + 8500 - 9000 & 32 & 2 & 6.25 + 9000 - 10000 & 7 & 0 & 0.00 + we can now compare this distribution of ordinary @xmath0 sct stars with the distribution of pulsating am stars in @xcite . we used the catalogue of @xcite and estimated the effective temperatures of the am stars using the @xcite calibrations . results are shown in table[dsct ] and fig.[distr ] . it is evident that the pulsating am stars are cooler than normal @xmath0 sct stars , a fact already mentioned by @xcite . this conclusion still holds even if the amplitude threshold in _ kepler _ data is lowered to a few micromagnitudes . scuti stars relative to all stars in the given effective temperature range is shown as the solid line histogram . the dashed line histogram is the percentage of pulsating am stars relative to all am stars in the given temperature range.,width=321 ] from this comparison , we may deduce that there is certainly a tendency for pulsating am stars to be confined more towards the red edge , but that this effect is far smaller than predicted by @xcite . in fact , table[dsct ] shows that the percentage of pulsating am stars among the am stars is about the same as the percentage of @xmath0 scuti stars in the instability strip . this tells us that driving in the heii ionization zone is practically unaffected . we may therefore conclude that there is no significant reduction of he in the ionization zone , contrary to the prediction of current diffusion models . we analyzed the spectrum of the am star hd27411 from the uves pop archive . our aim was to investigate if the atlas12 model atmosphere code provides more reliable results than the atlas9 code for chemically peculiar stars . we found that there is very little difference in the abundances derived from atlas9 and from atlas12 . since atlas12 demands considerably greater resources , it seems safe to use atlas9 , at least for moderate metallic enhancement . we find that the derived abundances in hd27411 are in good agreement with the predictions of diffusion models by @xcite . there were discrepancies for na , al , si , s , and cr , but these are resolved by using nlte model atmospheres . we investigated the reliability of effective temperatures and luminosities of am / fm stars determined by strmgren photometry by synthesizing spectra having the abundances of hd27411 for a range of effective temperatures . the resulting synthetic colours indicate that effective temperatures can be reliably determined from photometry , but owing to line blanketing in the @xmath24 passband , the resulting surface gravities are systematically to high , leading to lower luminosities . this result appears to be verified by comparing luminosities of am / fm stars obtained from their parallaxes and from photometry . * determination of reliable luminosities for am stars remains a difficult problem . at this stage , parallaxes offer the best results , but this can only be done for very few stars . as we have seen , luminosities obtained from strmgren photometry are subject to a systematic bias which depends on the overabundances of metals . the error in the surface gravity from high - resolution spectroscopy is typically 0.1 in @xmath3 . for a f main sequence and giants , this translates into an error of about 0.12 in @xmath41 when using the calibration of @xcite . for hd27411 , for example , we derive @xmath42 from spectroscopy , whereas the value derived from the parallax is @xmath43 ( table2 ) , although the two values only differ by two standard deviations , this is enough to cause a difference of 0.36 in @xmath41 . although spectroscopic determinations of luminosities may lead to quite large errors in the luminosity , they should at least not be biased . * by far the most serious problem confronting the diffusion model is that there seems to be no appreciable settling of he in the heii ionization zone , as predicted by the models . this is demonstrated by the fact that pulsating am / fm stars occur throughout the @xmath0 scuti instability strip , though they tend to be cooler than normal @xmath0 sct stars . in fact , the relative proportions of pulsating am stars to non - pulsating am stars is no different from the proportion of @xmath0 sct stars to constant stars in the @xmath0 sct instability strip . there is clearly a need to revise current ideas of diffusion to explain the am phenomenon . lab wishes to thank the national research foundation and the south african astronomical observatory for financial assistance . kurucz r. l. , 1997 , model atmospheres for individual stars with arbitrary abundances . in : the third conference on faint blue stars , a. g. d. philip , j. liebert , r. saffer and d. s. hayes ( eds . ) , published by l. davis press , p.33 kurucz r.l . , 1993 , a new opacity - sampling model atmosphere program for arbitrary abundances . in : peculiar versus normal phenomena in a - type and related stars , iau colloquium 138 , m.m . dworetsky , f. castelli , r. faraggiana ( eds . ) , a.s.p conferences series vol .	we analyze a high - resolution spectrum of the a3 m star hd27411 . we compare abundances derived from atlas9 model atmospheres with those using the more computationally - intensive atlas12 code . we found very little differences in the abundances , suggesting that atlas9 can be used for moderate chemical peculiarity . our abundances agree well with the predictions of diffusion theory , though for some elements it was necessary to calculate line profiles in non - thermodynamic equilibrium to obtain agreement . we investigate the effective temperatures and luminosities of am / fm stars using synthetic strmgren indices derived from calculated spectra with the atmospheric abundances of hd27411 . we find that the effective temperatures of am / fm stars derived from strmgren photometry are reliable , but the luminosities are probably too low . caution is required when deriving the reddening of these stars owing to line blanketing effects . a comparison of the relative proportions of pulsating and non - pulsating am stars with @xmath0 scuti stars shows quite clearly that there is no significant decrease of helium in the driving zone , contrary to current models of diffusion . [ firstpage ] stars : chemically peculiar stars : individual : hd27411 stars : abundances	8,313	315
fe - based superconductors ( febs ) represent a non - cuprate class of high-@xmath9 systems with the unconventional superconducting state . the origin of the latter is still debated . in general , febs can be divided into the two subclasses , pnictides and chalcogenides @xcite , with the square lattice of iron as the basic element , though with orthorhombic distortions in lightly doped materials . iron is surrounded by as or p situated in the tetrahedral positions within the first subclass and by se , te , or s within the second subclass . fermi surface ( fs ) is formed by fe @xmath10-orbitals and excluding the cases of extreme hole and electron dopings it consists of two hole sheets around the @xmath11 point and two electron sheets around the @xmath12 and @xmath13 points in the two - dimensional brillouin zone ( bz ) corresponding to one fe per unit cell ( the so - called 1-fe bz ) @xcite . in the 2-fe bz , electron pockets are centered at the @xmath14 point . nesting between these two groups of sheets leads to the enhanced antiferromagnetic fluctuations with the maximal scattering near the wave vector @xmath2 equal to @xmath12 or @xmath13 in the 1-fe bz or to @xmath15 in the 2-fe bz . different mechanisms of cooper pairs formation result in the distinct superconducting gap symmetry and structure @xcite . in particular , the rpa - sf ( random - phase approximation spin fluctuation ) approach gives the extended @xmath16-wave gap that changes sign between hole and electron fs sheets ( @xmath17 state ) as the main instability for the wide range of doping concentrations @xcite . on the other hand , orbital fluctuations promote the order parameter to have the sign - preserving @xmath6 symmetry @xcite . thus , probing the gap structure can help in elucidating the underlying mechanism . in this respect , inelastic neutron scattering ( ins ) is a useful tool since the measured dynamical spin susceptibility @xmath18 in the superconducting state carries information about the gap structure . there are been many reports of a well - defined peak in neutron spectra in 1111 , 122 , and 11 systems appearing only for @xmath19 at or around @xmath20 @xcite . the common explanation is that the peak is the spin resonance appearing due to the @xmath7 state . indeed , since @xmath2 connects fermi sheets with different signs of @xmath7 gaps , the resonance condition for the interband susceptibility is fulfilled and the spin resonance peak is formed at a frequency @xmath21 below @xmath22 with @xmath23 being the gap size @xcite . such simple explanation was indirectly questioned by the angle - resolved photoemission spectroscopy ( arpes ) results and recent measurements of gaps via andreev spectroscopy . latter clearly shows that there are at least two distinct gaps present in 11 , 122 , and 1111 systems @xcite and even three gaps in lifeas @xcite . larger gap ( @xmath0 ) is about 9mev and the smaller gap ( @xmath1 ) is about 4mev in baco122 materials . from arpes we know that electron fs sheets and the inner hole sheet are subject to opening the lager gap while the smaller gap is located at the outer hole fs @xcite . the very existence of the smaller gap rise the question what would be the spin resonance frequency in the system with two distinct gaps ? naive expectation is that the frequency shifts to the lower gap scale and @xmath24 . then the observed peak in ins in baco122 system at frequency @xmath25mev @xcite can not be the spin resonance since it is greater than @xmath26mev @xcite . thus the peak could be coming from the @xmath6 state @xcite , where it forms at frequencies _ above _ @xmath27 due to the redistribution of the spectral weight upon entering the superconducting state and a special form of scattering in the normal state . here we study this question in details and show that the naive expectation is wrong and that the true minimal energy scale is @xmath28 . latter is consistent with the maximal frequency of the observed peak in ins in baco122 and confirms that it is the true spin resonance evidencing the @xmath7 gap symmetry . the maximal energy scale is @xmath29 . whether the minimal or maximal energy scale will be realized depends on the relation between the exact band structure of a particular material and the wave vector of the spin resonance @xmath2 . to describe spin response in normal and superconducting states of febs , we use random phase approximation ( rpa ) with the local coulomb interactions ( hubbard and hund s exchange ) . in the multiband system , transverse dynamical spin susceptibility @xmath30 is the matrix in orbital ( or band ) indices . it can be obtained in the rpa from the bare electron - hole matrix bubble @xmath31 by summing up a series of ladder diagrams : @xmath32^{-1 } \hat\chi_{(0)+-}(\q,\omega ) , \label{eq : chi_s_sol}\end{aligned}\ ] ] where @xmath33 is the momentum , @xmath34 is the frequency , @xmath35 and @xmath36 are interaction and unit matrices in orbital ( or band ) space . exact form of @xmath35 and bare susceptibility @xmath31 depends on the underlying model . later we use two types of tight - binding models for the two - dimensional iron layer . first we study the four - band model of ref . with the following single - electron hamiltonian @xmath37 where @xmath38 is the annihilation operator of the @xmath10-electron with momentum @xmath39 , spin @xmath40 , and band index @xmath41 , @xmath42 are the on - site single - electron energies , @xmath43 is the electronic dispersion that yields hole pockets centered around the @xmath44 point , and @xmath45 is the dispersion that results in the electron pockets around the @xmath46 point of the 2-febz . parameters are the same as in ref . . in the superconducting state we assume either the @xmath6 state with @xmath47 or the @xmath7 state with @xmath48 . physical spin susceptibility @xmath49 obtained by calculating matrix elements @xmath50 via equation ( [ eq : chi_s_sol ] ) with the interaction matrix @xmath51 and with the bare spin susceptibility @xmath52 in the superconducting state ( see ref . for details ) . we assume here the effective hubbard interaction parameters to be @xmath53 and @xmath54 in order to stay in the paramagnetic phase @xcite . the model described above is simple enough to gain qualitative description of the spin response of superconductor with unequal gaps . but it lack for the orbital content of the bands that is important for the detailed structure of the susceptibility . that is why we also present results for the tight - binding model from ref . based on the fit to the dft band structure for lafeaso @xcite . the model includes all five iron @xmath10-orbitals ( @xmath55 , @xmath56 , @xmath57 , @xmath58 , @xmath59 ) enumerated by index @xmath60 and is given by @xmath61 d_{l \k \sigma}^\dagger d_{l ' \k \sigma } , \label{eq : h0}\ ] ] where @xmath62 is the annihilation operator of a particle with momentum @xmath39 , spin @xmath40 , and orbital index @xmath60 . later we use numerical values of hopping matrix elements @xmath63 and one - electron energies @xmath64 from ref . . this model for the undoped and moderately electron doped materials gives fs composed of two hole pockets , @xmath65 and @xmath66 , around the @xmath67 point and two electron pockets , @xmath68 and @xmath69 , centered around @xmath12 and @xmath13 points of the 1-fe bz . similar model for iron pnictides was proposed in ref . . the general two - particle on - site interaction would be represented by the hamiltonian @xcite : h_int & = & u _ f , m n_f m n_f m + u _ f , m < l n_f l n_f m + & & + j _ f , m < l _ , d_f l ^d_f m ^d_f l d_f m + & & + j _ f , m l d_f l ^d_f l ^d_f m d_f m . [ eq : hint ] where @xmath70 , @xmath71 is the number of particles operator at the site @xmath72 , @xmath73 and @xmath74 are the intra- and interorbital hubbard repulsion , @xmath75 is the hund s exchange , and @xmath76 is the so - called pair hopping . we choose the following values for the interaction parameters : @xmath77ev , @xmath78 , and make use of the spin - rotational invariance constraint @xmath79 and @xmath80 . green functions are diagonal in the band basis but not in the orbital basis . let us introduce creation and annihilation operators @xmath81 and @xmath82 of electrons with band index @xmath83 , in terms of which green functions are diagonal , @xmath84 . transformation from the orbital to the band basis is done via the matrix elements @xmath85 , @xmath86 , and for the transverse component of the bare spin susceptibility @xcite we have [ eq : chipmmu ] & & ^ll,mm_(0)+- ( , ) = -t _ , _ n , , , where @xmath87 and @xmath88 are matsubara frequencies , @xmath89 and @xmath90 are the normal and anomalous ( superconducting ) green s functions , respectively . components of the physical spin susceptibility @xmath91 are calculated using eq . ( [ eq : chi_s_sol ] ) with the interaction matrix @xmath92 from ref . . since calculation of the cooper pairing instability is not a topic of the present study , here we assume that the superconductivity is coming from some other theory and study either the @xmath6 state with @xmath93 or the @xmath7 state with @xmath94 , where @xmath83 is the band index . here we present results for susceptibilities at the wave vector @xmath95 as functions of frequency @xmath34 obtained via analytical continuation from matsubara frequencies ( @xmath96 with @xmath97 ) . imaginary part of bare and rpa spin susceptibilities in the four - band model ( [ eq : h04band ] ) are shown in fig . [ fig : imchi4band ] . first , we discuss result for equal gaps on electron ( @xmath98 , @xmath99 ) and hole ( @xmath100 , @xmath101 ) fss , @xmath102 . since @xmath103 describes particle - hole excitations and in the superconducting state all excitations are gapped below approximately @xmath104 ( at @xmath105 ) , then @xmath106 becomes finite only after that frequency . for the @xmath6 state , there is a gradual increase of the spin response for @xmath107 . for the @xmath7 state , @xmath2 connects fss with different signs of gaps , @xmath108 , and within rpa ( [ eq : chi_s_sol ] ) this results in the spin resonance peak divergence of @xmath109 at a frequency @xmath110 , see fig . [ fig : imchi4band ] , bottom panel . ( top ) and @xmath109 with @xmath111 in the 2-fe bz for the four - band model in the normal , @xmath6 and @xmath7 superconducting states . two cases of superconducting states are shown : equal @xmath23 s with @xmath102 , and unequal gaps with @xmath112 and @xmath113 . [ fig : imchi4band],title="fig : " ] ( top ) and @xmath109 with @xmath111 in the 2-fe bz for the four - band model in the normal , @xmath6 and @xmath7 superconducting states . two cases of superconducting states are shown : equal @xmath23 s with @xmath102 , and unequal gaps with @xmath112 and @xmath113 . [ fig : imchi4band],title="fig : " ] now let s consider the case of unequal gaps with a small gap scale on outer hole fs , @xmath113 , and a larger gap scale on all other fss , @xmath112 . as seen from fig . [ fig : imchi4band ] , top panel , for the @xmath7 state the discontinuous jump and , thus , @xmath114 , moved to lower frequencies . this new energy scale clearly tracked down in the @xmath6 state as the starting point of the susceptibility gradual increase . it is equal to @xmath115 , where @xmath0 and @xmath1 being the larger and smaller gap scales . consequently , the spin resonance peak in @xmath7 moved to lower frequencies , @xmath116 , see fig . [ fig : imchi4band ] , bottom panel . additional feature is the hump around the @xmath117 energy scale . note that the susceptibility in the @xmath6 state havent changed much compared to the equal gaps case . in the superconducting state , @xmath118 , as a function of momentum @xmath39 along the @xmath119 direction , i.e. @xmath120 . scattering wave vector @xmath2 entering the spin susceptibility is also shown . [ fig:5orbdelta ] ] with @xmath121 in the 1-fe bz for the five - orbital model in the normal , @xmath6 and @xmath7 superconducting states . two cases of superconducting states are shown : equal gaps with @xmath122 , and unequal gaps with @xmath123 and @xmath124 , where @xmath125 . latter case is shown in the inset , where gaps at the fs are plotted together with the wave vector @xmath2 . [ fig:5orbimchi ] ] to demonstrate where the new energy scale is coming from we turn our attention to the five - orbital model ( [ eq : h0 ] ) . its energy spectrum near the fermi level in the superconducting state , @xmath118 , is shown in fig . [ fig:5orbdelta ] . we consider here the case of unequal gaps with the smaller gap @xmath124 on the outer hole fs and larger gaps @xmath123 on inner hole and electron fss . to be consistent with the experimental data , we choose @xmath126 , see the inset in fig . [ fig:5orbimchi ] . naturally , the two energy scales , @xmath127 and @xmath128 , appear in the energy spectrum @xmath129 and they are connected with hole @xmath66 and electron @xmath130 bands , respectively . on the other hand , the susceptibility @xmath131 contains scattering _ between _ hole and electron bands with the wave vector @xmath2 . the energy gap that have to be overcome to excite electron - hole pair is the indirect gap with the scale @xmath4 . that is why spin excitations in the @xmath6 state start with the frequency proportional to the indirect gap @xmath132 , see fig . [ fig:5orbimchi ] . the same is true for the discontinuous jump in @xmath133 for the @xmath7 state it shifts to frequency @xmath134 . this , together with the corresponding @xmath135 singularity in @xmath136 , produce the spin resonance peak in rpa at frequency @xmath8 . such shift of resonance peak to lower frequencies compared to the equal gaps situation is seen in fig . [ fig:5orbimchi ] , where the spin response @xmath109 for the cases of equal and distinct gaps is shown . the changes in the band structure and/or doping level can result in the change of the indirect gap . in particular , since for the hole doping hole fss become larger the wave vector @xmath2 may connect states on the electron fs and on the _ inner _ hole fs . gaps on both these fss are determined by @xmath0 and thus the indirect gap will be equal to @xmath5 . this sets up a maximal energy scale for the spin resonance , i.e. @xmath29 . thus we conclude that depending on the relation between the wave vector @xmath2 and the exact fs geometry , the indirect gap in most febs can be either @xmath4 or @xmath5 . the peak in the dynamical spin susceptibility at the wave vector @xmath2 will be the true spin resonance if it appears below the indirect gap scale , @xmath8 . now we can compare energy scales extracted from arpes , andreev spectroscopy , and inelastic neutron scattering . latter gives peak frequency @xmath137mev in bafe@xmath138co@xmath139as@xmath140 with @xmath141k @xcite . for the same system , gap sizes extracted from arpes are @xmath142mev and @xmath143mev @xcite , and for a similar system with @xmath144k , @xmath145mev and @xmath146mev @xcite . gap sizes extracted from andreev spectroscopy are @xmath147mev and @xmath148mev in bafe@xmath149co@xmath150as@xmath140 with @xmath151k @xcite . evidently , @xmath152 and we can safely state that the peak in ins is the spin resonance . for the hole doped systems , peak frequency in ins is about @xmath153mev in ba@xmath154k@xmath155fe@xmath140as@xmath140 with @xmath156k @xcite . there is a slight discrepancy between gap sizes extracted from arpes and andreev spectra . former gives @xmath157mev and @xmath158mev in the same material with @xmath159k @xcite , thus @xmath152 . gap sizes from andereev spectroscopy are @xmath160mev and @xmath161mev in ba@xmath162k@xmath163fe@xmath140as@xmath140 with lower @xmath164k @xcite . in this case , @xmath165 but @xmath166 and we still can assume that the peak in ins is the spin resonance . however , in such a case definitive conclusion can be given only by the calculation of spin response for the particular experimental band structure . for more extensive review of available experimental data on @xmath167 and gap scales , see the supplemental material @xcite . on the separate note , we would like to mention that the appearance of a hump structure in the superconducting state at frequencies larger than the main peak frequency ( the so - called double resonance feature ) may be related to the @xmath128 energy scale , see fig . [ fig : imchi4band ] . such hump structure was observed in nafe@xmath168co@xmath169as @xcite and fete@xmath170se@xmath170 @xcite . somehow similar structure was found in polarized inelastic neutron studies of bafe@xmath171ni@xmath172as@xmath140 @xcite and ba(fe@xmath173co@xmath174)@xmath140as@xmath140 @xcite , but its origin may be related to the spin - orbit coupling @xcite rather than the simple @xmath128 energy scale . another explanation of the double resonance feature is related to the pre - existing magnon mode , i.e. the dispersive low - energy peak in underdoped materials is associated with the spin excitations of the magnetic order with the intensity enhanced below @xmath9 due to the suppression of the damping @xcite . we analysed the spin response of febs with two different superconducting gap scales , @xmath175 . spin resonance appears in the @xmath7 state below the indirect gap scale @xmath3 that is determined by the sum of gaps on two different fermi surface sheets connected by the scattering wave vector @xmath2 . in the @xmath6 state , spin excitations are absent below @xmath3 unless additional scattering mechanisms are assumed @xcite . for the fermi surface geometry characteristic to the most of febs materials , the indirect gap is either @xmath4 or @xmath5 . this gives the simple criterion to determine whether the experimentally observed peak in inelastic neutron scattering is the true spin resonance if the peak frequency @xmath21 is less than the indirect gap @xmath3 , then it is the spin resonance and , consequently , the superconducting state has the @xmath7 gap structure . comparison of energy scales extracted from ins , andreev spectroscopy , arpes and other techniques allowing to determine superconducting gaps , for most materials gives confidence that the observed feature in ins is the spin resonance peak . however , sometimes it is not always clear experimentally which gaps are connected by the wave vector @xmath2 . even without knowing this exactly , one can draw some conclusions . for example , if one of the gaps is @xmath0 , then there are three cases possible : ( 1 ) @xmath176 and the peak at @xmath21 is the spin resonance , ( 2 ) @xmath177 and the peak is definitely not a spin resonance , and ( 3 ) @xmath29 and the peak is most likely the spin resonance but the definitive conclusion can be drawn only from the calculation of the dynamical spin susceptibility for the particular experimental band structure . we would like to thank h. kontani , s.a . kuzmichev , t.e . kuzmicheva , v.m . pudalov , and i.s . sandalov for useful discussions . mmk is grateful to b. keimer and max - 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doped bafe@xmath178co@xmath179as@xmath140 system , nafe@xmath178co@xmath179as system , and fese , @xmath116 and , thus the peak in ins is the true spin resonance evidencing sign - changing gap . some hole doped ba@xmath178k@xmath179fe@xmath140as@xmath140 materials satisfy @xmath176 condition , and some satisfy @xmath180 condition . latter comes especially from newer tunneling @xcite and andreev reflection @xcite data reveling smaller gap values . the fact that @xmath180 is still consistent with the sign - changing gap , but as we mentioned before , the calculation of the spin response for the particular experimental band structure is required to make a final conclusion . 3 . the only case when @xmath181 is fete@xmath170se@xmath170 . according to our analysis , there should be no sign - changing gap structure . but before concluding this since this is the single case only , gap data coming from @xmath83sr @xcite should be double checked by independent techniques . interesting to note , that arpes in all cases gives gaps values larger than extracted from other techniques . natural question arise whether the arpes overestimates or all other methods underestimates superconducting gaps ? m. wang , m. yi , h.l . sun , p. valdivia , m.g . kim , z.j . xu , t. berlijn , a.d . christianson , s. chi , m. hashimoto , d.h . li , e. bourret - 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based materials within multiband models with two unequal gaps , @xmath0 and @xmath1 , on different fermi surface pockets . we show that due to the indirect nature of the gap entering the spin susceptibility at the nesting wave vector @xmath2 the total gap @xmath3 in the bare susceptibility is determined by the sum of gaps on two different fermi surface sheets connected by @xmath2 . for the fermi surface geometry characteristic to the most of iron pnictides and chalcogenides , the indirect gap is either @xmath4 or @xmath5 . in the @xmath6 state , spin excitations below @xmath3 are absent unless additional scattering mechanisms are assumed . the spin resonance appears in the @xmath7 superconducting state at frequency @xmath8 . comparison with available inelastic neutron scattering data confirms that what is seen is the true spin resonance and not a peak inherent to the @xmath6 state .	13,959	256
the presence of impurities usually deeply modify the nature of the spectrum of a quantum system , and thus its coherence and transport properties . in the absence of interactions , if the impurity distribution is completely random , all states of the spectrum are exponentially localized in dimensions one ( 1d ) and two ( 2d ) , while a mobility edge exists in dimensions three ( 3d)@xcite . if the impurity positions are correlated , as for instance if it exists a minimum distance between the impurities @xcite , some delocalized states can appear in the spectrum . this was demonstrated in 1d in the context of the random dimer model ( rdm ) and of the dual random dimer model ( drdm ) @xcite . in 1d , the effects of correlated disorder was studied in different physical contexts ( see for instance @xcite ) . in 2d , the effect of correlations is almost unexplored , except for the case of a speckle potential @xcite , and for the case of pseudo-2d random dimer lattices with separable dimensions @xcite . correlations in speckle potentials may mimic the presence of a mobility edge @xcite , but in the thermodynamic limit all states are localized @xcite . random dimers introduce a set of delocalized states in pseudo-2d lattices @xcite as in 1d @xcite . from a statistical point of view , the main difference between these two models is the decay of the correlation function that is algebraic for the first and exponential for the second . this `` short - range '' feature of the random dimer model is at the basis of the delocalization mechanism . in interacting systems , the presence of disordered impurities gives rise to a remarkable richness of phenomena . for instance , the condensate and the superfluid fraction are modified by the presence of the disorder @xcite , and this can shift the onset of superfluidity @xcite , and , on lattice systems , can induce exotic phases such as the bose glass @xcite . in this work we study the effect of a short - range correlated disorder on a bose gas confined on a 2d square lattice . first we introduce a 2d generalization of the drdm ( 2d - drdm ) . in such a model , impurities can not be first neighbours and each impurity also modifies the hopping with its nearest neighbor sites . using a decimation and renormalization procedure @xcite , we show that , in the non - interacting regime , it exists a resonance energy at which the structured impurity is transparent and the states around this energy are delocalized . it is remarkable that this resonance energy does not depend on the system dimensionality and it is the same as the drdm in 1d @xcite . then , we consider the case of a weakly interacting bose gas confined on such a potential . within a gutzwiller approach , we show that the effect of the 2d - drdm is to drive the homogeneity of the ground state . the disorder induces a non - monotonic behavior of the condensate spatial delocalization and of the condensate fraction as a function of the disorder strength , and enhances both in correspondence of the resonance energy of 2d - drdm single - particle hamiltonian . we show that the dependence of such quantities on the interaction strength can be explained by including the effect of the healing length in the resonance condition discussion . the manuscript is organized as follows . in sec . [ sec : model ] , we introduce the 2d - drdm potential and we demonstrate its single - particle delocalization properties in the region of the spectrum around the resonance energy . the effect of such a potential on a weakly - interacting bose gas is studied in sec . [ sec : results ] , where we also introduce a suitable inverse participation ratio for our many - body system and study it for the case of the 2d - drdm potential and for an uncorrelated random disorder . moreover , we compute the density distribution and the condensate fraction as functions of the disorder strength . our concluding remarks in sec . [ sec : concl ] complete this work . we consider the tight - binding single - particle hamiltonian @xmath0 where @xmath1 are the on - site energies , @xmath2 the first neighbor hopping terms , @xmath3 the number of sites and @xmath4 denotes the sum over first neighbor sites . we focus on a 2d square lattice of linear dimension @xmath5 ( @xmath6 lattice sites ) , and compare the ordered lattice with @xmath7 and @xmath8 @xmath9 @xmath10 , as schematized in fig . [ fig1](a ) with a lattice where we introduce an impurity at the site 0 , @xmath11 that modifies the hopping terms involving this site , @xmath12 [ fig . [ fig1](b ) ] . schematic representation of ( a ) the unperturbed hamiltonian ; ( b ) the hamiltonian in the presence of a single impurity ; ( c ) the effective hamiltonian after decimation of the site 0 in the hamiltonian ( a ) ; ( d ) the effective hamiltonian after decimation of the site 0 in the hamiltonian ( b ) . ] with the aim of understanding the effect of the impurity , we consider the green s function @xmath13 projected on the subspace @xmath14 , including all sites except the site @xmath15 with coordinates @xmath16 . using a decimation and renormalization technique @xcite , it can be shown that @xmath17 with @xmath18 & \phantom{bla}{\rm site\;of\;the\;site\;}0\\[1 mm ] h_{aa}&\phantom{bla}{\rm elsewhere } \end{array}\right.\ ] ] where @xmath19 . the effective hamiltonian for the unperturbed case in fig . [ fig1 ] ( a ) , is schematically illustrated in fig . [ fig1 ] ( c ) ; whereas the effective hamiltonian for the case with a single impurity in fig . [ fig1 ] ( b ) , is illustrated in fig.[fig1 ] ( d ) . the subspace @xmath14 does not `` feel '' the presence of the impurity if @xmath20 ( @xmath21 ) remains the same in the absence or in the presence of the impurity , namely if @xmath22 the condition ( [ condition ] ) is satisfied if @xmath23 . if @xmath24 is an allowed energy of the system , namely if @xmath25 , at @xmath26 the impurity will not affect the eigenstate at this energy ( in the subspace @xmath14 ) . if we add other impurities in the system , as the one in fig . [ fig1 ] ( b ) , with the supplementary condition that on - site impurities can not occupy first neighbor sites ( fig . [ fig2 ] ) , one can repeat the same argument as above , properly redefining the subspace @xmath14 , and one obtains exactly the same condition ( [ condition ] ) imposing that _ all _ the @xmath27 impurities do not perturb the system ( the subspace @xmath14 ) . thus at @xmath26 , the impurities are transparent as in the 1d drdm @xcite . indeed , with this procedure , we are defining a 2d - drdm , where at each `` isolated '' impurity correspond a structure of 4 hopping terms forming a cross , as shown in fig . let us remark that this definition of the model provides the same condition ( [ condition ] ) independently from the dimensionality of the system @xcite . however our model is fully 2d and the hamiltonian can not be mapped onto two 1d drdm as opposed to ref . @xcite . schematic representation of the 2d drdm . ] with the aim of analyzing the localization properties of this model , we consider the inverse participation ratio ( ipr ) @xmath28 the symbol @xmath29 denotes the average over different disorder configurations , and @xmath30 the wavefunction on site @xmath31 and at energy @xmath32 . if @xmath33 is an eigenvalue of the system and @xmath34 is an extended state , then @xmath35 decreases as a function of @xmath5 . on the other side , if @xmath36 is a localized state , then @xmath35 does not depend on @xmath5 ( if @xmath5 is larger than the localization length ) . in fig . [ fig3 ] we show the behavior of @xmath37 and @xmath38 ( column left and right respectively ) , for the hamiltonian illustrated in fig . [ fig2 ] . [ cols="^,^ " , ] we consider three set of parameters , ( i ) @xmath39 and @xmath40 , ( ii ) @xmath41 and @xmath42 , and ( iii ) @xmath43 and @xmath44 that give the same resonance energy , @xmath45 . in all the three cases , the curves @xmath46 collapse around @xmath26 meaning that the states are delocalized in this energy region . moreover , due to the large strength of the disorder , the spectrum varies considerably for the cases ( ii ) and ( iii ) , and an energy gap appears in ( iii ) . the inverse participation ratio , eq . ( [ eq : ipr ] ) , in two dimensions has the following asymptotic behavior @xcite @xmath47 thus , the asymptotic behavior of the function @xmath48 is @xmath49 with @xmath50 for localized states , and @xmath51 for extended states . in fig . [ fig : esponenti ] we have analyzed the exponent @xmath52 as a function of the energy for the set of parameters ( iii ) . we observe a high - energy band of localized states that has been created by the disorder ; the original ( without noise ) band has been distorted , and the states at its boundaries are localized . the center of the band , around @xmath24 , is mainly constituted of extended states . the width of the feature around @xmath24 corresponds to the width of the resonance dip of the inverse participation ratio at this energy value ( fig . [ fig3 ] ) . ) as a function of the energy for @xmath43 and @xmath44 . the exponent has been obtained using calculations for lattice - linear dimensions @xmath53 averaged over 20 realizations , the error bars correspond to the standard deviation of the fit of the data to eq . ( [ eq : asympatty ] ) . the vertical dashed line indicates @xmath24 . ] these results confirm that our 2d extension of the drdm introduced by dunlap and collaborators in ref . @xcite for 1d systems introduces a set of delocalized states even at higher dimensions . we now consider the case of weakly interacting bosons in the presence of the potential defined in sec . [ sec : model ] . this system is described by the bose - hubbard hamiltonian in the grand canonical ensemble @xmath54 where @xmath55 is the creation operator defined at the lattice site @xmath31 , @xmath56 , @xmath57 the interparticle on - site interaction strength , and @xmath58 denotes the chemical potential fixing the average number of bosons . we use a gutzwiller approach to find the ground state wavefunction for a given set of parameters and average number of particles . the gutzwiller ansatz is given by the site product wavefunction in the occupation number representation @xmath59 where @xmath60 are the probability amplitudes of finding @xmath61 particles on site @xmath31 . the ansatz provides an interpolating approximation correctly describing both the bose - condensed and mott - insulating phases for low and high @xmath57 , respectively , in dimensions larger than one . in addition , the approximation becomes exact for all @xmath57 in the limit of infinite dimensions @xcite . we minimize the average energy given by hamiltonian ( [ eq : bhgw ] ) as a function of the set of amplitudes @xmath60 with the normalization and average number of particle constraint for at least 30 disorder realizations for each set of parameters . the minimization is done using standard conjugate - gradient and/or broyden - fisher techniques @xcite which provides reasonable performance for moderate lattice sizes . to quantify the extent of delocalization of the ground state @xmath62 in the interacting regime , we decompose it onto the localized basis @xmath63 , @xmath64 , representing the distribution of a homogeneous condensate with average density @xmath65 on the lattice @xcite . we define the many - body ground - state ipr @xmath66 with respect to this basis as @xmath67 @xmath66 measures the homogeneity of the ground state in the condensation regime : the smaller @xmath66 the more spatially delocalized is the condensate . in fig . [ fig : igs_l20 ] we show the behavior of @xmath66 as a function of @xmath68 , by fixing @xmath69 , @xmath70 and @xmath71 , for several values of @xmath72 . ( color online ) @xmath73 as a function of @xmath74 for @xmath69 , @xmath70 and @xmath71 particles per site . the different curves correspond to different values of @xmath72 as indicated in the figure . the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] we compare the case of @xmath75 of correlated impurities @xmath27 with the one of the same percentage of uncorrelated impurities , where there is no restriction for the position distribution of the on - site impurities @xmath68 and no correlations between them and the additional hopping @xmath72 ( un - rand ) . we note that due to the correlations present in the 2d - drdm the maximum percentage of allowed impurities is @xmath76 ( in this limit the system would be an ordered checkerboard ) . we can observe that , in the case of the 2d - drdm potential , @xmath66 has a minimum as a function of @xmath68 , whose position depends on the value of @xmath72 . this non - monotonic behavior is a signature of the resonance induced by the correlations of the disordered potential . indeed , it disappears for the case of the un - rand potential and for large values of @xmath72 ( strong disorder ) . the dip in the @xmath66 for the un - rand potential and weak disorder ( @xmath40 ) indicates that some drdm impurities may still statistically appear , in the absence of correlations . the effect of such impurities is not fully destroyed by the other defects only if the strength of the disorder is weak . in the perturbative regime for negligible interactions , one would expect that correlations modify the ground state if @xmath77 , @xmath78 being the ground - state energy per particle , which corresponds to @xmath79 in the weak disorder regime . this condition , that can be written @xmath80 , \label{cond - pat}\ ] ] determines the location of the minimum of @xmath66 at @xmath81 for @xmath40 , @xmath82 for @xmath42 , and @xmath83 for @xmath44 . however , in the limit of strong disorder , due to the interactions these values strongly differ from those shown in fig . [ fig : igs_l20 ] . in fact , we calculate @xmath66 for smaller values of @xmath65 and verify that the minimum location of @xmath66 depends on @xmath24 and that the shift observed is indeed an effect of the interactions . the results are illustrated in fig . [ fig : igs_l20_n ] , where we focus on the case @xmath42 . ( color online ) @xmath73 as a function of @xmath74 for @xmath69 , @xmath84 and @xmath42 . the different curves correspond to different values of the average density @xmath65 as indicated in the figure . all the curves correspond to the 2d - drdm potential . the vertical dashed line indicates the non - interacting resonance condition given in eq . ( [ cond - pat ] ) . ] by decreasing the value of @xmath65 , the minimum position @xmath85 of @xmath66 shifts from @xmath86 to about @xmath87 as expected by the perturbative argument . this shift can be understood as follows . the interactions introduce the so - called healing length @xmath88 @xcite that represents a coherence length over which the system feels the effect of an impurity , or in other words , the distance a site affects its neighborhood . for @xmath70 and @xmath65 from 20 to 5 , the value of @xmath89 ranges approximately from @xmath90 to 3 lattice spacing @xmath91 , which shows that , already for this @xmath57 value , the role of the interactions is important , effectively reducing the coherence length . to quantify this effect , we can partition the system into independent boxes of dimension @xmath92 , and use a mode - matching argument to determine their ground states : the condensate is more homogeneous if the lowest eigenvalue of each box is the same despite the presence of an impurity . ( color online)boxes of different sizes , in the presence and in the absence of an impurity . ] therefore , this mode - matching argument fixes the value of @xmath68 . for the case @xmath70 and @xmath71 , @xmath93 , and this gives @xmath94 while for @xmath95 , @xmath96 , and we expect to find @xmath97 , in good agreement with the results showed in fig . [ fig : igs_l20_n ] . namely , the larger is @xmath89 , the better we recover the non - interacting condition eq . ( [ cond - pat ] ) . this effect is summarized in table [ tab : uloco ] . ccc @xmath98 & fig . & @xmath68 + + @xmath91 & [ figboxes](a)&@xmath99 $ ] + @xmath100 & [ figboxes](b ) & @xmath101 $ ] + @xmath102 & [ figboxes](c ) & @xmath103 $ ] + we remark that this mode - matching condition is equivalent to match the resonance energy @xmath24 with the lowest eigenvalue of the unperturbed system of size @xmath92 . these simple arguments , allow us to understand the shift of @xmath68 as a function of the interaction energy @xmath104 and the role of the structured impurities in the presence of the interactions . we study the scaling behavior of @xmath73 with respect to @xmath5 . analogously to the case of the single - particle ipr @xmath105 [ see eq . ( [ eq : asympatty ] ) ] , we expect that @xmath106 with @xmath50 for a condensate localized on few sites , and @xmath51 for a homogeneous extended condensate . the behavior of @xmath73 for different values of @xmath5 is shown in fig . [ fig : igs_l ] . ( color online ) @xmath73 as a function of @xmath74 for @xmath42 , @xmath70 and @xmath71 particles per site . the different curves correspond to different values of @xmath5 as indicated in the figure . the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] we observe that the minima , corresponding to different system sizes , collapse all together , meaning that the ground state corresponds to a spatial homogeneous condensate in the parameter regime where the correlations are dominant . at lower values of @xmath68 , @xmath73 scales as @xmath107 , and larger values of @xmath68 , @xmath73 scales as @xmath108 , with @xmath109 and @xmath110 @xmath111 . this sort of `` super - delocalization '' , in the low @xmath68 region , is determined by the large value of @xmath72 that compensates , in the structured impurities , the effect of the site defect . indeed , we observe an analogous behavior for the un - rand potential . for such a potential , where the effect of @xmath72 is no more dominant , all the curves collapse together . thus we expect that in this region the effect of the uncorrelated impurities on the ground state density distribution does not depend on the system size . with the aim of characterizing the ground state configurations in the different regions , we show in figs . [ fig : denlatt2.1][fig : denlatt15.1 ] the spatial density distribution @xmath61 for @xmath69 , @xmath71 at @xmath112 ( fig . [ fig : denlatt2.1 ] ) , @xmath113 ( fig . [ fig : denlatt6.6 ] ) , and @xmath114 ( fig . [ fig : denlatt15.1 ] ) together with a pattern showing the locations of impurities . ( color online ) lattice density plots together with site and bond impurities locations for @xmath42 , @xmath115 and drdm disorder ( top ) and un - rand ( bottom).,title="fig : " ] + ( color online ) lattice density plots together with site and bond impurities locations for @xmath42 , @xmath115 and drdm disorder ( top ) and un - rand ( bottom).,title="fig : " ] ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath116.,title="fig : " ] + ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath116.,title="fig : " ] ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath117.,title="fig : " ] + ( color online ) same as fig . [ fig : denlatt6.6 ] for @xmath117.,title="fig : " ] the addition of a hopping term @xmath72 favors the delocalization of the density both for the 2d - drdm and un - rand disorders . however , in the case of the 2d - drdm , it is more beneficial as it tends to partially compensate the decrease in the density caused by the site impurity , reducing the decrease by means of the structured disorder . for small values of @xmath68 ( see fig . [ fig : denlatt2.1 ] ) , in the region where the effect of @xmath72 is dominant , the density in the impurity regions is even larger with respect to the density elsewhere . for large values of @xmath68 ( see fig . [ fig : denlatt15.1 ] ) , the effect of both types of disorder is similar as the change in the on - site energies dominates . this limit gives rise to a strongly depleted density at the impurity location plus a rather uniform background . the largest differences among the 2d - drdm and un - rand results are seen at the minimum of @xmath66 ( see fig.[fig : denlatt6.6 ] ) , where we can clearly observe a more homogeneous density spread over the lattice ( lower @xmath66 ) , and a consequently larger delocalization for the 2d - drdm than for the un - rand potential . the density behavior determines the condensate fraction @xmath118 , as shown in fig . [ fig : n0_l ] . ( color online ) condensate fraction @xmath119 as a function of @xmath74 for @xmath42 , @xmath70 and @xmath71 particles per site . the different curves correspond to different values of @xmath5 as indicated in the figure . the filled symbols correspond to the 2d - drdm potential and the empty symbols correspond to the un - rand potential . ] in correspondence of the minimum of the function @xmath66 , we observe that the condensate fraction @xmath119 does not depends of the system size , in the presence of the 2d - drdm potential . the resonance condition minimizes the fluctuations with respect the chosen homogeneous basis @xmath63 and fixes @xmath119 . at lower value of @xmath68 , we observe a `` super - delocalization '' ( @xmath73 scales as @xmath107 ) , and for both the 2d - drdm and the un - rand potentials , the large value of @xmath72 enhances the coherence and @xmath119 increases with system size . at larger values of @xmath68 , where @xmath73 scales as @xmath108 , the 2d - drdm impurities create holes in the system , and @xmath119 decreases with system size . for the case of the un - rand potential , one can observe a monotonic behavior of @xmath119 as a function of @xmath68 . as for the case of the 2d - drdm , the region where all the curves @xmath73 collapse together corresponds to a region where @xmath119 does not depend on the system size . the difference with the 2d - drdm is a larger decrease of @xmath119 in this region . for 2d - drdm , only one value of @xmath68 has this peculiarity , and the maximum position of the condensate fraction foregoes this point . let us remark that the minimum of @xmath120 corresponds to the minimum deviation with respect a homogeneous condensate , and , because of border effects , this target state is not necessarily the one that ensures a maximum value of @xmath119 in finite systems . the predicted condensate fraction enhancement for the drdm at low @xmath68 , being it very small , could be very difficult to be measured . however the non - diminishing of the coherence in a range of about @xmath121 should be observable , and could be directly compared with the result for the un - rand where the decrease of the coherence should be sizable . in summary , we introduce a correlated disorder model that is the natural extension of the drdm in 2d . we show that , in the non - interacting regime , such a disorder introduces some delocalized states if the resonance energy characterizing these structures belongs to the spectrum of the unperturbed system . in the presence of weak interactions , the 2d - drdm drives the density spatial fluctuations . by means of a mode - matching argument that includes the effect of the interactions , we show that the resonance energy is at the origin of these phenomena . a direct consequence is a non - monotonic behavior of the condensate fraction as a function of the disorder strength , and its enhancement for values close to the resonance condition . this work shows that short - range correlations in a disordered potential can modify and enhance the coherence of a many - body system in the weak interacting regime . such effects could be measured in the context of ultracold atoms with an accurate measurement of the density and coherence , via , for instance , a fringes contrast interference experiment . our results could also be extended to homogeneous systems provided one is able to engineer suitable impurities that are transparent for a given energy .	we introduce a two - dimensional short - range correlated disorder that is the natural generalization of the well - known one - dimensional dual random dimer model [ phys . rev . lett * 65 * , 88 ( 1990 ) ] . we demonstrate that , as in one dimension , this model induces a localization - delocalization transition in the single - particle spectrum . moreover we show that the effect of such a disorder on a weakly - interacting boson gas is to enhance the condensate spatial homogeneity and delocalisation , and to increase the condensate fraction around an effective resonance of the two - dimensional dual dimers . this study proves that short - range correlations of a disordered potential can enhance the quantum coherence of a weakly - interacting many - body system .	7,477	208
the scope of the growing field of complexity science `` ( or complex systems '' ) includes a broad variety of problems belonging to different scientific areas . examples for complex systems `` can be found in physics , biology , computer science , ecology , economy , sociology and other fields . a recurring theme in most of what is classified as complex systems '' is that of _ emergence_. emergent properties are those which arise spontaneously from the collective dynamics of a large assemblage of interacting parts . a basic question one asks in this context is how to derive and predict the emergent properties from the behavior of the individual parts . in other words , the central issue is how to extract large - scale , global properties from the underlying or microscopic degrees of freedom . in the physical sciences , there are many examples of emergent phenomena where it is indeed possible to relate the microscopic and macroscopic worlds . physical systems are typically described in terms of equations of motion of a huge number of microscopic degrees of freedom ( e.g. atoms ) . the microscopic dynamics is often erratic and complex , yet in many cases it gives rise to patterns with characteristic length and time scales much larger than the microscopic ones ( e.g. the pressure and temperature of a gas ) . these large scale patterns often posses the interesting , physically relevant properties of the system and one would like to model them or simulate their behavior . an important problem in physics is therefore to understand and predict the emergence of large scale behavior in a system , starting from its microscopic description . this problem is a fundamental one because most physical systems contain too many parts to be simulated directly and would become intractable without a large reduction in the number of degrees of freedom . a useful way to address this issue is to construct coarse - grained models , which treat the dynamics of the large scale patterns . the derivation of coarse - grained models from the microscopic dynamics is far from trivial . in most cases it is done in a phenomenological manner by introducing various ( often uncontrolled ) approximations . the problem of predicting emergent properties is most severe in systems which are modelled or described by _ undecidable _ mathematical algorithms@xcite . for such systems there exists no computationally efficient way of predicting their long time evolution . in order to know the system s state after ( e.g. ) one million time steps one must evolve the system a million time steps or perform a computation of equivalent complexity . wolfram has termed such systems _ computationally irreducible _ and suggested that their existence in nature is at the root of our apparent inability to model and understand complex systems @xcite . it is tempting to conclude from this that the enterprise of physics itself is doomed from the outset ; rather than attempting to construct solvable mathematical models of physical processes , computational models should be built , explored and empirically analyzed . this argument , however , assumes that infinite precision is required for the prediction of future evolution . as we mentioned above , usually coarse - grained or even statistical information is sufficient . an interesting question that arises is therefore : is it possible to derive coarse - grained models of undecidable systems and can these coarse - grained models be decidable and predictable ? in this work we address the emergence of large scale patterns in complex systems and the associated predictability problems by studying cellular - automata ( ca ) . ca are spatially and temporally discrete dynamical systems composed of a lattice of cells . they were originally introduced by von neumann and ulam @xcite in the 1940 s as a possible way of simulating self - reproduction in biological systems . since then , ca have attracted a great deal of interest in physics @xcite because they capture two basic ingredients of many physical systems : 1 ) they evolve according to a local uniform rule . 2 ) ca can exhibit rich behavior even with very simple update rules . for similar and other reasons , ca have also attracted attention in computer science @xcite , biology @xcite , material science @xcite and many other fields . for a review on the literature on ca see refs . . the simple construction of ca makes them accessible to computational theoretic research methods . using these methods it is sometimes possible to quantify the complexity of ca rules according to the types of computations they are capable of performing . this together with the fact that ca are caricatures of physical systems has led many authors to use them as a conceptual vehicle for studying complexity and pattern formation . in this work we adopt this approach and study the predictability of emergent patterns in complex systems by attempting to systematically coarse - grain ca . a brief preliminary report of our project can be found in ref . . there is no unique way to define coarse - graining , but here we will mean that our information about the ca is locally coarse - grained in the sense of being stroboscopic in time , but that nearby cells are grouped into a supercell according to some specified rule ( as is frequently done in statistical physics ) . below we shall frequently drop the qualifier `` local '' whenever there is no cause for confusion . a system which can be coarse - grained is _ compact - able _ since it is possible to calculate its future time evolution ( or some coarse aspects of it ) using a more compact algorithm than its native description . note that our use of the term compact - able refers to the phase space reduction associated with coarse - graining , and is agnostic as to whether or not the coarse - grained system is decidable or undecidable . accordingly , we define _ predictable _ to mean that a system is decidable or has a decidable coarse - graining . thus , it is possible to calculate the future time evolution of a predictable system ( or some coarse aspects of it ) using an algorithm which is more compact than both the native and coarse - grained descriptions . our work is organized as follows . in section [ ca_intro ] we give an introduction to ca and their use in the study of complexity . in section [ cg_procedure ] we present a procedure for coarse - graining ca . section [ cg_results ] shows and discusses the results of applying our procedure to one dimensional ca . most of the ca that we attempt to coarse - grain are wolfram s 256 elementary rules for nearest - neighbor ca . we will also consider a few other rules of special interest . in section [ kolmogorov_complexity ] we consider whether the coarse - grain - ability of many ca that we found in the elementary rule family is a common property of ca . using computational theoretic arguments we argue that the large scale behavior of local processes must be very simple . almost all ca can therefore be coarse - grained if we go to a large enough scale . our results are summarized and discussed in [ conclusions ] . cellular automata are a class of homogeneous , local and fully discrete dynamical systems . a cellular automaton @xmath0 is composed of a lattice @xmath1 of cells that can each assume a value from a finite alphabet @xmath2 . we denote individual lattice cells by @xmath3 where the indexing reflects the dimensionality and geometry of the lattice . cell values evolve in discrete time steps according to the pre - prescribed update rule @xmath4 . the update rule determines a cell s new state as a function of cell values in a finite neighborhood . for example , in the case of a one dimensional , nearest - neighbor ca the update rule is a function @xmath5 and @xmath6 $ ] . at each time step , each cell in the lattice applies the update rule and updates its state accordingly . the application of the update rule is done in parallel for all the cells and all the cells apply the same rule . we denote the application of the update rule on the entire lattice by @xmath7 . in early work @xcite , wolfram proposed that ca can be grouped into four classes of complexity . class 1 consists of ca whose dynamics reaches a steady state regardless of the initial conditions . class 2 consists of ca whose long time evolution produces periodic or nested structures . ca from both of these classes are simple in the sense that their long time evolution can be deduced from running the system a small number of time steps . on the other hand , class 3 and class 4 consist of complex " ca . class 3 ca produce structures that seem random . class 4 ca produce localized structures that propagate and interact in a complex way above a regular background . this classification is heuristic and the assignment of ca to the four classes is somewhat subjective . successive works on ca attempted to improve it or to find better alternatives@xcite . to the best of our knowledge there is , to date , no universally agreed upon classification scheme of ca . based on numerical experiments , wolfram hypothesized that most of class 3 and 4 ca are _ computationally irreducible_@xcite . namely , the evolution of these ca can not be predicted by a process which is drastically more efficient than themselves . in order to calculate the state of a computationally irreducible ca after @xmath8 time steps , one must run the ca for @xmath8 time steps or perform a computation of equivalent complexity . this definition is somewhat loose because it is not always clear how to compare computation running times and efficiency on different architectures . in addition , wolfram recognized that even computationally irreducible systems may have some superficial reducibility `` ( see page 746 in ref . ) and can be reduced to a limited extent . the difference between superficial '' and true reducibility however is not well defined . it is nevertheless clear that the asymptotic @xmath9 behavior of a computationally irreducible system can not be predicted by any computation of finite size . wolfram further argued that computationally irreducible systems are abundant in nature and that this fact explains our inability as physicists to deal with complex systems @xcite . it is difficult in general to tell whether a ca , behaving in an apparently complex way , is computationally irreducible . more concrete properties of ca which are related to computational irreducibility are _ undecidability _ and _ universality_. mathematical processes are said to be undecidable when there can be no algorithm that is guaranteed to predict their outcome in a finite time . equivalently , ca are said to be undecidable when aspects of their dynamics are undecidable . computationally irreducible ca are therefore undecidable and in the weak asymptotic definition that we gave above , computational irreducibility is equivalent to undecidability . for lack of a better choice we adopt this asymptotic definition and in the reminder of this work we will use the two terms interchangeably . some ca are known to be universal turing machines@xcite and are capable of performing all computations done by other processes . a famous two dimensional example is conway s game of life@xcite ; several examples in one dimension are lindgren and nordahl @xcite , albert and culik @xcite and wolfram s rule 110 @xcite . universal ca are , in a sense , maximally complex because they can emulate the dynamics of all other ca . being universal turing machines , these ca are subject to undecidable questions regarding their dynamics@xcite . for example whether an initial state will ever decay into a quiescent state is the ca equivalence of the undecidable halting problem@xcite . universal ca are therefore undecidable . wolfram s classification of ca is topological in the sense that ca are classified according to the properties of their trajectories . a different , more ambitious , approach is to classify ca according to a parameter derived directly from their rule tables . langton @xcite suggested that ca rules can be parameterized by his @xmath10 parameter which measures the fraction of non - quiescence rule table entries . he showed a strong correlation between the value of @xmath10 and the complexity found in the ca trajectories . for small values of @xmath10 one characteristically finds class 1 and 2 behavior while for @xmath11 a class 3 behavior is usually observed . langton identified a narrow region of intermediate values of @xmath10 where he found class 4 characteristic behavior . based on these observations langton proposed the _ edge of chaos _ hypothesis@xcite . this hypothesis claims that in the space of dynamical systems , interesting systems which are capable of computation are located at the boundary between simple and chaotic systems . this appealing hypothesis however was criticized in later works @xcite . recently , a different parametrization of ca rule tables was proposed by dubacq et al . this new approach is based on the information content of the rule table as measured by its kolmogorov complexity . as we will show below , our results lend support to this notion and indicate that rule tables with low kolmogorov complexity lead to simple behavior and vice versa . in addition to attempts to find order and hierarchy in the space of ca rules , much research has been devoted to the study of ca classes with special properties . additive ca ( or linear ) @xcite , commuting ca @xcite and ca with certain algebraic properties @xcite are a few examples . unsurprisingly , the dynamics of ca which enjoy such special properties can in most cases be understood and predictable at some level . in this work we will mostly be concerned with the family of one dimensional , nearest neighbor binary ca that were the subject of wolfram s investigations . these 256 elementary rules are among the simplest imaginable ca and thus present us with the least computational challenges when attempting to coarse - grain them . we will use wolfram s notation@xcite for identifying individual rules . the update function of an elementary rule is described by a rule number between 0 and 255 . the eight bit binary representation of the rule number specifies the update function outcome for the eight possible three cell configurations ( where 000 `` is the least significant and 111 '' is the most significant bit ) . ca are often conveniently visualized with different colors denoting different cell values . when dealing with binary ca we will use the convention @xmath12 , @xmath13 and use the two notations interchangeably . we now turn to study the emergence of large scale patterns in ca and the associated predictability problems by attempting to coarse - grain ca . there are many ways to define a coarse - graining of a dynamical system . in this work we define it as a ( real - space ) renormalization scheme where the original ca @xmath0 is coarse - grained to a renormlized ca @xmath14 through the lattice transformation @xmath15 . the projection function @xmath16 projects the value of a block of @xmath17 cells in @xmath18 , which we term a _ supercell _ , to a single cell in @xmath19 . by writing @xmath20 we denote the block - wise application of @xmath21 on the entire lattice @xmath22 . only non - trivial cases where @xmath21 is irreversible are considered because we want @xmath19 to provide a partial account of the full dynamics of @xmath18 . in order for @xmath19 and @xmath21 to provide a coarse - grained emulation of @xmath18 they must satisfy the commutativity condition @xmath23 for every initial condition @xmath24 of @xmath18 . the constant @xmath25 in the above equation is a time scale associated with the coarse - graining . a repeated application of eq.([coarse - graining_def ] ) shows that @xmath26 for all @xmath8 . namely , running the original ca for @xmath27 time steps and then projecting is equivalent to projecting the initial condition and then running the renormalized ca for @xmath8 time steps . thus , if we are only interested in the projected information we can run the more efficient ca @xmath19 . renormalization group transformations in statistical physics are usually performed with projection operators that arise from a physical intuition and understanding of the system in question . majority rules and different types of averages are often the projection operators of choice . in this work we have the advantage that the ca we wish to coarse - grain are fully discrete systems and the number of possible projections of a supercell of size @xmath17 is finite . we will therefore consider all possible ( at least with small supercells ) projection operators and will not restrict ourselves to coarse - graining by averaging . in addition , the discrete nature of ca makes it very difficult to find useful approximate solutions of eq.([coarse - graining_def ] ) because there is no natural small parameter that can be used to construct perturbative coarse - graining schemes . we therefore require that eq . ( [ coarse - graining_def ] ) is satisfied exactly . we now define a simple procedure for coarse - graining ca . other constructions are undoubtedly possible . for simplicity we limit our treatment to one - dimensional systems with nearest neighbor interactions . generalizations to higher dimensions and different interaction radii are straightforward . the commutativity condition eq . ( [ coarse - graining_def ] ) implies that the renormlized ca @xmath19 is homomorphic to the dynamics of @xmath18 on the scale defined by the supercell size @xmath17 . to search for explicit coarse - graining rules , we define the @xmath17th supercell version @xmath28 of @xmath18 . each cell of @xmath29 represents @xmath17 cells of @xmath18 and accepts values from the alphabet @xmath30 which includes all possible configurations of @xmath17 cells in @xmath18 . the transition function @xmath31 of the supercell ca can be defined in many ways depending on our choice of the supercells interaction radius . here we choose @xmath29 to be a nearest neighbor ca and compute @xmath32 by running @xmath18 for @xmath17 time steps on all possible initial conditions of length @xmath33 . in this way @xmath29 follows the dynamics of @xmath18 and each application of @xmath29 computes the evolution of a block of @xmath17 cells of @xmath18 , for @xmath17 time steps . this choice will later result in a coarse - grained ca @xmath19 which is itself nearest - neighbor . this is convenient because it enables us to compare the original and coarse - grained systems . another convenient feature of this construction is that it renders the coarse - graining time scale @xmath25 equal to the supercell size @xmath17 . other constructions however are undoubtedly possible . note that @xmath29 is not a coarse - graining of @xmath18 because no information was lost in the cell translation . next we attempt to generate the coarse ca @xmath19 by projecting the alphabet of @xmath29 on a subset @xmath34 which will serve as the alphabet of @xmath19 . this is the key step where information is being lost . the transition function @xmath35 is constructed from @xmath31 by projecting its arguments and outcome : @xmath36= p\left(f_{a^n}\left[x_1,x_2,x_3\right]\right ) . \label{fb_mv_construction}\ ] ] here @xmath37 denotes the projection of the supercell value @xmath38 . this construction is possible only if @xmath39\right)&= & p\left(f_{a^n}\left[y_1,y_2,y_3\right]\right ) , \nonumber \\ & & \forall \left(x , y\|p(x_i)=p(y_i)\right ) . \label{trans_func_projection}\end{aligned}\ ] ] otherwise , @xmath35 is multi - valued and our coarse - graining attempt fails for the specific choice of @xmath17 and @xmath21 . equations ( [ fb_mv_construction ] ) and ( [ trans_func_projection ] ) can also be cast in the matrix form @xmath40 which may be useful . here @xmath29 is an @xmath41 matrix which specify the @xmath17 cell block output for every possible combination of @xmath33 cells . @xmath21 is an @xmath42 matrix that project from @xmath43 to @xmath44 . @xmath45 is a @xmath46 matrix which projects 3 consecutive super cells and is a ( simple ) function of @xmath21 . the coarse - grained ca @xmath19 is an @xmath47 matrix and is also a function of @xmath21 . this is a greatly over determined equation for the projection operator @xmath21 . for a given value of @xmath17 and @xmath44 the equation contains @xmath48 constraints while @xmath21 is defined by @xmath43 free parameters . in cases where eq . ( [ trans_func_projection ] ) is satisfied , the resulting ca @xmath19 is a coarse - graining of @xmath29 with a time scale @xmath49 . for every step @xmath50 $ ] of @xmath29 , @xmath19 makes the move @xmath51 \\ & = & p\left(f_{a^n}\left[a^n_{n-1}(t),a^n_n(t),a^n_{n+1}(t)\right]\right ) \nonumber \\ & = & p\left(a^n_n(t+1)\right)\ ; , \nonumber\end{aligned}\ ] ] and therefore satisfies eq . ( [ coarse - graining_def ] ) . since a single time step of @xmath29 computes @xmath17 time steps of @xmath18 , @xmath19 is also a coarse - graining of @xmath18 with a coarse - grained time scale @xmath52 . analogies of these operators have been used in attempts to reduce the computational complexity of certain stochastic partial differential equations @xcite . similar ideas have been used to calculate critical exponents in probabilistic ca @xcite . to illustrate our method let us give a simple example . rule 128 is a class 1 elementary ca defined on the @xmath53 alphabet with the update function @xmath54= } \nonumber \\ & & \left\ { \begin{array}{l } \square\;,\;x_{n-1},x_n , x_{n+1}\neq \blacksquare,\blacksquare,\blacksquare \\ \blacksquare\;,\;x_{n-1},x_n , x_{n+1}=\blacksquare,\blacksquare,\blacksquare\\ \end{array } \right . \;. \label{f128}\end{aligned}\ ] ] figure [ cgof146figure ] b ) shows a typical evolution of this simple rule where all black regions which are in contact with white cells decay at a constant rate . to coarse - grain rule 128 we choose a supercell size @xmath55 and calculate the supercell update function @xmath56= } \nonumber \\ & & \left\ { \begin{array}{l } \blacksquare\blacksquare\;,\;y_{n-1},y_n , y_{n+1}=\blacksquare\blacksquare,\blacksquare\blacksquare,\blacksquare\blacksquare \\ \square\blacksquare\;,\;y_{n-1},y_n , y_{n+1}=\square\blacksquare,\blacksquare\blacksquare,\blacksquare\blacksquare \\ \blacksquare\square\;,\;y_{n-1},y_n , y_{n+1}=\blacksquare\blacksquare,\blacksquare\blacksquare,\blacksquare\square \\ \square\square\;,\;\mbox{all other combinations } \end{array } \right.\;. \label{rule128supercellf}\end{aligned}\ ] ] next we project the supercell alphabet using @xmath57 namely , the value of the coarse - grained cell is black only when the supercell value corresponds to two black cells . applying this projection to the supercell update function eq.([rule128supercellf ] ) we find that @xmath58\right)= } \nonumber \\ & & \left\ { \begin{array}{l } \blacksquare\;,\;p(y_{n-1}),p(y_n),p(y_{n+1})=\blacksquare,\blacksquare,\blacksquare \\ \end{array } \right . \;,\end{aligned}\ ] ] which is identical to the original update function @xmath59 . rule 128 can therefore be coarse - grained to itself , an expected result due to the scale invariant behavior of this simple rule . it is interesting to notice that the above coarse - graining procedure can lose two very different types of dynamic information . to see this , consider eq.([trans_func_projection ] ) . this equation can be satisfied in two ways . in the first case @xmath60&=&f_{a^n}\left[y_1 , y_2,y_3\right],\nonumber \\ & & \forall \left(x , y\|p(x_i)=p(y_i)\right)\ ; , \label{irrelevant_cond}\end{aligned}\ ] ] which necessarily leads to eq . ( [ trans_func_projection ] ) . @xmath31 in this case is insensitive to the projection of its arguments . the distinction between two variables which are identical under projection is therefore _ irrelevant _ to the dynamics of @xmath29 , and by construction to the long time dynamics of @xmath18 . by eliminating irrelevant degrees of freedom ( dof ) , coarse - graining of this type removes information which is redundant on the microscopic scale . the coarse ca in this case accounts for all possible long time trajectories of the original ca and the complexity classification of the two ca is therefore the same . in the second case eq . ( [ trans_func_projection ] ) is satisfied even though eq . ( [ irrelevant_cond ] ) is violated . here the distinction between two variables which are identical under projection is _ relevant _ to the dynamics of @xmath18 . replacing @xmath38 by @xmath61 in the initial condition may give rise to a difference in the dynamics of @xmath18 . moreover , the difference can be ( and in many occasions is ) unbounded in space and time . coarse - graining in this case is possible because the difference is constrained in the cell state space by the projection operator . namely , projection of all such different dynamics results in the same coarse - grained behavior . note that the coarse ca in this case can not account for all possible long time trajectories of the original one . it is therefore possible for the original and coarse ca to fall into different complexity classifications . coarse - graining by elimination of relevant dof removes information which is not redundant with respect to the original system . the information becomes redundant only when moving to the coarse scale . in fact , redundant " becomes a subjective qualifier here since it depends on our choice of coarse description . in other words , it depends on what aspects of the microscopic dynamics we want the coarse ca to capture . let us illustrate the difference between coarse - graining of relevant and irrelevant dof . consider a dynamical system whose initial condition is in the vicinity of two limit cycles . depending on the initial condition , the system will flow to one of the two cycles . coarse - graining of irrelevant dof can project all the initial conditions on to two possible long time behaviors . now consider a system which is chaotic with two strange attractors . coarse - graining irrelevant dof is inappropriate because the dynamics is sensitive to small changes in the initial conditions . coarse - graining of relevant dof is appropriate , however . the resulting coarse - grained system will distinguish between trajectories that circle the first or second attractor , but will be insensitive to the details of those trajectories . in a sense , this is analogous to the subtleties encountered in constructing renormalization group transformations for the critical behavior of antiferromagnets@xcite . the coarse - graining procedure we described above is not constructive , but instead is a self - consistency condition on a putative coarse - graining rule with a specific supercell size @xmath17 and projection operator @xmath21 . in many cases the single - valuedness condition eq.([trans_func_projection ] ) is not satisfied , the coarse - graining fails and one must try other choices of @xmath17 and @xmath21 . it is therefore natural to ask the following questions . can all ca be coarse - grained ? if not , which ca can be coarse - grained and which can not ? what types of coarse - graining transitions can we hope to find ? to answer these questions we tried systematically to coarse - grain one dimensional ca . we considered wolfram s 256 elementary rules and several non - binary ca of interest to us . our coarse - graining procedure was applied to each rule with different choices of @xmath17 and @xmath21 . in this way we were able to coarse - grain 240 out of the 256 elementary ca . these 240 coarse - grained - able rules include members of all four classes . the 16 elementary ca which we could not coarse - grain are rules 30 , 45 , 106 , 154 and their symmetries . rules 30 , 45 and 106 belong to class 3 while 154 is a class 2 rule . we do nt know if our inability to coarse - grain these 16 rules comes from limited computing power or from something deeper . we suspect ( and give arguments in section [ kolmogorov_complexity ] ) the former . the number of possible projection operators @xmath21 grows very fast with @xmath17 . even for small @xmath17 , it is computationally impossible to scan all possible @xmath21 . in order to find valid projections , we therefore used two simple search strategies . in the first strategy , we looked for coarse - graining transitions within the elementary ca family by considering @xmath21 which project back on the binary alphabet . excluding the trivial projections @xmath62 and @xmath63 there are @xmath64 such projections . we were able to scan all of them for @xmath65 and found many coarse - graining transitions . figure [ mapfigure ] shows a map of the coarse - graining transitions that we found within the family of elementary rules . an arrow in the map indicates that each rule from the origin group can be coarse - grained to each rule from the target group . the supercell size @xmath17 and the projection @xmath21 are not shown and each arrow may correspond to several choices of @xmath17 and @xmath21 . as we explained above , only coarse - grainings with @xmath66 are shown due to limited computing power . other transitions within the elementary rule family may exist with larger values of @xmath17 . this map is in some sense an analogue of the familiar renormalization group flow diagrams from statistical mechanics . several features of fig . [ mapfigure ] are worthy of a short discussion . first , notice that the map manifests the left``@xmath67right '' and 0``@xmath671 '' symmetries of the elementary ca family . for example rules 252 , 136 and 238 are the 0``@xmath671 '' , left``@xmath67right '' and the 0``@xmath671 '' and left``@xmath67right '' symmetries of rule 192 respectively . second , coarse - graining transitions are obviously transitive , i.e. if @xmath18 goes to @xmath19 with @xmath68 and @xmath19 goes to @xmath69 with @xmath70 then @xmath18 goes to @xmath69 with @xmath71 . for some transitions , the map in fig . [ mapfigure ] fails to show this property because we did not attain large enough values of @xmath17 . another interesting feature of the transition map is that the apparent rule complexity never increases with a coarse - graining transition . namely , we never find a simple behaving rule which after being coarse - grained becomes a complex rule . the transition map , therefore , introduces a hierarchy of elementary rules and this hierarchy agrees well with the apparent rule complexity . the hierarchy is partial and we can not relate rules which are not connected by a coarse - graining transition . as opposed to the wolfram classification , this coarse - graining hierarchy is well defined and is therefore a good candidate for a complexity measure@xcite . finally notice that the eight rules 0 , 60 , 90 , 102 , 150 , 170 , 204 , 240 , whose update function has the additive form @xmath72&=&\alpha\cdot x_{n-1}\oplus\beta\cdot x_n \oplus \gamma\cdot x_{n+1}\;,\nonumber \\ & & \alpha,\beta,\gamma\in \{0,1\}\;,\end{aligned}\ ] ] where @xmath73 denotes the xor operation , are all fixed points of the map . this result is not limited to elementary rules . as showed by barbe et.al @xcite , additive ca in arbitrary dimension whose alphabet sizes are prime numbers coarse - grain themselves . we conjecture that there are situations where reducible fixed points exist for a wide range of systems , analogous to the emergence of amplitude equations in the vicinity of bifurcation points . when projecting back on the binary alphabet , one maximizes the amount of information lost in the coarse - graining transition . at first glance , this seems to be an unlikely strategy , because it is difficult for the coarse - grained ca to emulate the original one when so much information was lost . in terms of our coarse - graining procedure such a projection maximizes the number of instances @xmath74 of eq . ( [ trans_func_projection ] ) . on second examination , however . this strategy is not that poor . the fact that there are only two states in the coarse - grained alphabet reduces the probability that an instance @xmath74 of eq.([trans_func_projection ] ) will be violated to 1/2 . the extreme case of this argument would be a projection on a coarse - grained alphabet with a single state . such a trivial projection will never violate eq . ( [ trans_func_projection ] ) ( but will never show any patterns or dynamics either ) . a second search strategy for valid projection operators that we used is located on the other extreme of the above tradeoff . namely , we attempt to lose the smallest possible amount of information . we start by choosing two supercell states @xmath75 and @xmath76 and unite them using @xmath77 where the subscript in @xmath78 denotes that this is an initial trial projection to be refined later . the refinement process of the projection operator proceeds as follows . if @xmath79 ( starting with @xmath80 ) satisfies eq . ( [ trans_func_projection ] ) then we are done . if on the other hand , eq . ( [ trans_func_projection ] ) is violated by some @xmath81\right)&\neq & p_n\left(f_{a^n}\left[y_1,y_2,y_3\right]\right)\ ; , \nonumber \\ & & p_n(x_i)=p_n(y_i)\;,\end{aligned}\ ] ] the inequality is resolved by refining @xmath79 to @xmath82\;\ ; , r_2=f_{a^n}\left[y_1,y_2,y_3\right]\;.\end{aligned}\ ] ] this process is repeated until eq . ( [ trans_func_projection ] ) is satisfied . a non - trivial coarse - graining is found in cases where the resulting projection operator is non - constant ( more than a single state in the coarse - grained ca ) . by trying all possible @xmath83 initial pairs , the above projection search method is guaranteed to find a valid projection if such a projection exist on the scale defined by the supercell size @xmath17 . using this method we were able to coarse - grain many ca . the resulting coarse - grained ca that are generated in this way are often multicolored and do not belong to the elementary ca family . for this reason it is difficult to graphically summarize all the transitions that we found in a map . instead of trying to give an overall view of those transitions we will concentrate our attention on several interesting cases which we include in the examples section bellow . as our first example we choose a transition between two class 2 rules . the elementary rule 105 is defined on the alphabet @xmath53 with the transition function @xmath84=\overline{x_{n-1}\oplus x_{n } \oplus x_{n+1}}\;,\ ] ] where the over - bar denotes the not operation , and @xmath12 , @xmath13 . we use a supercell size @xmath55 and calculate the transition function @xmath85 , defined on the alphabet @xmath86 . now we project this alphabet back on the @xmath53 alphabet with @xmath87 a pair of cells in rule 105 are coarse - grained to a single cell and the value of the coarse cell is black only when the pair share a same value . using the above projection operator we construct the transition function of the coarse ca . the result is found to be the transition function of the additive rule 150 : @xmath88=x_{n-1}\oplus x_{n } \oplus x_{n+1}\;.\ ] ] figure [ 105to150figure ] shows the results of this coarse - graining transition . in fig . [ 105to150figure ] ( a ) we show the evolution of rule 105 with a specific initial condition while fig . [ 105to150figure ] ( b ) shows the evolution of rule 150 from the coarse - grained initial condition . the small scale details in rule 105 are lost in the transformation but extended white and black regions are coarse - grained to black regions in rule 150 . the time evolution of rule 150 captures the overall shape of these large structures but without the black - white decorations . as shown in fig . [ mapfigure ] , rule 150 is a fixed point of the transition map . rule 105 can therefore be further coarse - grained to arbitrary scales . as a second example of coarse - grained - able elementary ca we choose rule 146 . rule 146 is defined on the @xmath53 alphabet with the transition function @xmath89= } \nonumber \\ & & \left\ { \begin{array}{l } \blacksquare\;,\;x_{n-1}x_nx_{n+1}=\square\square\blacksquare;\blacksquare\square\square;\blacksquare\blacksquare\blacksquare \\ \square\;,\;\mbox{all other combinations } \\ \end{array}\right . \;.\end{aligned}\ ] ] it produces a complex , seemingly random behavior which falls into the class 3 group . we choose a supercell size @xmath90 and calculate the transition function @xmath91 , defined on the alphabet @xmath92 . now we project this alphabet back on the @xmath53 alphabet with @xmath93 namely , a triplet of cells in rule 146 are coarse - grained to a single cell and the value of the coarse cell is black only when the triplet is all black . using the above projection operator we construct the transition function of the coarse ca . the result is found to be the transition function of rule 128 which was given in eq . ( [ f128 ] ) . rule 146 can therefore be coarse - grained by rule 128 , a class 1 elementary ca . in figure [ cgof146figure ] we show the results of this coarse - graining . [ cgof146figure ] ( a ) shows the evolution of rule 146 with a specific initial condition while fig . [ cgof146figure ] ( b ) shows the evolution of rule 128 from the coarse - grained initial condition . our choice of coarse - graining has eliminated the small scale details of rule 146 . only structures of lateral size of three or more cells are accounted for . the decay of such structures in rule 146 is accurately described by rule 128 . note that a class 3 ca was coarse - grained to a class 1 ca in the above example . our gain was therefore two - fold . in addition to the phase space reduction associated with coarse - graining we have also achieved a reduction in complexity . our procedure was able to find predictable coarse - grained aspects of the dynamics even though the small scale behavior of rule 146 is complex , potentially irreducible . rule 146 can also be coarse - grained by non elementary ca . using a supercell size of @xmath94 we found that the difference between the combinations @xmath95 and @xmath96 is irrelevant to the long time behavior of rule 146 . it is therefore possible to project these two combinations into a single coarse grained state . the same is true for the combinations @xmath97 and @xmath98 which can be projected to another coarse - grained state . the end result of this coarse - graining ( fig . [ cgof146figure ] ( c ) ) is a 62 color ca which retains the information of all other 6 cell combinations . the amount of information lost in this transition is relatively small , 2/64 of the supercell states have been eliminated . more impressive alphabet reductions can be found by going to larger scales . for @xmath17=7,8,9,10 and 11 we found an alphabet reduction of 9/128 , 33/256 , 97/512 , 261/1024 and 652/2048 respectively . [ cgof146figure ] ( d ) shows the percentage of states that can be eliminated as a function of the supercell size @xmath17 . all of the information lost in those coarse - grainings corresponds to irrelevant dof . the two different coarse - graining transitions of rule 146 that we presented above are a good opportunity to show the difference between relevant and irrelevant dof . as we explained earlier , a transition like 146@xmath99128 where the rules has different complexities must involve the elimination of relevant dof . indeed if we modify an initial condition of rule 146 by replacing a @xmath100 segment with @xmath101 we will get a modified evolution . as we show in figure [ rule146rdoffigure ] , the difference in the trajectories has a complex behavior and is unbounded in space and time . however , since @xmath100 and @xmath101 are both projected by eq . ( [ rule146projection ] ) to @xmath102 , the projections of the original and modified trajectories will be identical . in contrast , the coarse graining of rule 146 to the 62 state ca of fig . [ cgof146figure ] ( c ) involves the elimination of irrelevant dof only . if we replace a @xmath97 in the initial condition with a @xmath98 we find that the difference between the modified and unmodified trajectories decays after a few time steps . the elementary ca rule 184 is a simplified one lane traffic flow model . its transition function is given by @xmath103= } \nonumber \\ & & \left\ { \begin{array}{l } \square\;,\;x_{n-1}x_nx_{n+1}=\square\square\square;\square\square\blacksquare;\square\blacksquare\square;\blacksquare\blacksquare\square \\ \blacksquare\;,\;x_{n-1}x_nx_{n+1}=\square\blacksquare\blacksquare;\blacksquare\square\square;\blacksquare\square\blacksquare;\blacksquare\blacksquare\blacksquare \\ \;.\end{aligned}\ ] ] identifying a black cell with a car moving to the right and a white cell with an empty road segment we can rewrite the update rule as follows . a car with an empty road segment to its right advances and occupies the empty segment . a car with another car to its right will avoid a collision and stay put . this is a deterministic and simplified version of the more realistic nagel schreckenberg model @xcite . rule 184 can be coarse - grained to a 3 color ca using a supercell size @xmath55 and the local density projection @xmath104{0.4,0.4,0.4}\blacksquare\color{black}}\;,\;x=\square\blacksquare;\blacksquare\square \\ \blacksquare\;,\;x=\blacksquare\blacksquare \\ \end{array}\right.\;.\ ] ] the update function of the resulting ca is given by @xmath105= } \\ & & \left\ { \begin{array}{l } \square\;,\;y_{n-1}y_ny_{n+1}=\square\square\square;\square\square{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}};\square\square\blacksquare;\square{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}\square;\square{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}\\ \blacksquare\;,\;y_{n-1}y_ny_{n+1}=\square\blacksquare\blacksquare;{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}\blacksquare;{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}\blacksquare\blacksquare;\blacksquare{\color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}\blacksquare;\blacksquare\blacksquare\blacksquare \\ { \color[rgb]{0.4,0.4,0.4}\blacksquare\color{black}}\;,\;\mbox{all other combinations } \\ \end{array}\right .. \nonumber\end{aligned}\ ] ] figure [ rule184figure ] shows the result of this coarse - graining . [ rule184figure ] ( a ) shows a trajectory of rule 184 while fig . [ rule184figure ] ( b ) shows the trajectory of the coarse ca . from this figure it is clear that the white zero density regions correspond to empty road and the black high density regions correspond to traffic jams . the density 1/2 grey regions correspond to free flowing traffic with an exception near traffic jams due to a boundary effect . by using larger supercell sizes it is possible to find other coarse - grained versions of rule 184 . as in the above example , the coarse - grained states group together local configurations of equal car densities . the projection operators however are not functions of the local density alone . they are a partition of such a function and there could be several coarse - grained states which correspond to the same local car density . we found ( empirically ) that for even supercell sizes @xmath106 the coarse - grained ca contain @xmath107 states and for odd supercell sizes @xmath108 they contain @xmath109 states . figure [ rule184figure ] ( c ) shows the amount of information lost in those transitions as a function of @xmath17 . most of the lost information corresponds to relevant dof but some of it is irrelevant . rule 110 is one of the most interesting rules in the elementary ca family . it belongs to class 4 and exhibits a complex behavior where several types of particles `` move and interact above a regular background . the behavior of these particles '' is rich enough to support universal computation @xcite . in this sense rule 110 is maximally complex because it is capable of emulating all computations done by other computing devices in general and ca in particular . as a consequence it is also undecidable @xcite . we found several ways to coarse - grain rule 110 . using @xmath94 , it is possible to project the 64 possible supercell states onto an alphabet of 63 symbols . figure [ rule110figure ] ( a ) and ( b ) shows a trajectory of rule 110 and the corresponding trajectory of the coarse - grained 63 states ca . a more impressive reduction in the alphabet size is obtained by going to larger values of @xmath17 . for @xmath110 we found an alphabet reduction of @xmath111 , @xmath112 , @xmath113 , @xmath114 , @xmath115 and @xmath116 respectively . only irrelevant dof are eliminated in those transitions . [ rule110figure ] ( c ) shows the percentage of reduced states as a function of the supercell size @xmath17 . we expect this behavior to persist for larger values of @xmath17 . another important coarse - graining of rule 110 that we found is the transition to rule 0 . rule 0 has the trivial dynamics where all initial states evolve to the null configuration in a single time step . the transition to rule 0 is possible because many cell sequences can not appear in the long time trajectories of rule 110 . for example the sequence @xmath117 is a so called garden of eden `` of rule 110 . it can not be generated by rule 110 and can only appear in the initial state . coarse - graining by rule 0 is achieved in this case using @xmath118 and projecting @xmath117 to @xmath119 and all other five cell combinations to @xmath102 . another example is the sequence @xmath120 . this sequence is a garden of eden '' of the @xmath121 supercell version of rule 110 . it can appear only in the first 12 time steps of rule 110 but no later . coarse - graining by rule 0 is achieved in this case using @xmath121 and projecting @xmath120 to @xmath119 and all other 13 cell combinations to @xmath102 . these examples are important because they show that even though rule 110 is undecidable it has decidable and predictable coarse - grained aspects ( however trivial ) . to our knowledge rule 110 is the only proven undecidable elementary ca and therefore this is the only ( proven ) example of undecidable to decidable transition that we found within the elementary ca family . it is interesting to note that the number of garden of eden `` states in supercell versions of rule 110 grows very rapidly with the supercell size @xmath17 . as we show in fig.[rule110figure ] ( d ) , the fraction of garden of eden '' states out of the @xmath122 possible sequences , grows almost linearly with @xmath17 . in addition , at every scale @xmath17 there are new garden of eden `` sequences which do not contain any smaller gardens of eden '' as subsequences . these results are consistent with our understanding that even though the dynamics looks complex , more and more structure emerges as one goes to larger scales . we will have more to say about this in section [ kolmogorov_complexity ] . the garden of eden `` states of supercell versions of rule 110 represent pieces of information that can be used in reducing the computational effort in rule 110 . the reduction can be achieved by truncating the supercell update rule to be a function of only non garden of eden '' states . the size of the resulting rule table will be much smaller ( @xmath123 with @xmath124 ) than the size of the supercell rule table . efficient computations of rule 110 can then be carried out by running rule 110 for the first @xmath17 time steps . after @xmath17 time steps the system contains no garden of eden `` sequences and we can continue to propagate it by using the truncated supercell rule table without loosing any information . note that we have not reduced rule 110 to a decidable system . at every scale we achieved a constant reduction in the computational effort . wolfram has pointed out that many irreducible systems have pockets of reducibility and termed such a reduction as superficial reducibility '' ( see page 746 in ref . ) . it will be interesting to check how much superficial reducibility `` is contained in rule 110 at larger scales . it will be inappropriate to call it superficial '' if the curve in fig.[rule110figure ] ( d ) approaches 100% in the large @xmath17 limit . it might be argued that the coarse - graining of rule 110 by rule 0 is a trivial example of an undecidable to a decidable coarse - graining transition . the fact that certain configurations can not be arrived at in the long time behavior is not very surprising and is expected of any irreversible system . in order to search for more interesting examples we studied other one dimensional universal ca that we found in the literature . lindgren and nordahl @xcite constructed a 7 state nearest neighbor and a 4 state next - nearest neighbor ca that are capable of emulating a universal turing machine . the entries in the update tables of these ca are only partly determined by the emulated turing machine and can be completed at will . we found that for certain completion choices these two universal ca can be coarse - grained to a trivial ca which like rule 0 decay to a quiescent configuration in a single time step . another universal ca that can undergo such a transition is wolfram s 19 state , next - nearest neighbor universal ca @xcite . these results are essentially equivalent to the rule 110 @xmath99 rule 0 transition . a more interesting example is albert and culik s @xcite universal ca . it is a 14 state nearest - neighbor ca which is capable of emulating all other ca . the transition table of this ca is only partly determined by its construction and can be completed at will . we found that when the empty entries in the transition function are filled by the copy operation @xmath125=x_n\ ; , \label{copy_operation}\ ] ] the resulting undecidable ca has many coarse - graining transitions to decidable ca . in all these transitions the coarse - grained ca performs the copy operation eq . ( [ copy_operation ] ) for all @xmath126 . different transitions differ in the projection operator and the alphabet size of the coarse - grained ca . figure [ albertculik_figure ] shows a coarse - graining of albert and culik s universal ca to a 4 state copy ca . the coarse - grained ca captures three types of persistent structures that appear in the original system but is ignorant of more complicated details . the supercell size used here is @xmath55 . in the previous section we showed that a large majority of elementary ca can be coarse - grained in space and time . this is rather surprising since finding a valid projection operator is equivalent to solving eq.([cg_matrix_form ] ) which is greatly over constrained . solutions for this equation should be rare for random choices of the matrix @xmath29 . in this section we show that solutions of eq.([cg_matrix_form ] ) are frequent because @xmath29 is not random but a highly structured object . as the supercell size @xmath17 is increased , @xmath29 becomes less random and the probability of finding a valid projection approaches unity . to appreciate the high success rate in coarse - graining elementary ca consider the following statistics . by using supercells of size @xmath55 and considering all possible projection operators @xmath127 we were able to coarse - grain approximately one third of all 256 elementary ca rules . recall that the coarse - graining procedure that we use involves two stages . in the first stage we generate the supercell version @xmath29 , a 4 color ca in the @xmath55 case . in the second stage we look for valid projection operators . 4 color ca that are @xmath55 supercell versions of elementary ca are a tiny fraction of all possible ( @xmath128 ) 4 color ca . if we pick a random 4 color ca and try to project it ; i.e. attempt to solve eq . ( [ cg_matrix_form ] ) with @xmath29 replaced by an arbitrary 4 color ca , we find an average of one solvable instance out of every @xmath129 attempts . this large difference in the projection probability indicates that 4 color ca which are supercells versions of elementary rules are not random . the numbers become more convincing when we go to larger values of @xmath17 and attempt to find projections to random @xmath122 color ca . to put our arguments on a more quantitative level we need to quantify the information content of supercell versions of ca . an accepted measure in algorithmic information theory for the randomness and information content of an individual@xcite object is its kolmogorov complexity ( algorithmic complexity ) @xcite . the kolmogorov complexity @xmath130 of a string of characters @xmath38 with respect to a universal computer @xmath131 is defined as @xmath132 where @xmath133 is the length of @xmath38 in bits and @xmath134 is the bit length of the minimal computer program that generates @xmath38 and halts on @xmath131 ( irrespective of the running time ) . this definition is sensitive to the choice of machine @xmath131 only up to an additive constant in @xmath134 which do not depend on @xmath38 . for long strings this dependency is negligible and the subscript @xmath131 can be dropped . according to this definition , strings which are very structured require short generating programs and will therefore have small kolmogorov complexity . for example , a periodic @xmath38 with period @xmath135 can be generated by a @xmath136 long program and @xmath137 . in contrast , if @xmath38 has no structure it must be generated literally , i.e. the shortest program is print(x ) " . in such cases @xmath138 , @xmath139 and the information content of @xmath38 is maximal . by using simple counting arguments @xcite it is easy to show that simple objects are rare and that @xmath139 for most objects @xmath38 . kolmogorov complexity is a powerful and elegant concept which comes with an annoying limitation . it is uncomputable , i.e. it is impossible to find the length of the minimal program that generates a string @xmath38 . it is only possible to bound it . it is easy to see that supercell ca are highly structured objects by looking at their kolmogorov complexity . consider the ca @xmath140 and its @xmath17th supercell version @xmath141 ( for simplicity of notation we omit the subscript @xmath18 from the alphabet size ) . the transition function @xmath31 is a table that specifies a cell s new state for all @xmath142 possible local configurations ( assuming a is nearest neighbor and one dimensional ) . @xmath31 can therefore be described by a string of @xmath142 symbols from the alphabet @xmath143 . the bit length @xmath144 of such a description is @xmath145 if @xmath29 was a typical ca with @xmath146 colors we could expect that @xmath147 , the length of the minimal program that generates @xmath31 , will not differ significantly from @xmath144 . however , since @xmath29 is a super cell version of @xmath18 we have a much shorter description , i.e. to construct @xmath29 from @xmath18 . this construction involves running @xmath18 , @xmath17 time steps for all possible initial configurations of @xmath33 cells . it can be conveniently coded in a program as repeated applications of the transition function @xmath4 within several loops . up to an additive constant@xcite , the length of such a program will be equal to the bit length description of @xmath4 : @xmath148 note that we have used @xmath149 to indicate that this is an upper bound for the length of the minimal program that generates @xmath31 . this upper bound , however , should be tight for an update rule @xmath4 with little structure . the kolmogorov complexity of @xmath31 can consequently be bounded by @xmath150 this complexity approaches zero at large values of @xmath17 . our argument above shows that the large scale behavior of ca ( or any local process ) must be simple in some sense . we would like to continue this line of reasoning and conjecture that the small kolmogorov complexity of the large scale behavior is related to our ability to coarse - grain many ca . at present we are unable to prove this conjecture analytically , and must therefore resort to numerical evidence which we present below . ideally , in order to show that such a connection exists one would attempt to coarse - grain ca with different alphabets and on different length scales ( supercell sizes ) , and verify that the success rate correlates with the kolmogorov complexity of the generated supercell ca . this , however , is computationally very challenging and going beyond ca with a binary alphabet and supercell sizes of more than @xmath151 is not realistic . a more modest experiment is the following . we start with a ca @xmath18 with an alphabet @xmath152 , and check whether its @xmath17 supercell version @xmath29 contains all possible @xmath146 states . namely , if there exist @xmath153 such that @xmath154 such a missing state of @xmath29 is sometimes referred to as a garden of eden `` configuration because it can only appear in the initial state of @xmath29 . note that by the construction of @xmath29 , a garden of eden '' state of @xmath29 can appear only in the first @xmath155 time steps of @xmath18 and is therefore a generalized garden of eden `` of @xmath18 . in cases where a state of @xmath29 is missing , @xmath18 can be trivially coarse - grained to the elementary ca rule 0 by projecting the missing state of @xmath29 to 1 '' and all other combinations to 0 `` . this type of trivial projection was discussed earlier in connection with the coarse - graining of rule 110 . finding a garden of eden '' state of @xmath29 is computationally relatively easy because there is no need to calculate the supercell transition function @xmath31 . it is enough to back - trace the evolution of @xmath18 and check if all @xmath17 cell combinations has a @xmath33 cell ancestor combination , @xmath17 time steps in the past . figure [ missing_colors ] ( a ) shows the statistics obtained from such an experiment . it exhibits the fraction @xmath156 of ca rules with different alphabet sizes @xmath152 , whose @xmath17th supercell version is missing at least one state . each data point in this figure was obtained by testing 10,000 ca rules . the fraction @xmath156 approaches unity at large values of @xmath17 , an expected behavior since most of the ca are irreversible . figure [ missing_colors ] ( b ) shows the same data as in ( a ) when plotted against the variable @xmath157 where @xmath152 is the alphabet size , @xmath158 is the upper bound for the kolmogorov complexity of the supercell ca from eq.([upper_bound_kc ] ) and @xmath69 is a constant . the excellent data collapse imply a strong correlation between the probability of finding a missing state and the kolmogorov complexity of a supercell ca . this figure also shows that the data points can be accurately fitted by @xmath159 with @xmath160 a constant and @xmath161 ( solid line in fig . [ missing_colors ] ( b ) ) . having the scaling form @xmath162 we can now study the behavior of @xmath156 with large alphabet sizes . assuming @xmath163 and @xmath164 to be continuous we define @xmath165 as the point where @xmath166 . for a fixed value of @xmath152 , the slope of @xmath156 at the transition region can be calculated by @xmath167 where @xmath168 putting together eqs . ( [ rge_slope ] ) and ( [ nh ] ) we find that the slope of @xmath156 at the transition region grows as @xmath169 for large values of @xmath152 . an indication of this phenomena can be seen in fig . [ missing_colors ] ( a ) which shows sharper transitions at large values of @xmath152 . in the limit of large @xmath152 , @xmath156 becomes a step function with respect to @xmath17 . this fact introduces a critical value @xmath170 such that for @xmath171 the probability of finding a missing state is zero and for @xmath172 the probability is one . the value of this critical @xmath17 grows with the alphabet size as @xmath173 . note that @xmath170 is an emergent length scale , as it is not present in any of the ca rules , but according to the above analysis will emerge ( with probability one ) in their dynamics . a direct consequence of the emergence of @xmath174 is that a measure 1 of all ca can be coarse - grained to the elementary rule 0 " on the coarse - grained scale @xmath174 . generalized garden of eden " states are a specific form of emergent pattern that can be encountered in the large scale dynamics of ca . is the kolmogorov complexity of ca rules related to other types of coarse - grained behavior ? to explore this question we attempted to project ( solve eq . ( [ cg_matrix_form ] ) ) random ca with bounded kolmogorov complexities . to generate a random ca @xmath175 with a bounded kolmogorov complexity we view the update rule @xmath4 as a string of @xmath176 bits , denote the @xmath177th bit by @xmath178 and apply the following procedure : 1 ) randomly pick the first @xmath179 bits of @xmath4 . 2 ) randomly pick a generating function @xmath180 . 3 ) set the values of all the empty bits of @xmath4 by applying @xmath181 : @xmath182\;,\ ] ] starting at @xmath183 and finishing at @xmath184 . up to an additive constant , the length of such a procedure is equal to @xmath185 , the number of random bits chosen . the kolmogorov complexity of the resulting rule table can therefore be bounded by @xmath186 for small values of @xmath179 this is a reasonable upper bound . however for large values of @xmath179 this upper bound is obviously not tight since the size of @xmath181 can be much larger than the length of @xmath4 . using the above procedure we studied the probability of projecting ca with different alphabets and different upper bound kolmogorov complexities @xmath187 . for given values of @xmath152 and @xmath179 we generated 10,000 ( 200 for the @xmath188 case ) ca and tried to find a valid projection on the @xmath189 alphabet . figure [ projection_probability ] ( a ) shows the fraction @xmath190 of solvable instances as a function of @xmath191 . the constant @xmath69 used for this data collapse is @xmath192 , very close to @xmath193 . as valid projection solutions we considered all possible projections @xmath194 . in doing so we may be redoing the missing states experiment because many low kolmogorov complexity rules has missing states and can thus be trivially projected . in order to exclude this option we repeated the same experiment while restricting the family of allowed projections to be equal partitions of @xmath195 , i.e. @xmath196 the results are shown if fig . [ projection_probability ] ( b ) . it seems that in both cases there is a good correlation between the kolmogorov complexity ( or its upper bound ) of a ca rule and the probability of finding a valid projection . in particular , the fraction of solvable instances goes to one at the low @xmath187 limit . as shown by the solid lines in fig.[projection_probability ] , this fraction can again be fitted by @xmath197 where @xmath160 is a constant and in this case @xmath198 . how many of the ca rules that we generate and project show a complex behavior ? does the fraction of projectable rules simply reflect the fraction of simple behaving rules ? to answer this question we studied the rules generated by our procedure . for each value of @xmath152 and @xmath179 we generated 100 rules and counted the number of rules exhibiting complex behavior . a rule was labelled complex " if it showed class 3 or 4 behavior and exhibited a complex sensitivity to perturbations in the initial conditions . [ projection_probability ] ( c ) shows the statistics we obtained with different alphabet sizes as a function of @xmath187 while the inset shows it as a function of @xmath179 . we first note that our statistics support dubacq et al . @xcite , who proposed that rule tables with low kolmogorov complexities lead to simple behavior and rule tables with large kolmogorov complexity lead to complex behavior . moreover , our results show that the fraction of complex rules does not depend on the alphabet size and is only a function of @xmath179 . rules with larger alphabets show complex behavior at a lower value of @xmath187 . as a consequence , a large fraction of projectable rules are complex and this fraction grows with the alphabet size @xmath152 . as we explained earlier , the kolmogorov complexity of supercell versions of ca approaches zero as the supercell size @xmath17 is increased . our experiments therefore indicate that a measure one of all ca are coarse - grained - able if we use a coarse enough scale . moreover , the data collapse that we obtain and the sharp transition of the scaling function suggest that it may be possible to know in advance at what length scales to look for valid projections . this can be very useful when attempting to coarse - grain ca or other dynamical systems because it can narrow down the search domain . as in the case of garden of eden " states that we studied earlier , we interpret the transition point as an emergent scale which above it we are likely to find self organized patterns . note however that this scale is a little shifted in fig . [ projection_probability ] ( b ) when compared with fig . [ projection_probability ] ( a ) . the emergence scale is thus sensitive to the types of large scale patterns we are looking for . in this work we studied emergent phenomena in complex systems and the associated predictability problems by attempting to coarse - grain ca . we found that many elementary ca can be coarse - grained in space and time and that in some cases complex , undecidable ca can be coarse - grained to decidable and predictable ca . we conclude from this fact that undecidability and computational irreducibility are not good measures for physical complexity . physical complexity , as opposed to computational complexity should address the interesting , physically relevant , coarse - grained degrees of freedom . these coarse - grained degrees of freedom maybe simple and predictable even when the microscopic behavior is very complex . the above definition of physical complexity brings about the question of the objectivity of macroscopic descriptions @xcite . is our choice of a coarse - grained description ( and its consequent complexity ) subjective or is it dictated by the system ? our results are in accordance with shalizi and moore @xcite : it is both . in many cases we discovered that a particular ca can undergo different coarse - graining transitions using different projection operators . in these cases the system dictates a set of valid projection operators and we are restricted to choose our coarse - grained description from this set . we do however have some freedom to manifest our subjective interest . the coarse - graining transitions that we found induce a hierarchy on the family of elementary ca ( see fig . [ mapfigure ] ) . moreover , it seems that rule complexity never increases with coarse - graining transitions . the coarse - graining hierarchy therefore provides a partial complexity order of ca where complex rules are found at the top of the hierarchy and simple rules are at the bottom . the order is partial because we can not relate rules which are not connected by coarse - graining transitions . this coarse - graining hierarchy can be used as a new classification scheme of ca . unlike wolfram s , classification this scheme is not a topological one since the basis of our suggested classification is not the ca trajectories . nor is this scheme parametric , such as langton s @xmath10 parameter scheme . our scheme reflects similarities in the algebraic properties of ca rules . it simply says that if some coarse - grained aspects of rule @xmath18 can be captured by the detailed dynamics of rule @xmath19 then rule @xmath18 is at least as complex as rule @xmath19 . rule @xmath18 maybe more complex because in some cases it can do more than its projection . note that our hierarchy may subdivide wolfram s classes . for example rule 128 is higher on the hierarchy than rule 0 . these two rules belong to class 1 but rule 128 can be coarse - grained to rule 0 and it is clear that an opposite transition can not exist . it will be interesting to find out if class 3 and 4 can also be subdivided . in the last part of this work we tried to understand why is it possible to find so many coarse - graining transitions between ca . at first blush , it seems that coarse - graining transitions should be rare because finding valid projection operators is an over constrained problem . this was our initial intuition when we first attempted to coarse - grain ca . to our surprise we found that many ca can undergo coarse - graining transitions . a more careful investigation of the above question suggests that finding valid projection operators is possible because of the structure of the rules which govern the large scale dynamics . these large scale rules are update functions for supercells , whose tables can be computed directly from the single cell update function . they thus contain the same amount of information as the single cell rule . their size however grows with the supercell size and therefore they have vanishing kolmogorov complexities . in other words , the large scale update functions are highly structured objects . they contain many regularities which can be used for finding valid projection operators . we did not give a formal proof for this statement but provided a strong experimental evidence . in our experiments we discovered that the probability to find a valid projection is a universal function of the kolmogorov complexity of the supercell update rule . this universal probability function varies from zero at large kolmogorov complexity ( small supercells ) to one at small kolmogorov complexity ( large supercells ) . it is therefore very likely that we find many coarse - graining transitions when we go to large enough scales . our interpretation of the above results is that of emergence . when we go to large enough scales we are likely to find dynamically identifiable large scale patterns . these patterns are emergent ( or self organized ) because they do not explicitly exist in the original single cell rules . the large scale patterns are forced upon the system by the lack of information . namely , the system ( the update rule , not the cell lattice ) does not contain enough information to be complex at large scales . finding a projection operator is one specific type of an over constrained problem . motivated by our results we looked into other types of over constrained problems . the satisfyability@xcite problem ( k - sat ) is a generalized ( np complete ) form of constraint satisfaction system . we generated random 3-sat instances with different number of variables deep in the un - sat region of parameter space . the generated instances however were not completely random and were generated by generating functions . the generating functions controlled the instance s kolmogorov complexity , in the same way that we used in section [ proj_prob_with_bounded_k ] . we found@xcite that the probability for these instances to be satisfiable obeys the same universal probability function of eq . ( [ rproj_fit ] ) . it will be interesting to understand the origin of this universality and its implications . in this work , we have restricted ourselves to deal with ca because it is relatively easy to look for valid projection operators for them . a greater ( and more practical ) challenge will now be to try and coarse - grain more sophisticated dynamical systems such as probabilistic ca , coupled maps and partial differential equations . these types of systems are among the main work horses of scientific modelling , and being able to coarse - grain them will be very useful , and is a topic of current research , e.g. in material science@xcite . it will be interesting to see if one can derive an emergence length scale for those systems like the one we found for garden of eden " sequences in ca ( section [ gardensofeden ] ) . such an emergence length scale can assist in finding valid projection operators by narrowing the search to a particular scale . ng wishes to thank stephen wolfram for numerous useful discussions and his encouragement of this research project . ni wishes to thank david mukamel for his help and advice . this work was partially supported by the national science foundation through grant nsf - dmr-99 - 70690 ( ng ) and by the national aeronautics and space administration through grant nag8 - 1657 .	we study the predictability of emergent phenomena in complex systems . using nearest neighbor , one - dimensional cellular automata ( ca ) as an example , we show how to construct local coarse - grained descriptions of ca in all classes of wolfram s classification . the resulting coarse - grained ca that we construct are capable of emulating the large - scale behavior of the original systems without accounting for small - scale details . several ca that can be coarse - grained by this construction are known to be universal turing machines ; they can emulate any ca or other computing devices and are therefore undecidable . we thus show that because in practice one only seeks coarse - grained information , complex physical systems can be predictable and even decidable at some level of description . the renormalization group flows that we construct induce a hierarchy of ca rules . this hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method . finally we argue that the large scale dynamics of ca can be very simple , at least when measured by the kolmogorov complexity of the large scale update rule , and moreover exhibits a novel scaling law . we show that because of this large - scale simplicity , the probability of finding a coarse - grained description of ca approaches unity as one goes to increasingly coarser scales . we interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics .	19,764	358
the first gravitationally lensed quasar @xcite has been discovered more than 30 years ago , turning gravitational lensing from an obscure theoretical field into a mainstream observational one . more than 100 strongly lensed quasars have been discovered to date , and it has convincingly been demonstrated that these objects provide insights into various topics in astrophysics and cosmology , as well as being a unique tool for studying the dark universe . applications include the study of the quasar host galaxies at high redshift ( e.g. , * ? ? ? * ) , dark matter substructures and luminous satellites ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the structure and evolution of massive galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and microlensing applied in the study of the structure of quasar accretion disk ( e.g. , * ? ? ? * ; * ? ? ? * ) , broad line regions(e.g . ? * ; * ? ? ? * ) , as well as to measure the stellar mass fractions in the lens ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? following early work by e.g. @xcite , analyses of statistically well - defined samples of lensed quasars ( i.e. samples in which lens candidates are selected by a homogeneous method whose completeness is known ) can now constrain the cosmological constant / dark energy by comparing the number and distribution of image separation of lensed quasars with theoretical models ( e.g. , * ? ? ? time delay measurements between quasar images constrain the hubble constant free of the calibration in the cosmic distance ladder ( e.g. , * ? ? ? finally , the distribution of lensed image separations , from galaxy to cluster mass scales , reflects the hierarchical structure formation and the effects of baryon cooling ( e.g. , * ? ? ? the sloan digital sky survey quasar lens search ( sqls ; * ? ? ? * ; * ? ? ? * ) is a systematic survey for lensed quasars , aiming to construct a large sample of gravitationally lensed quasars at optical wavelengths . it relies on the large homogeneous sample of spectroscopically - confirmed quasars from the sloan digital sky survey ( sdss ; * ? ? ? the techniques employed by the sqls to identify lensed quasar candidates are described in the references above . we followed up all candidates with deeper imaging ( usually with the university of hawaii 2.2 m telescope ; uh88 ) to detect the lensing galaxy . we then performed follow - up spectroscopy of the most promising candidates , to confirm their lensing nature . sqls is at present the prominent search for lensed quasars in the optical , comprising of 62 lensed quasars to date ( december 2014 ) , @xmath12/3 of which are new discoveries ( @xcite ) . it has also produced the largest statistically complete sample of lensed quasars ( 26 objects ; @xcite ) . a disadvantage of sqls , like other ground - based optical strong lens surveys , is its poor detail in imaging lensed quasars . even when performing follow - up observations with the uh88 telescope , the pixel scale @xmath2 is large , and the seeing @xmath3 is similar to the image separation of a galaxy - scale strong lens ( @xmath4 ) . therefore , high - resolution imaging of these quasar lenses is the key to turning each lens into a highly useful astrophysical and cosmological probe . this is necessary for obtaining accurate relative astrometry and point / surface photometry for the quasar images , lensing galaxy , and quasar host galaxy ( in case the latter is detected ) , which are used to constrain both the light and the mass distribution in these systems . in the following , we highlight three of the applications enabled by the high - resolution images of a large sample of objects , such as the sample provided by our work . _ estimating the hubble constant from high resolution and time delay measurements : _ although monitoring observations , for which only relative fluxes are relevant , can be performed with small telescopes aided by image deconvolution , to determine time delays measurements between multiple images , ( e.g. , * ? * ; * ? ? ? * ) , high resolution single epoch observations are still required to construct an accurate lens model ( e.g. , * ? ? ? * ) . as time delays are currently being measured by the cosmological monitoring of gravitational lenses ( cosmograil ; * * ) for many of the sqls lenses , high resolution data is in demand . for example , @xcite obtained high resolution images of a four - image lensed quasar with an early adaptive optics systems , resulting in a relative lens galaxy position three times more precise than before , which allowed to measure the hubble constant two times more precisely than in previous studies . _ quasar galaxy hosts and the correlation with @xmath5 : _ the tight correlations found between the masses of supermassive black holes @xmath5 and overall properties of the host galaxy bulges , such as velocity dispersion , luminosity and stellar mass ( e.g. , * ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) suggest that the black hole growth is coupled to the galaxy evolution . these correlations have been established for local galaxies , based on spatially resolved kinematics . since spatially resolved kinematics are very difficult to obtain at @xmath6 , the most straightforward correlation that can be studied at large redshifts is the one between @xmath5 and the bulge galaxy luminosity . however , in this case agns or quasars must be employed , as the nuclear activity allows the estimation of @xmath5 using the virial technique ( e.g. , * ? ? ? * ; * ? ? ? the difficulty then lies in accurately decomposing the faint quasar host , which is subject to the cosmological surface brightness dimming , from the bright nuclear source . as was demonstrated by @xcite , this task is facilitated for gravitationally lensed quasars by two effects . first , strong gravitational lensing produces natural magnification typically by several factors , by increasing the total flux and the apparent size of the host galaxy , while conserving surface brightness . second , lensing distorts the host into an arc and gives it a dramatically different morphology from that of the point spread function ( psf ) , making it significantly less susceptible to systematic problems in the psf model ( an inaccurately known psf ) . high - resolution observations are paramount in measuring the surface brightness of the lensed host galaxy . _ the ellipticity and orientation of mass and light in the lensing galaxies : _ an ongoing question is to what extent there are correlations between the azimuthal luminous profile of the lens galaxies and their overall ( dark + luminous ) mass profile . strong lensing provides a unique tool for testing these correlations . there exists a known correlation between the mass and light orientation , within @xmath7 deg ( e.g. , * ? ? ? * ; * ? ? ? it is not presently known however if the scatter in the relation is intrinsic , or due to measurement errors . on the other hand , there is less consensus on a correlation between the mass and light ellipticity . for example , such a tight correlation was found by @xcite for the sloan lens acs survey ( slacs ; * ? ? ? * ) sample of galaxy - galaxy lenses ( i.e. foreground galaxies acting as strong lenses for background galaxies ) , although @xcite did not find a correlation for a selected subsample of the slacs lenses . a correlation was also found for the sl2s galaxy - galaxy lenses @xcite , but has significantly more scatter . there are also recent results from different samples ( e.g. , * ? ? ? * ; * ? ? ? the difference between the slacs and sl2s lenses is that the former lie in an environment with significantly less average external shear @xmath8 , and also that they have smaller einstein radius @xmath9 relative to the characteristic scale ( effective radius ) of the galaxy @xmath10 , resulting in the more relevant role played by stellar mass in defining the potential within the critical curve ( @xmath11 , 0.5 for sl2s and slacs , respectively ) . for much larger values of @xmath12 , studies based on weak lensing ( e.g. * ? ? ? * ) found that the dark matter haloes , which dominate the mass profile at these radii , are flatter than the luminous profile , in agreement with theoretical expectations . for the lenses of strongly lensed quasars , initial hubble space telescope ( @xmath13 ) observations have failed to find a correlation between the mass and light ellipticity @xcite . more recent results employing a uniform analysis technique and deconvolution of @xmath13 data are presented in @xcite and show that there is a correlation for their sample of four - image lensed quasars ( quads ) . the slacs lenses and the lenses of strongly lensed quasars are known to probe different populations of lenses , with lensed quasars typically lying in richer environments ( e.g. , * ? ? ? in addition , for most lensed quasars @xmath9 is on the order of a few @xmath10 , therefore probing an intermediate range . the new correlation found by @xcite , after eliminating the outliers , holds only for ellipticities @xmath14 . for these small ellipticities , as the authors note , one possible bias resulting in the correlation is that the effect of a rounder dark matter halo on the total mass distribution may be harder to detect . the goal of this work is to enhance the value of the sqls gravitationally lensed quasar sample through the use of high - resolution observations . among the 62 lensed quasars in the sqls sample , only @xmath15 have available high - resolution observations , mainly supplied by @xmath13 . however , recent progress of adaptive optics ( ao ) , especially the use of laser guide stars ( lgs ) , makes it possible to obtain high - resolution images using ground - based telescopes . due to the smaller diffraction limit , near - infrared imaging with subaru telescope ao can potentially achieve _ three times _ the spatial resolution of @xmath13 , while also employing a finer pixel scale . ao has previously been used in the literature in the study of gravitationally lensed quasars . @xcite , @xcite and @xcite used ao to study galaxy scale two - image lensed quasars ( doubles ) . @xcite discovered additional lensing galaxies in known lensed quasars , using ao . @xcite observed a quad in order to look for substructure , which @xcite was successful in detecting for another quad . in addition , @xcite and @xcite have used ao to study the arcs of strongly lensed galaxies , reconstruct the source , or study in high detail the lensing galaxy . however , these have been isolated studies focused on single systems . the observational campaign described in this paper is the largest dedicated study of a large sample of lensed quasars with ao . in section [ section : obsall ] we present our observing strategy and describe acquired data including data reduction . in section [ section : technique ] we present the morphological modelling technique we employed , and in section [ section : lensmodels ] the mass modelling technique . we continue with a comparison of our results with those found for the same systems in the literature ( section [ sect : discuss ] ) , and a concluding discussion of the technique we used , in section [ sect : discutionoverall ] . section [ section : hostfacts ] shows results obtained on the quasar host galaxies . finally , section [ section : lightmass ] studies the lens environment and the relation between mass and light for the systems in our sample , and section [ sect : concl ] concludes this work . in the main appendix , we describe in detail the analysis of each individual system . throughout this work , the concordance cosmology with @xmath16 , @xmath17 , @xmath18 and @xmath19 is assumed . for sdss j1001 + 5027 and sdss j1206 + 4332 , where time delay measurements are available , the @xcite result , @xmath20 , is assumed . magnitudes are given in the vega system . all object coordinates assume j2000 . all observed images and residual plots are in logarithmic scale . sky coverage is a limiting factor when using ao , as all targets must be located close to a tip - tilt ( tt ) star of suitable brightness in order to perform ao correction , even in the case that lgs is used . subaru telescope however presently has the most relaxed constraints on tt star , among comparable systems on 8 - 10 m telescopes ( section [ section : ircsao ] ) . this makes it an ideal tool to use in the observational campaign of the sqls quasars . indeed , 54 of the 62 sqls quasars are accessible to the subaru telescope ao system . the ao correction functions better at longer wavelengths ( e.g. * ? ? ? * ) , making @xmath21-band the natural choice among the widely used @xmath22 near - infrared bands . an additional reason to use @xmath21-band in the study of quasars and their host galaxies is that for redshifts @xmath23 , @xmath21-band falls in a wavelength region where the quasar host galaxy is brightest compared to the nuclear source ( figure [ fig : qsogal ] ) . moreover , for the purpose of modelling gravitationally lensed quasars , extinction , intrinsic variability and microlensing are all weaker at progressively longer wavelengths , contaminating the true image flux ratio less ( e.g. * ? ? ? * ) . however , exposure times in the @xmath21-band are limited by the strong sky background emission . on the other hand , @xmath24-band provides a good compromise between background level , sensitivity , and ao performance , and we therefore used this band throughout the observation campaign . band at redshifts 0 and 2.5 marked . the composite quasar spectrum is from @xcite . for the galaxy template , the mean spectrum of local galaxies from @xcite is extrapolated using the evolutionary models of @xcite . [ fig : qsogal],width=321 ] as one of the goals of the campaign is to measure accurate lens galaxy morphologies and potentially resolve the quasar host galaxies from the point - like nuclear source , a good knowledge of the psf is essential . for the majority of the systems , we used the sdss casjobs search query to identify bright psf and guide star pairs . these were selected such that the guide stars of the psf and the target have similar brightness , and the separations between the target and its guide star , as well as the psf and its respective guide star , are similar . the psfs are close on the sky to the target ( within @xmath25 ) , and they were observed immediately before or after the target , in the same natural guide star ( ngs ) or lgs mode . this was done to ensure as much as possible that the atmospherical turbulence characteristics and therefore the ao correction is relatively constant between the psf and the target . however , as we will discuss below , these separately observed psf stars turn out not to be suitable for the accurate characterisation of the target image psfs . sdss j1515 + 1511 and sdss j1002 + 4449 are the only observed systems that have a bright star in the detector field of view ( fov ) , which can potentially be used as psf . here we centred the fov , and therefore the lgs , between the star and the target . this was done , again , so that the ao correction at the location of the target and the psf star is similar . the ao imaging observations were performed with the infrared camera and spectrograph ( ircs ; * ? ? ? * ) at the subaru telescope , along with the laser guide star adaptive optics system ( lgs@xmath26ao188 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ircs uses two @xmath27 alladin iii arrays for imaging and spectroscopy , and in imaging mode provides two plate scales , of 52 and 20 mas per pixel . after the distortion correction described in section [ section : distort ] , the pixel scales are @xmath28 and @xmath29 , respectively . the available fovs are @xmath30 and @xmath31 , respectively . ao188 uses a curvature sensor with 188 control elements , operating at 2000 hz , and a 188 element bimorph deformable mirror ( bim188 ) . it operates at the nasmyth focus of the subaru telescope . both ngs and lgs modes are available . the recommended guide star constraints in ngs mode are brightness larger than @xmath32 mag , within @xmath33 from the target . the lgs mode uses an artificial sodium laser guide star @xcite for high - order wavefront sensing . the lgs is typically less bright than @xmath34 mag in @xmath35-band . the recommended tip - tilt star constraints in lgs mode are brightness larger than @xmath36 mag , within @xmath37 ( acceptable up to @xmath38 ) from the target . more information about ao188 is provided on the instrument web page . the observations were performed between 2011 - 2014 . in total , @xmath39 nights were assigned for the present campaign , about half of which were lost due to telescope / instrument trouble or cloud coverage . observation priority was given based on the scientific interest of each target and the degree to which the targets are suitable for ao observations in terms of tt star brightness and separation , and distance to zenith / airmass . the 28 objects that were successfully observed , the observation modes , exposure times , filters , pixel scales , typical airmass , photometric stars and observation dates are given in table [ tab : followup - data ] . two of the objects , sdss j0820 + 0812 and sdss j1206 + 4332 , were observed as back - up , without ao . all observations were performed with 5 or 9-point dithering , in order to remove bad pixels and cosmic rays , and to allow for flat frame and sky frame creation from the data . although dome flats were obtained for some observations , these have proven to be of inferior quality . although the targets were chosen to be accessible to ao , many of these border the recommended limit in terms of the tt brightness and separation . as such , low strehl ratios @xmath40 ( table [ tab : analpsf ] ) were typically obtained . the 52 mas pixel scale mode was typically used , in order to increase the signal - to - noise ( s / n ) and not to avoid significantly oversampled psfs . the list of observed tt and psf stars is given in table [ tab : ao ] . for 12 objects , psf star observations were skipped because of rapidly changing atmospheric conditions . typical seeing during the observations can be inferred from table [ tab : analpsf ] , and was generally @xmath41 . data reduction was performed with iraf , using the ircs imgred package designed to reduce data obtained with ircs . the reduction consisted of the following steps : 1 . each frame was checked for dark patterns by dividing it to the next observed frame . also , the number of counts at the location of the target was checked for linearity . all targets are well inside linearity limit , except for the core of the star inside the fov for sdss j0743 + 2457 ( within 5% ) , and cores of the bright quasar images in sdss j0904 + 1512 ( within 3% ) , sdss j1322 + 1052 ( within 2% ) and sdss j1353 + 1138 ( within 4% ) . as persistent fringe artifacts were found in the dome flats , sky flats were created by masking bad pixels and bright objects , then median - combining the dithered frames which were divided by the mean count value . each raw frame was corrected for different pixel sensitivities by division to the sky flat . the geometric distortion correction map described in section [ section : distort ] was applied . sky - background frames were generated by median - combining the flat - fielded frames . as the near - infrared sky is highly time - variable , the sky - background frames were generated for each dither sequence individually . pixel values for the masked pixels were interpolated from the combined ones . before subtracting the sky frame , each frame was normalized so that their median values match each other . position offsets between the images were obtained via cross - correlations between regions that contain bright objects , typically the targets . cosmic rays falling in the regions used to perform the cross - correlations were manually masked . the full - width at half - maximum ( fwhm ) of star - like objects in each frame was measured by fitting a moffat function to the radial profile with the iraf imexam task , and the frames with significantly different values were discarded . the frames were average - combined using rejection algorithms ( average sigma clipping or clipping based on the ccd parameters ) . pixel binning was performed for a few frames with significantly oversampled psf , as mentioned in appendix [ section : objectsdescript ] . photometric zero - point calibration was performed , typically using standard stars . all objects were corrected for galactic extinction @xcite , and atmospheric extinction relative to the standard star . it must be noted that the standard star catalogue magnitudes at @xmath42band were used to calibrate the photometry at @xmath43band . however , the expected differences based on interpolation are small ( @xmath44 mag ) . after reducing the data , it became clear that none of the separately - observed psf stars represent suitable psfs to model the corresponding systems . this was also concluded by other studies employing separately observed psf calibrators ( e.g. , * ? ? ? the reason is psf variability , both in the science targets and the separately - observed stars . this is caused by the rapid change of the atmospheric turbulence characteristics , which induce different responses from the ao system . figure [ fig:0946psf ] shows the time variability of the core component fwhm in the psf for both the target and the separately - observed star in sdss j0946 + 1835 , as well as the outstanding residuals obtained if the star is used as a psf to model the system . it is imperative that we construct a well - characterised psf for each system , in order to reach the science goal , i.e. obtain an accurate astrometrical , photometrical and morphological characterisation of the multiple components of each observed system . in order to guide the reader , here we provide a short summary of the following subsections , in which we illustrate in detail our psf reconstruction and modelling technique . in section [ aopsf ] we give general considerations on the ao psf . in section [ section : newmethod ] we briefly mention the ao psf modelling techniques encountered in the literature . in section [ hostlens ] we introduce our technique of modelling analytically all components of a gravitationally lensed quasar simultaneously , as their light profiles overlap , and derive the most suitable analytical profile . we describe the implementation of this technique , where we take special care to properly explore the parameter space . in section [ subsection : hybrid ] we show that for selected systems we can go beyond an analytical psf , and we derive a hybrid psf that specifically accounts for the non - analytical components . in the next sections ( [ section : hostfit ] and [ section : simulate ] , respectively ) , we describe how we need to modify out technique to model systems where we detect a quasar host galaxy , as its profile fitting requires a mass model . we then insure that the error estimates for each derived parameter accounts for systematics introduced by our modelling technique . finally , in section [ section : distort ] we address the issue of how the instrument field distortion impacts our derived astrometry . for a seeing - limited image , spatial resolution is usually characterised by the fwhm of a point - like stellar profile . this characterisation becomes unreliable near the diffraction limit , as the fwhm measurement is complicated due to the diffraction rings . in the diffraction limit , the shape of the psf is described by a two - dimensional airy function ( ignoring the telescope spider ) , with the first diffraction ring at an angular distance of @xmath45 from the centre ; here @xmath46 denotes wavelength , and @xmath47 is the diameter of the telescope mirror . in this case the strehl ratio @xcite is used , and is defined as the ratio of the intensity at the peak of the observed seeing disc divided by the intensity at the peak of the theoretical airy disc . in general the ao psf is described as two components : a nearly diffraction - limited core of fwhm @xmath48 , and a seeing limited halo / wing of fwhm @xmath49 , where @xmath50 is fried s parameter @xcite . the core is typically approximated by a gaussian ( e.g. , * ? ? ? * ) , whereas the halo is approximated by a moffat profile ( * ? ? ? * section [ hostlens ] ) , which has a more extended wing than a gaussian . as a product of the atmosphere as well as the ao system , the ao psf exhibits both temporal and spatial variability . the spatial variability ( anisoplanatism ) causes the psf to vary across the fov . in lgs ao , there are three sources of anisoplanatism ( e.g. , * ? ? ? angular anisokinetism ( or tip - tilt angular anisoplanatism ) results from the difference between the tip - tilt component of the wavefronts of the tt star and the science object , and is radially symmetric around the tt star . focal anisoplanatism ( the cone effect ) occurs because the lgs ( located at a finite altitude ) samples the cone of turbulence between the lgs and the telescope , while the turbulence experienced by the science object is distributed in a cylinder . finally , angular anisoplanatism results from the difference in the higher order wavefront terms between lgs and the science object . in the course of this work , the psf is assumed not to vary spatially for objects in close proximity of each other , such as the multiple components of a gravitationally lensed system . indeed , these objects are separated by @xmath51 , much less than the isoplanatic angle in the lgs mode . there are several examples in the literature of modelling ao lensed quasars without an a priori known psf . these made use of the typical structure of a lensed quasar : two or four point - like images with a lensing galaxy in between . while the quasar point - source is safely unresolved , since it is typically located at high redshift , the quasar host galaxy may or may not be resolved . in all cases , only doubles where the host galaxy is unresolved were modelled in the literature , and in the case of quads , the host was actively removed during modelling , without estimating any of its physical parameters . @xcite modelled the double sdss j0806 + 2006 with the psf estimated from the more isolated , brighter image , and employing image deconvolution . they do not mention however if any systematics are introduced by this technique . @xcite however argued for the use of convolution against deconvolution , since the psf is poorly defined , the s / n is limited , and smooth extended emission ( in the case of hosts ) makes the results uncertain . when there is a physical model to test , they advise that the best approach is to convolve that model with the best estimate of the psf and then compare the result to the observations . @xcite took advantage of the cross configuration of the quad sdss j0924 + 0219 , where the host galaxy has a different morphology at the location of each image , and estimated the psf by combining the superposed and normalised images , in an iterative procedure . @xcite analysed the quad b0128 + 437 by iteratively fitting a psf modelled as three gaussian components , together with the lensing galaxy modelled with a sersic profile . @xcite modelled the psf for a lensed double by using a hybrid composed of the observed brightness distribution of the bright image at the centre , and analytical wings . there is enough information ( e.g. , * ? ? ? * ) in the multiple images of a lensed system to reconstruct the psf . in this paper , we combined the approach of @xcite and @xcite into a new technique , by adopting an analytical psf model and simultaneously optimising for the global psf parameters as well as the individual objects comprising the system : multiple point sources , and the lensing galaxy modelled as a sersic profile convolved with the psf . where possible , we take an additional step where we refine this fit by using a hybrid psf ( section [ subsection : hybrid ] ) . the optimisation is performed using a downhill simplex method @xcite to find the minimum goodness - of - fit @xmath52 in parameter space , and is implemented in a new software , hostlens , which was developed by one of us ( m. oguri ) . hostlens combines the functionality of galfit with that of glafic @xcite . like galfit , it allows the simultaneous fitting of both point and extended sources , but in addition can simultaneously optimise the parameters of an analytical psf . like glafic , it can fit an observed lensing configuration with a mass model , while simultaneously fitting analytically the morphology of an extended lensed source , such as a quasar host galaxy ( section [ section : hostfit ] ) . we note that most of the functionalities of hostlens are currently implemented in glafic . we considered several analytical psf models to use with hostlens . according to section [ aopsf ] , the ao psf consists of two components , one describing the compact psf core , and the other one the extended seeing wings . @xcite used a single moffat profile , whose wings are more extended than that of a gaussian , to approximate the full ao psf . @xcite stated that the psf is generally matched quite well by the sum of a narrow gaussian for the core , and either a moffat or another gaussian for the halo . @xcite used a gaussian for the core and an exponential function for the wings . @xcite , mentioned above , used three gaussians : one component represents the diffraction - limited core , another represents the seeing - limited diffuse psf , and the third encodes structure in the psf . in order to find the most appropriate model , we fitted the majority of bright stars observed during the observation runs ( i.e. the psf stars ) with analytical profiles . two of these fits are shown in figures [ fig : starresid2d ] and [ fig : starresid1d ] . the first figure shows a two - dimensional fit and the subsequent residuals after subtracting elliptical but concentric 1 moffat , 1 gaussian + 1 moffat , 2 moffat , and 3 gaussian profiles , respectively . the second shows the fitted radial plot corresponding to the same models . as can be seen , both in terms of the formal @xmath53d.o.f . , as well as in terms of the appearance of residuals , and radial profile matches , the 2 moffat profile provides the best fit . this is found in the overwhelming majority of fitted stars , although in some cases , such as the second example shown , the 1 gaussian + 1 moffat profile provides a similarly good fit . the residuals in the other models all show a concentric halo , and do not match the overall radial profiles . on the other hand , none of the models ( including 2 moffat ) is able to model the non - analytical components close to the centre . this entirely empirical analysis resulted in the decision to use the 2 moffat profile as the analytical psf for hostlens . the better fit of a moffat core , which is more extended than a gaussian , may be due to the very small strehl ratios @xmath54% typically obtained in the observations . indeed , the typical core fwhm is @xmath55 , about three times that of the diffraction - limited core . where @xmath57 is the elliptical radial distance , @xmath58 is the distance to the centre along the semimajor ( semiminor ) axis and @xmath59 is the ratio of the semiminor to the semimajor axis ; @xmath60 is a parameter specifying the shape of the profile , related to the core - to - halo flux ratio . the moffat profile results into a gaussian profile as the @xmath61 case . in hostlens , the psf is modelled with 9 analytical parameters : two for each of the four components of an individual moffat profile : fwhm , ellipticity , position angle and @xmath60 , as well as one parameter specifying the relative distribution of flux , flux1/(flux1 + flux2 ) . the two moffat profiles are concentric . each point source is modelled with three parameters : the positions along x and y , and the total flux . in addition , every sersic profile has ellipticity , orientation , effective radius and sersic index parameters . finally , there is a parameter for the sky value . therefore , a typical two - image lensed system requires 24 parameters to model analytically . we modelled each system with hostlens , at least at an initial stage . the derived analytical parameters of all psfs are shown in table [ tab : analpsf ] . we selected suitably large cut regions ( typically @xmath62 ) around the targets , and masked all objects that were not simultaneously fitted by hostlens , in order to allow reliable sky value estimates and model fitting . due to the large number of parameters and the possibility that the parameters might get trapped in a local @xmath52 minimum , we chose 500 sets of different initial parameter values from a range of plausible values , e.g. from a flat distribution in the case of position angles , and a gaussian distribution around the iraf imexam - derived value for the core fwhm component . we chose the model with the lowest value of @xmath52 ( we also checked the second - best model to be similar ) , and in addition we performed a parameter search around it using markov - chain monte carlo ( mcmc ) , for the purpose of both refining the parameter values , and estimating confidence intervals ( error bars ) . we ran ten mcmc chains , and removed a fraction of @xmath63 of the points from the head of each chain ( the burn - in stage " points ; i.e sensitive to the start values ) . to test the convergence of these chains , we compared the variance of the distribution of points in each chain to the variance of the combined distribution , following @xcite . after applying a smoothing technique , the combined chains provided 68% confidence intervals on the analytical parameters ( @xmath64 in the gaussian approximation ) . throughout this work , unless otherwise stated , for all the systems where we did not detect a quasar host galaxy , and which we fitted with a hostlens - produced analytical psf , we also checked the modelling results for consistency , with galfit . in order to produce with galfit error bars that account for the uncertainties of the analytical psf , we produced 500 realisations of the hostlens - produced psf , by drawing at random from the probability distribution of the analytical parameters of the psf , obtained during the mcmc runs above ; we then ran galfit with each of the 500 psfs , and took the standard deviations of the resulting parameters as an error estimate . we did not use galfit for the systems with host galaxy detections , as it lacks the functionality to fit this component ( see section [ section : hostfit ] ) . we note that hostlens and glafic measure the effective radius of a sersic profile using a circularised radius , while galfit measures the radius along the semi - major axis . the effective radii measured in this paper are reported following the galfit definition . for the lensing galaxies , we considered both an unconstrained sersic index @xmath65 and @xmath66 ( table [ tab : lensmorphology ] ) , in order to ease comparison with other samples of lenses where the light profile is modelled as a de vaucouleurs profile ( see * ? ? ? * and references therein ) . the observed systems can be classified into two categories , according to the degree in which the bright image a is separated from the other objects , and the visual appearance of the modelling residuals . for systems with small separation where the non - analytical residuals at the centre clearly overlap with a large fraction of the galaxy light ( e.g. sdss j1254 + 2235 and sdss j1334 + 3315 ) , as well as for objects where , due to the low s / n , no conspicuous core residuals are seen , data modelling stops as described above . for the other systems , it is possible to model away the faint quasar image and the lensing galaxy using the analytical hostlens profiles , and obtain a clean image a to be used as the `` actual psf '' ( panels a - g in figure [ fig : hybrid ] ) . the advantage in this approach is that the estimated psf is the `` true '' psf ( accounting for all non - analytical features ) , albeit a noisy one . next , we tested whether this psf produces better statistical results . a simple comparison with the analytical fit in terms of @xmath52 can not be used , as image a is fitted perfectly by design . therefore , we compared the standard deviation of the residual pixel values in a region containing the rest of the objects ( typically b and g ) . we found that in almost all cases , the analytical psf produced better results . this is due to large noise in the wing of the `` actual psf '' . therefore , we used another approach , previously employed by @xcite . this consists in using the observed brightness distribution of the bright image only in its central regions , and replacing the wings after a certain cut radius with analytical wings from the 2 moffat fit ( panels h - i in figure [ fig : hybrid ] ) . this `` hybrid psf '' approach succeeded in producing fits at least as good as the analytical psf in terms of the standard deviation of the residuals . we considered two sources of systematics for the hybrid psf approach . first , because the contributions from b and g are initially modelled using the analytical psf , they may be improperly subtracted at the location of a , and therefore affect the hybrid psf estimation . this can be checked by building the hybrid psf in an iterative approach , where in each subsequent step the hybrid psf is used to model away b and g , resulting in a better hybrid psf estimate . we did this for sdss j1405 + 0959 and obtained that subsequent psfs are virtually identical . this is expected , as we applied the hybrid psf technique only to systems with isolated bright images . secondly , the final models may depend on the size of the hybrid cut region . we chose the original cut radius as the radius at which all non - analytical components disappear , of @xmath7 pixels . for all hybrid psf systems ( excluding the ones with a quasar host galaxy detection , as described below , or otherwise specified ) , we included into the error budget the scatter between the results obtained from this cut size , and another one @xmath67 larger . we used the analytical and/or hybrid psf approaches described above for all systems where a quasar host galaxy was not detected . table [ tab : morphologytechnique ] shows whether the analytical or the hybrid psf was used to model each system . a novel undertaking in this paper is the fitting of quasar host galaxies , for several systems , without an a - priori known psf . this is only possible for gravitationally lensed quasars , because the host galaxies are tangentially stretched around the critical curves by the lensing effect , resulting in arcs that are distinct from the psf shape , and can be modelled analytically ( figure [ fig : simulatedsystem ] ) . in order to model the spatially extended flux distribution of the lensed images , it is necessary to model both the lens mass distribution and the extended source distribution . this is done using the approach described in @xcite , and makes use of the fact that gravitational lensing conserves surface brightness . hostlens was designed with the functionality to fit host galaxies modelled with a sersic profile . when fitting a host galaxy with hostlens , the parameters typically used to model the individual images are replaced by a single source profile . this profile consists of the parameters of a sersic model , its flux as well as the flux of the central point source , and the lensing parameters : the einstein radius , shear / ellipticity and their orientation angles , where a singular isothermal mass model is assumed ( section [ section : lensmodels ] ) . the luminous lens galaxy is fitted simultaneously , as before , with a sersic profile . there are five cases where the quasar host galaxy was clearly detected and fitted . we define a clear detection as a system for which there are clear visible arcs similar in orientation to the critical curves of the lensing model , and which are convincingly removed by fitting a host galaxy . in the case of two systems with host detections , sdss j0904 + 1512 and sdss j1322 + 1052 , there were outstanding residuals when using an analytical psf , and therefore we employed a hybrid psf . this was possible through an iterative process , where an analytical psf is initially used to obtain a first rough approximation of the host profile . the host galaxy is then subtracted from the original observed frame , and subsequently a hybrid psf is created as described in section [ subsection : hybrid ] . this hybrid psf is used in the next step of the iteration to refit the original system , and refine the host model . subsequent steps lead to new hybrid psfs . the iteration stops once the @xmath52 stops decreasing . to summarise , the general approach to the morphological modelling of the imaging data in this paper , whether the quasar host galaxy is detected or not , is described in the flowchart of figure [ fig : flowchart ] . to test the reliability of the morphological fitting with hostlens using the analytical or hybrid psf methods , we performed a series of simulations for each system . in these simulations , we used a separately observed psf star , or a star in the fov that is the most similar in terms of analytical parameters to the psf of the respective system ( prioritising for the core fwhm parameter ) . table [ tab : morphologytechnique ] shows which star was used as psf template for each system . with hostlens and/or galfit , we simulated each system using this corresponding psf template , and the best - fit astrometry / morphology / photometry parameters we derived when modelling that particular observed system . such a simulated system is shown in figure [ fig : simulatedsystem ] . from the final science frame of each observed ( i.e. real ) system , we identified 100 blank sky cuts , and added them as noise realisations to the simulated system . we then remodelled each of the 100 simulated noisy systems in the same way as we did the observed system . the scatter in the resulting 100 values for each fitted parameter , with respect to the known input value , represents an error estimate . the only exception is for the case when we simulated the system using the best - fit parameters , but remodelled it using sersic index @xmath68 . in that case , we considered the scatter around the mean fitted value ( table [ tab : lensmorphology ] ) . we used two time - saving approximations in remodelling the simulated systems . first , we did not perform a search starting from random values in the parameter space for each individual simulated system , as we did for the original system , but instead we started the fit from the known ( correct ) values . second , in cases where we used a hybrid psf , we did not create it in each of the 100 simulations from that very same image , but created it from only one of the simulations , and then used it as the hybrid psf to model all simulated frames . also , for the few cases when a hybrid psf is used to fit a detected host galaxy , we only performed a small number of simulations , as these are time consuming . these simulations can account for the systematics introduced by the analytical modelling of a non - analytical psf , or the use of hybrid psfs . the orientation of the detector was typically the same for the science objects and the psf stars , therefore diffraction patterns due to the telescope spider are similar . the noise cuts from the actual science frames , previously employed by ( e.g. * ? ? ? * ) , account for how the model parameters would depend on the existence of background objects , too faint to be visible , and are more realistic than artificially - generated noise . as final errors on astrometry / morphology / photometry , we used the largest error bars among those resulting from these simulations and all other methods described in the previous subsections . this is because the methods are not necessarily independent , and one typically dominates the error budget . this system is reported in @xcite as a two - image lensed quasar at @xmath69 , with an image separation @xmath70 . in the new ao data , the inferred image separation is @xmath71 , making this system the smallest - separation lensed quasar in the sqls . however , the lensing galaxy is undetected in the new observations ( figure [ fig:1128resid ] ) , with the residuals being identical whether a lens galaxy is modelled or not . without visual confirmation of improvement in the residuals , the extracted physical parameters of the lens are unreliable , as it is likely that noise is being fitted . considering the very small separation of this system and the fact that there are non - analytical residuals which can not be well fit , we conclude that the faint lensing galaxy , visible in @xcite , can not be reliably resolved from the bright quasar images with the current technique . the discovery of sdss j1320 + 1644 and subsequent data analysis based on non - ao data is reported in @xcite . this is a large - separation quasar pair , likely to be images of a lensed quasar source at @xmath72 , by a lensing galaxy group / cluster at @xmath73 . however the lensing nature of this object is not secure . @xcite noted that the most efficient way to test the lensing hypothesis is by performing ao observations sensitive enough to detect the quasar host galaxy , and search for evidence of lensed arcs . the acquired ao data was affected by cloud coverage , and as such did not reach the intended depth , being in fact shallower than the original non - ao data . even after @xmath74 pixel binning to increase s / n , there are no visible signs of a host galaxy detection ( figure [ fig:1320resid ] ) . both an analytical psf and a psf built on image a were used , leading to clean residuals for image b. the only new insight brought by the new ao data is on the quasar flux ratio . this is a / b @xmath75 , and is significantly larger than the values of 1.4 and 1.0 , considered in @xcite . while this has little impact on their inferred lens models , since the main model constraints were astrometric , it does impact their discussion on the intrinsic flux ratio . @xcite argued that , since typical changes due to intrinsic variability and microlensing are small in the near - infrared , the flux ratio @xmath76 observed in the spectra at long wavelengths and in the @xmath77bands is representative of the true flux ratio . an alternative hypothesis was that the flux ratio @xmath78 found at longer infrared wavelengths is more accurate , and the changes due to microlensing at shorter wavelengths occur on time scales longer than the @xmath79 year spanned by the observations available at that time . the new flux ratio supports this second hypothesis in the sense that larger flux variations do indeed occur on time scales of @xmath80 years , assuming , of course , that the gravitational lens hypothesis is correct for this system .	we present the results of an imaging observation campaign conducted with the subaru telescope adaptive optics system ( ircs+ao188 ) on 28 gravitationally lensed quasars and candidates ( 23 doubles , 1 quad , 1 possible triple and 3 candidates ) from the sdss quasar lens search . we develop a novel modelling technique that fits analytical and hybrid point spread functions ( psfs ) , while simultaneously measuring the relative astrometry , photometry , as well as the lens galaxy morphology . we account for systematics by simulating the observed systems using separately observed psf stars . the measured relative astrometry is comparable with that typically achieved with the hubble space telescope , even after marginalizing over the psf uncertainty . we model for the first time the quasar host galaxies in 5 systems , without a - priory knowledge of the psf , and show that their luminosities follow the known correlation with the mass of the supermassive black hole . for each system , we obtain mass models far more accurate than those previously published from low - resolution data , and we show that in our sample of lensing galaxies the observed light profile is more elliptical than the mass , for ellipticity @xmath0 . we also identify eight doubles for which the sources of external and internal shear are more reliably separated , and should therefore be prioritized in monitoring campaigns aimed at measuring time - delays in order to infer the hubble constant . [ firstpage ] adaptive optics gravitationally lensed quasars quasar host galaxies	13,217	395
neutrino masses , baryogenesis , dark matter and the acoustic peaks in the power spectrum of the cosmic microwave background ( cmb ) radiation require an extension of the standard model of particle physics . the supersymmetric standard model with right - handed neutrinos and spontaneously broken @xmath0@xmath1@xmath2 , the difference of baryon and lepton number , provides a minimal framework which can account for all these phenomena @xcite . @xmath0@xmath1@xmath2 breaking at the grand unification ( gut ) scale leads to an elegant explanation of the small neutrino masses via the seesaw mechanism and explains baryogenesis via leptogenesis @xcite . the lightest supersymmetric particle is an excellent candidate for dark matter @xcite and the spontaneous breaking of @xmath0@xmath1@xmath2 requires an extended scalar sector , which automatically yields hybrid inflation @xcite , explaining the inhomogeneities of the cmb . recently , we have suggested that the decay of a false vacuum of unbroken @xmath0@xmath1@xmath2 symmetry generates the initial conditions of the hot early universe : nonthermal and thermal processes produce an abundance of heavy neutrinos whose decays generate primordial entropy , baryon asymmetry via leptogenesis and gravitino dark matter from scatterings in the thermal bath @xcite . in this context , tachyonic preheating after hybrid inflation @xcite sets the stage for a matter dominated phase whose evolution is described by boltzmann equations , finally resulting in a radiation dominated phase . it is remarkable that the initial conditions of this radiation dominated phase are not free parameters but are determined by the parameters of a lagrangian , which in principle can be measured by particle physics experiments and astrophysical observations . our work is closely related to previous studies of thermal leptogenesis @xcite and nonthermal leptogenesis via inflaton decay @xcite , where the inflaton lifetime determines the reheating temperature . in supersymmetric models with global @xmath0@xmath1@xmath2 symmetry the scalar superpartner @xmath5 of the lightest heavy majorana neutrino @xmath6 can play the role of the inflaton in chaotic @xcite or hybrid @xcite inflation models . one of the main motivations for nonthermal leptogenesis has been that the ` gravitino problem ' for heavy unstable gravitinos @xcite can be avoided by means of a low reheating temperature . in the following we shall assume that the gravitino is the lightest superparticle . gravitino dark matter can then be thermally produced at a reheating temperature compatible with leptogenesis @xcite . the present work is an extension of ref . we discuss in detail the effect of all supersymmetric degrees of freedom on the reheating process and restrict the parameters of the lagrangian such that they are compatible with hybrid inflation and the production of cosmic strings during spontaneous symmetry breaking . this implies in particular that @xmath0@xmath1@xmath2 is broken at the gut scale . the consistency of hybrid inflation , leptogenesis and gravitino dark matter entails an interesting connection between the lightest neutrino mass @xmath7 and the gravitino mass @xmath8 . as we shall see , the final results for baryon asymmetry and dark matter are rather insensitive to the effects of superparticles and details of the reheating process . due to the restrictions on the parameter space compared to ref . @xcite the lower bound on the gravitino mass increases to about @xmath4 . the paper is organized as follows . in section [ sec_2 ] we briefly recall field content and superpotential of our model , in particular the froggatt - nielsen flavour structure on which our analysis is based . we then discuss the time - dependent masses of all particles during the spontaneous breaking of @xmath0@xmath1@xmath2 symmetry in the supersymmetric abelian higgs model , the restrictions of hybrid inflation and cosmic strings on the parameters , and the particle abundances produced during tachyonic preheating . section [ sec_tools ] deals with the time evolution after preheating and the required set of boltzmann equations for all particles and superparticles . the detailed description of the reheating process is given in section [ sec : example ] with emphasis on the various contributions to the abundance of @xmath6 neutrinos , the lightest of the heavy majorana neutrinos , whose decays eventually generate entropy and baryon asymmetry . particularly interesting is the emerging plateau of a reheating temperature which determines the final gravitino abundance . in section [ sec_parameterspace ] a systematic scan of the parameter space is carried out , and relations between neutrino and superparticle masses are determined . three appendices deal with important technical aspects : the full supersymmetric lagrangian for an abelian gauge theory in unitary gauge , which is used to describe the time - dependent @xmath0@xmath1@xmath2 breaking ( appendix [ app_sqed ] ) , @xmath9 violation in all supersymmetric @xmath10 scattering processes ( appendix [ app_cp ] ) and the definition of the reheating temperature ( appendix [ app : trh ] ) . our study is based on an extension of the minimal supersymmetric standard model ( mssm ) which offers solutions to a series of problems in particle physics and cosmology . its main features are right - handed neutrinos , a @xmath11 factor in the gauge group and three chiral superfields , needed for @xmath0@xmath1@xmath2 breaking and allowing for supersymmetric hybrid inflation . in this section , we give a review of this model , presented earlier in ref . @xcite , thereby focussing on the aspects which are especially relevant for this paper . a characteristic feature of the model is that inflation ends in a phase transition which breaks the extra @xmath12 symmetry . during this phase transition the system experiences the decay from the false into the true vacuum . at the same time , this phase transition is responsible for the production of entropy , matter and dark matter through tachyonic preheating and subsequent leptogenesis . finally , it yields masses for the right - handed neutrinos , thereby setting the stage for the seesaw mechanism , which can explain the observed light neutrino masses . the superpotential is given by @xmath13 where @xmath14 and @xmath15 are the chiral superfields containing the higgs field responsible for breaking @xmath0@xmath1@xmath2 , @xmath16 contains the inflaton , i.e. the scalar field driving inflation , and @xmath17 denote the superfields containing the charge conjugates of the right - handed neutrinos . in the following , we will refer to the components of @xmath14 , @xmath15 and @xmath16 as the symmetry breaking sector , whereas the components of @xmath17 form the neutrino sector . @xmath18 is the scale at which @xmath0@xmath1@xmath2 is broken . the @xmath0@xmath1@xmath2 charges are @xmath19 , @xmath20 , and @xmath21 . @xmath22 and @xmath23 denote coupling constants , and @xmath24 represents the mssm superpotential , @xmath25 for convenience , all superfields have been arranged in @xmath26 multiplets , @xmath27 and @xmath28 , and @xmath29 are flavour indices . we assume that the colour triplet partners of the electroweak higgs doublets @xmath30 and @xmath31 have been projected out . the vacuum expection values @xmath32 and @xmath33 break the electroweak symmetry . in the following we will assume large @xmath34 , implying @xmath35 . for notational convenience , we will refer to @xmath30 as @xmath36 from now on . in addition to these chiral superfields , the model also contains a vector supermultiplet @xmath37 ensuring invariance under local @xmath0@xmath1@xmath2 transformations and the gravity supermultiplet consisting of the graviton @xmath38 and the gravitino @xmath39 . the flavour structure of the model is parametrized by a froggatt - nielsen flavour model based on a global @xmath40 group , following refs . @xcite . according to this model , the couplings in the superpotential can be estimated up to @xmath41 factors as powers of a common hierarchy parameter @xmath42 , with the exponent given by the sum of the flavour charges @xmath43 of the fields involved in the respective operators . setting the charges of all higgs fields to zero , this implies @xmath44 the numerical value of the parameter @xmath45 is deduced from the quark and lepton mass hierarchies . this remarkably simple flavour model can reproduce the experimental data on standard model masses and mixings , while at the same time it remains flexible enough to incorporate the phenomena beyond the standard model mentioned above . further details on the predictive power of this model can be found in ref . @xcite , where we recently performed a monte - carlo study to examine the impact of the @xmath41 factors . in the following , we will restrict our analysis to the case of a hierarchical heavy ( s)neutrino mass spectrum , @xmath46 , where @xmath47 . furthermore we assume the heavier ( s)neutrino masses to be of the same order of magnitude as the common mass @xmath48 of the particles in the symmetry breaking sector , for definiteness we set @xmath49 . with this , the froggatt - nielsen flavour charges are fixed as denoted in tab . [ tab_flavourcharges ] . .froggatt - nielsen flavour charge assignments . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] taking the @xmath0@xmath1@xmath2 gauge coupling to be @xmath50 , the model can now , up to @xmath41 factors , be parametrized by the @xmath40 charges @xmath51 and @xmath52 . the @xmath0@xmath1@xmath2 breaking scale @xmath18 , the mass of the lightest of the heavy ( s)neutrinos @xmath53 , and the effective light neutrino mass parameter @xmath54 are related to these by @xmath55 here , @xmath56 , the geometric mean of the two light neutrino mass eigenvalues @xmath57 and @xmath58 , characterizes the light neutrino mass scale , which , with the charge assignments above , can be fixed to @xmath59 ev . to obtain this result , we exploited the seesaw formula @xmath60 with @xmath61 . furthermore , it can be shown that @xmath54 is bounded from below by the lightest neutrino mass @xmath7 @xcite . in the following , we will study the model in terms of the more physical quantities @xmath18 and @xmath53 instead of the @xmath40 charges . to partly account for the @xmath41 uncertainties in the neutrino mass matrices , we will additionally vary @xmath54 . apart from this , we ignore any further uncertainties of the model and simply set the @xmath41 prefactors to one . furthermore , when considering the production of dark matter in form of gravitinos , cf . section [ sec_mssm ] , the gravitino ( @xmath62 ) and gluino ( @xmath63 ) masses will be additional parameters . before the spontaneous breaking of @xmath0@xmath1@xmath2 , supersymmetry is broken by the vacuum energy density @xmath64 , which drives inflation . during this time , the dynamics of the system is governed by the slowly rolling scalar component @xmath65 of the inflaton multiplet @xmath16 . the scalar components of the higgs superfields @xmath66 are stabilized at zero . the right - handed sneutrinos and the scalar mssm particles obtain their masses due to supergravity contributions . as the field value of the inflaton decreases , so do the effective masses in the higgs sector , until a tachyonic direction develops in the effective scalar potential . the subsequent phase transition can best be treated in unitary gauge , in which the physical degrees of freedom are manifest . in particular , performing a super - gauge transformation relates the higgs superfields @xmath66 and the vector superfield @xmath37 to the respective fields @xmath67 and @xmath68 in unitary gauge , @xmath69 note that the chiral superfield @xmath70 playing the role of the gauge transformation parameter is chosen such that @xmath14 and @xmath15 are mapped to the same chiral superfield @xmath67 . this reflects the fact that one chiral superfield is ` eaten ' by the vector superfield in order to render it massive . the supermultiplet @xmath67 contains two real scalar degrees of freedom , @xmath71 , where @xmath72 remains massive throughout the phase transition and @xmath73 is the actual symmetry - breaking higgs field . it acquires a vacuum expectation value proportional to @xmath74 which approaches @xmath18 at large times . in the lagrangian , we account for symmetry breaking by making the replacement @xmath75 , where @xmath76 denotes the fluctuations around the homogeneous higgs background . the fermionic component @xmath77 of the supermultiplet @xmath67 pairs up with the fermionic component @xmath78 of the inflaton supermultiplet @xmath16 to form a dirac fermion @xmath79 , the higgsino , which becomes massive during the phase transition . due to supersymmetry , the corresponding scalar fields ( @xmath76 , @xmath72 and inflaton @xmath65 ) end up having the same mass as the higgsino in the supersymmetric true vacuum . likewise , the gauge supermultiplet @xmath68 ( gauge boson @xmath80 , real scalar @xmath81 , dirac gaugino @xmath82 ) and the ( s)neutrinos @xmath83 ( @xmath84 ) acquire masses . note that the choice of unitary gauge , cf . eq . , forbids us to use the wess - zumino gauge , so @xmath68 denotes a full massive gauge multiplet with four scalar and four fermionic degrees of freedom . the capital @xmath85 refers to the physical majorana particle @xmath86 built from the two weyl spinors contained in the superfields @xmath87 and @xmath88 . @xmath89 denotes the complex scalar superpartner of the left - chiral fermion @xmath88 . for an overview of the particle spectrum , see fig . [ fig : productiondecay ] . at the end of the phase transition , supersymmetry is restored . an explicit calculation of the lagrangian describing this phase transition is given in appendix [ app_sqed ] . we can read off the mass eigenvalues during the phase transition , cf . to : @xmath90 here we have ignored corrections which arise due to thermal effects and due to supersymmetry breaking before the end of inflation in some hidden sector , leading to a mass for the gravitino . @xmath1@xmath2 breaking . the higgs field @xmath76 and particles coupled to it are produced during tachyonic preheating , as marked by the red boxes . the gauge degrees of freedom then decay nearly instantaneously ( black , dashed arrows ) , whereas the decay and production of the other degrees of freedom can be described by boltzmann equations ( blue , solid arrows ) . the numbers in parentheses denote the respective internal degrees of freedom.,scaledwidth=100.0% ] the spontaneous breaking of @xmath0@xmath1@xmath2 discussed in the previous section marks the end of a stage of hybrid inflation , which is governed by the first term in the superpotential in eq . . as the symmetry breaking proceeds very rapidly and abruptly , it represents what is often referred to as a ` waterfall ' phase transition . it is accompanied by the production of local topological defects in the form of cosmic strings as well as the nonadiabatic production of particles coupled to the higgs field , a process commonly known as tachyonic preheating @xcite . in the following , we shall first discuss cosmic strings and then tachyonic preheating . due to the nontrivial topology of its vacuum manifold , the abelian higgs model underlying the @xmath0@xmath1@xmath2 phase transition gives rise to solitonic field configurations . these are called cosmic strings ( for a review , cf . e.g. @xcite ) . their energy per unit length is @xmath91 with @xmath92 and @xmath93^{-1}$ ] for @xmath94 . according to ref . @xcite , the characteristic length separating two strings formed during tachyonic preheating is @xmath95 here @xmath96 is the velocity of the radial component of the inflaton field , @xmath97 , at the onset of the phase transition , which can be determined from the scalar potential by exploiting the equation of motion for @xmath98 in the slow - roll approximation , @xmath99 . according to refs . @xcite , in the region of parameter space we are interested in , the slope of the scalar potential is determined by the coleman - weinberg one - loop corrections , cf . e.g. ref . @xcite for the explicit formulas resulting from the superpotential eq . . with this , the energy density stored in strings just after the end of the phase transition can be calculated as @xmath100 inserting eqs . and and the expression for the coleman - weinberg one - loop potential from ref . @xcite , we find that the fraction of energy stored in cosmic strings directly after the phase transition increases strongly with the coupling parameter @xmath23 . this is due to the higher energy density per cosmic string as well as the shorter average distance between two strings . for instance , for @xmath101 gev and @xmath102 , we find @xmath103 and @xmath104 . for @xmath105 , this is reduced to @xmath106 and @xmath107 . as the universe evolves , the cosmic strings intercommute , forming closed loops which are separated from the infinite strings . these oscillate , loosing energy into gravitational waves as well as into the higgs and gauge degrees of freedom until they eventually decay , cf . refs . @xcite . after a relaxation time , which is roughly given by @xmath108 @xcite , there is only @xmath41 cosmic string per hubble volume left and the energy density stored in the cosmic strings scales as @xmath109 . these relic cosmic strings can in principal be observed today , e.g. via string induced gravitational lensing effects in the cmb . the nonobservation of these effects implies an upper bound on the energy per unit length @xcite . in the following , we will work with @xmath110 where @xmath111 is newton s constant with @xmath112 gev denoting the planck mass . inserting this into eq . puts an upper bound on @xmath18 , which weakly depends on @xmath23 , @xmath113 in ref . @xcite , the authors discuss hybrid inflation and cosmic string production in a setup very similar to ours . taking into account current experimental bounds inferred from the spectrum of fluctuations in the cmb @xcite and from the nonobservation of cosmic strings @xcite , they find viable inflation for @xmath114 this significantly constrains the allowed parameter space . with the scale of @xmath0@xmath1@xmath2 breaking basically fixed , @xmath115 gev , eq . implies @xmath116 and a factor of proportionality of about 5 . this is still consistent with the froggatt - nielsen model , since three @xmath41 factors enter in the calculation of @xmath18 . the bounds on @xmath23 restrict the second free @xmath40 charge , @xmath117 , cf . eq . , and therefore @xmath53 . in the following , we will consider the restricted parameter space @xmath118 here , the variation of @xmath54 accounts for the uncertainties of the froggatt - nielsen model . the chosen range easily covers the expected values for @xmath54 in this setup , cf . @xcite for a recent analysis . the production and decay of cosmic strings can in principle have a large influence on the state of the universe just after the phase transition . however , as we will argue in the following , for our purposes it is not necessary to treat these processes in detail , as long as we restrict ourselves to the parameter space in eq . . according to eq . , it is possible to have as much as @xmath119 cosmic strings per hubble volume for large values of the coupling parameter @xmath23 . for the maximal value of the coupling constant , @xmath102 , roughly half of the total energy density just after the phase transition is stored in cosmic strings . however , since in this case the relaxation time of the cosmic strings , @xmath120 , is much smaller than a hubble time , the major component of this energy has been converted back into higgs and gauge degrees of freedom before the processes which we describe by means of boltzmann equations , cf . section [ sec_mssm ] , become relevant . since the exact mechanism of energy loss of cosmic strings is not yet fully understood , we will in the following omit the effects from cosmic strings , keeping in mind that at the very most , they will convert about half of the initial energy density of the higgs bosons into particles of the higgs and gauge multiplets . typically the effects from cosmic strings are much less important , e.g. for @xmath121 their relative energy contribution is at the level of at most @xmath122 . due to supersymmetry , the extra higgsinos produced will decay into the same supermultiplet as the higgs bosons would have , thus inducing no significant change in the following discussion . the extra gauge particles will decay predominantly into radiation , which is quickly diluted at this early stage of the matter dominated phase governed by the nonrelativistic higgs bosons . thus it can be expected that our setup is insensitive to a modification of the contribution from cosmic strings to the initial conditions of the following reheating phase . we also confirmed this in a numerical study . considering the case of extremal string production , we shifted half of the energy initially stored in the higgs bosons at the end of preheating into the gauge degrees of freedom and calculated the resulting entropy , baryon asymmetry and gravitino dark matter . we find no deviations from the results presented in section [ sec : example ] above the percent level . let us now consider the production of particles coupled to the higgs field . as the value of the inflaton field decreases , the scalar potential develops a tachyonic instability in the direction of the higgs field . quantum fluctuations of the higgs field @xmath123 with wave number @xmath124 begin to grow exponentially , while its average value remains zero . the strong population of the long wavelength higgs modes leads to a large abundance of nonrelativistic higgs bosons . other particles coupled to the higgs field are nonperturbatively produced due to the rapid change of their effective masses @xcite . the mode equations for the gauge , higgs , inflaton , and neutrino supermultiplets are governed by the time - dependent masses proportional to @xmath125 given by eq . . according to ref . @xcite , this leads to particle production , with the energy and number densities for bosons and fermions after tachyonic preheating given by @xmath126 with @xmath127 and @xmath128 , where @xmath129 denotes the mass of the respective particle in the true vacuum ; @xmath130 counts the spin degrees of freedom of the respective particle . just as the higgs bosons themselves , these particles are produced with very low momentum , i.e. nonrelativistically . a deviation from this mechanism is found for the imaginary component @xmath72 of the complex field @xmath131 , since fixing the gauge to unitary gauge yields a constant contribution to its mass . neglecting the expansion of the universe , the mode equation for @xmath72 reads @xmath132 we can absorb the constant mass contribution in the momentum @xmath133 . in the language of ref . @xcite , this is equivalent to a shift in the ` asymptotic _ in _ frequency ' @xmath134 . to excite a given mode more energy is necessary , the production is thus less efficient . with the ingredients discussed so far , the stage is now set for the emergence of the hot early universe . in other words , @xmath0@xmath1@xmath2 breaking after hybrid inflation provides the initial conditions for the successful generation of entropy , matter and dark matter . before we demonstrate this numerically , we will first introduce the necessary tools , i.e. decay rates and supersymmetric boltzmann equations , in section [ sec_tools ] . in this section , we will discuss the evolution of the particle abundances from the initial conditions set by @xmath0@xmath1@xmath2 symmetry breaking to the radiation dominated era . a schematic overview of all relevant processes is given in fig . [ fig : productiondecay ] . during tachyonic preheating , most of the vacuum energy is converted into higgs bosons . at the same time , particles coupled to the higgs bosons , i.e. particles of the gauge , higgs , inflaton and neutrino supermultiplets are produced ( cf . red boxes in fig . [ fig : productiondecay ] ) , with the resulting abundances given by eq . . among these particles , the members of the gauge supermultiplet have by far the shortest lifetime . due to their large couplings they decay basically instantaneously into ( s)neutrinos and mssm particles ( cf . dashed , black arrows in fig . [ fig : productiondecay ] ) . this sets the initial conditions for the following phase of reheating , which we will describe with boltzmann equations ( cf . solid , blue arrows in fig . [ fig : productiondecay ] ) . due to our choice of a hierarchical ( s)neutrino mass spectrum , the decay of particles from the symmetry breaking sector into the two heavier ( s)neutrino generations is kinematically forbidden . these particles can hence only decay into particles of the @xmath6 supermultiplet . these ( s)neutrinos , just as the neutrinos produced through gauge particle decays and thermally produced ( s)neutrinos , decay into mssm particles , thereby generating the entropy of the thermal bath as well as a lepton asymmetry . note that these different production mechanisms for the ( s)neutrinos yield ( s)neutrinos with different energies , which due to relativistic time - dilatation , decay at different rates . finally , the thermal bath produces a thermal gravitino abundance , which will turn out to be in the right ball - park to yield the observed dark matter abundance . in the following we list the total and partial vacuum decay rates necessary to quantify the processes described above . the total vacuum decay rates for the particles of the symmetry breaking , gauge and neutrino sectors are @xmath135_{ii } m_{i } = \frac{1}{4 \pi } \ , \frac{\widetilde{m}_i m_i^2}{v_{\text{ew}}^2 } \,,\end{aligned}\ ] ] with @xmath136 denoting the superfields of the model carrying @xmath0@xmath1@xmath2 charges @xmath137 . the relevant partial decay rates at leading order are given by @xmath138 \gamma^0_{\tau \rightarrow n_1 n_1 \,\ , } & = \gamma^0_{\phi \rightarrow \tilde n_1 \tilde n_1 } = \gamma^0_{\psi \rightarrow \tilde n_1^ * n_1 } = \gamma_s^0 \ , , \nonumber \\[6pt ] \gamma^0_{a \rightarrow \phi_x \phi_x } & = \frac{1}{2 } \ , \gamma^0_{a \rightarrow \psi_x \psi_x } = \frac{1}{3 } \ , \gamma^0_{c \rightarrow \phi_x \phi_x } = \frac{1}{3 } \ , \gamma^0_{\tilde a \rightarrow \phi_x \psi_x } \nonumber \\[6pt ] & = \frac{1}{3 } \ , \frac{q_x^2 \left ( 1 - 4 \frac{m_x^2}{m_s^2 } \right)^{1/2}}{\sum_x q_x^2 \left ( 1 - 4 \frac{m_x^2}{m_s^2 } \right)^{1/2 } } \ , \gamma^0_g\ , , \end{aligned}\ ] ] with @xmath139 and @xmath140 denoting the scalar and fermionic components of a superfield @xmath136 . at tree level the pseudoscalar @xmath72 decays exclusively into fermionic neutrinos , similar to its scalar partner @xmath76 , whose branching ratio into scalar neutrinos is suppressed by two powers of the mass ratio @xmath141 . the production of @xmath72 particles during tachyonic preheating , cf . sec . [ sec_tachyonic_preheating ] , is however negligible compared to the production of @xmath76 particles . we can thus neglect the contribution from the pseudoscalar @xmath72 in the following . another important consequence of eq . is that taking into account that the production of the different gauge degrees of freedom during tachyonic preheating is proportional to the respective spin degrees of freedom , cf . eq . , the decay products of the gauge degrees of freedom consist to equal shares of scalar and fermionic degrees of freedom . in this section , we present the boltzmann equations describing the evolution of the universe after the decay of the gauge degrees of freedom , as depicted by the blue , solid arrows in fig . [ fig : productiondecay ] . this analysis is a supersymmetric extension of the study performed earlier in ref . @xcite , exploiting the techniques explained there in detail . in general , the evolution of the phase space density @xmath142 of a particle species @xmath136 is determined by @xmath143 with @xmath144 denoting the liouville operator and the @xmath145 denoting the collision operators of all relevant processes involving the particle @xmath136 : @xmath146 \ , , \end{split}\ ] ] where @xmath147 denotes the sum over all internal degrees of freedom of the initial and final states and the momentum space element @xmath148 is given by @xmath149 @xmath150 is a statistical factor to prevent double counting of identical particles . the quantum statistical factors due to bose enhancement and pauli blocking have been omitted , since typically they only yield minor corrections @xcite . in the following , we will often work with integrated boltzmann equations , which are obtained by integrating eq . over @xmath151 . in a friedmann - lematre - robertson - walker universe , the resulting equation can be simplified to @xmath152 with @xmath51 denoting the scale factor , @xmath36 the hubble rate , @xmath153 the effective production rate of @xmath136 particles , and @xmath154 the comoving number density , i.e. the number of @xmath136 particles in a volume @xmath155 . performing a rescaling of @xmath51 in eq . leaves the physical number density @xmath156 invariant . for convenience , we will thus set @xmath157 at the end of preheating in the following . another useful quantity next to the number density @xmath156 is the energy density @xmath158 , @xmath159 * evolution of the gravitational background * the time - dependence of the scale factor @xmath160 is governed by the friedmann equation . for a flat universe and a constant equation of state @xmath161 between some reference time @xmath162 and time @xmath163 , the friedmann equation yields @xmath164 ^{\frac{2}{3(1+\omega ) } } \ , . \label{eq : scalefac}\ ] ] after preheating , the universe is dominated by nonrelativistic higgs bosons , i.e. @xmath165 . after the end of the reheating process , the universe is radiation dominated , @xmath166 . in the intermediate region , the equation of state changes continuously . we approximate this by implementing a piecewise constant effective equation of state with coefficients @xmath167 in the intervals @xmath168 $ ] with @xmath169 . the @xmath167 are determined iteratively by requiring self - consistency of the friedmann equation , @xmath170 in our numerical calculations , we approximate the total energy density by its two dominant components , the energy density of the higgs bosons and the energy density of the neutrinos produced in higgs , higgsino and inflaton decays , @xmath171 , for which we will obtain analytical expressions below , cf . eqs . and . in the following we will calculate the hubble rate @xmath172 using eq . . * massive degrees of freedom * the boltzmann equations describing the massive degrees of freedom introduced above are @xmath173 @xmath76 , @xmath65 and @xmath79 , the particles of the symmetry breaking sector , are produced via tachyonic preheating only , hence their initial number densities are given by eq . and their initial phase space distributions are peaked at low momenta , and thus can be taken to be proportional to @xmath174 . the collision operators on the right - hand side of eqs . to describe the decay of these particles . the resulting ordinary differential equations are solved by @xmath175 \ , , \qquad x = \sigma , \ , \phi , \ , \psi \ , , \label{eq_fx}\ ] ] with @xmath176 denoting the time at the end of preheating . we fix the origin of the time axis by setting @xmath177 . also the abundances of all heavy ( s)neutrinos obtain contributions from tachyonic preheating . the corresponding phase space distribution functions are of the same form as @xmath178 in eq . . the collision operators for the lightest ( s)neutrinos are more involved . just as in ref . @xcite , they can be treated best by separating the phase space density into the contributions due to thermal ( th ) and nonthermal ( nt ) ( s)neutrinos . introducing @xmath179 , the energy of a particle @xmath136 at time @xmath163 which was produced with energy @xmath180 at time @xmath181 , @xmath182 \left(\frac{m_{x}}{e_0 } \right)^2 \right\}^{1/2 } \,,\ ] ] we find for the comoving number densities of nonthermally produced ( s)neutrinos : @xmath183 \label{eq : nxntsol } \nonumber\\ & + n^{\text{ph}}_{x } ( t_{\text{ph } } ) \ , e^{-\gamma^0_{n_1 } ( t - t_{\text{ph } } ) } + \ , n^{g}_{x } ( t_g ) \ , \exp\left[-\int_{t_g}^t dt ' \frac{m_1 \gamma^0_{n_1 } } { { \cal e}_{x } ( m_g/2 ; t_g , t ' ) } \right ] \ , , \nonumber\end{aligned}\ ] ] with @xmath184 and @xmath185 here @xmath186 denotes the initial @xmath136 abundance from nonperturbative particle production during tachyonic preheating , whereas @xmath187 refers to the initial @xmath136 abundance from the decay of the gauge degrees of freedom . note that this notation is valid throughout this paper : the lower indices on number densities , decay rates , etc . indicate the respective particle , whereas the upper index refers to the origin of this particle . the time @xmath188 denotes the lifetime of the gauge particles after preheating , @xmath189 , cf . eq . , and corresponds to the value @xmath190 of the scale factor , @xmath191 . furthermore , also the ( s)neutrinos of the second and third generation are produced in the decays of gauge particles . the corresponding comoving number densities of these ( s)neutrino species are of the same form as @xmath192 in eq . . inserting eq . and eq . into eq . yields the same time - dependence , @xmath193 , for both neutrinos and sneutrinos . . hence implies a constant ratio between neutrinos and sneutrinos produced via decays of particles from the symmetry breaking sector throughout the reheating phase . for instance , @xmath194 the precise value and the dependence on @xmath53 arises due to the initial conditions set by tachyonic preheating , cf . eq . , and the branching ratios denoted in eq . for increasing @xmath53 , we find a weak increase of @xmath195 . unlike the two heavier ( s)neutrino generations , ( s)neutrinos of the first generation are also produced thermally from the bath . assuming kinetic equilibrium , their comoving number densities are determined by the integrated boltzmann equation @xmath196 with @xmath197 denoting the comoving number density in thermal equilibrium and @xmath198 is the vacuum decay width weighted with the average inverse time dilatation factor , @xmath199 in eq . we are interested in the decay width of the thermally produced neutrinos , @xmath200 . in this case eq . can be evaluated to @xmath201 , where @xmath202 denotes the modified bessel function of the second kind of order @xmath88 . note however that eq . is not restricted to this case but also allows the calculation of , for example , the decay width of the neutrinos produced by the decay of the higgs bosons , @xmath203 . * mssm degrees of freedom * the boltzmann equations governing the lepton number asymmetry and the abundance of mssm particles in the thermal bath are @xmath204 with @xmath205 denoting the collision operators responsible for the production , decay and scattering of ( anti)(s)leptons and @xmath206 describing the number of radiation quanta produced in the respective processes . a subtle but important point concerning the boltzmann equation for the lepton asymmetry is the correct treatment of @xmath207 scattering processes with heavy ( s)neutrinos in the intermediate state . the collision operator for ( s)neutrino decay takes care of the on - shell contributions to these processes , so we need to add the off - shell contributions . the @xmath9-conserving part is negligible compared to the on - shell contribution , so we shall concentrate on the @xmath9-violating part . this can be obtained by calculating the @xmath9-violating contribution of the full @xmath208 scattering process and then subtracting the on - shell @xmath9-violating contribution ( reduced collision operator ) . by exploiting unitarity and @xmath209 invariance , we prove in appendix [ app_cp ] that the @xmath9-violating contribution of the full @xmath208 scattering process vanishes up to corrections of @xmath210 , so that we can replace the @xmath9-violating off - shell contribution by the negative of the @xmath9-violating on - shell contribution . with this , the integrated boltzmann equation up to @xmath211 obtained from eq . reads @xmath212 with the washout rate @xmath213 and the effective ( non)thermal production rates for the lepton asymmetry @xmath214 given by @xmath215 in eq . we have introduced @xmath216 and @xmath217 as the nonthermal and thermal contributions to the total lepton asymmetry @xmath218 , respectively . the decay rate of the thermally produced ( s)neutrinos , @xmath219 , as well as the decay rates @xmath220 , @xmath221 , and @xmath222 for nonthermally produced ( s)neutrinos , are given by eq . . note that eq . relates decay rates @xmath223 and effective production rates @xmath224 . the latter describe the relative increase of the respective particle species due to a given production process and can directly be compared with the hubble rate @xmath36 in order to determine the efficiency of the respective process . @xmath225 parametrizes the @xmath9 asymmetry in the @xmath83 and @xmath226 decays , which , in the froggatt - nielsen model , can be estimated as @xcite @xmath227 in the following , we will set @xmath225 to its maximal value , thus obtaining an upper bound for the produced lepton asymmetry . analogously , this time neglecting terms of @xmath228 , eq . yields the integrated boltzmann equation for the relativistic degrees of freedom of the mssm , @xmath229 with @xmath230 denoting the effective rates of ( non)thermal radiation production , @xmath231 here @xmath232 denotes the effective increase of radiation quanta in the thermal bath by adding a particle @xmath136 stemming from the production mechanism @xmath233 with energy @xmath234 , @xmath235 another important quantity in this context is the total radiation production rate @xmath236 . it counts the radiation quanta produced per unit time and is obtained by dividing the right - hand side of the boltzmann equation for radiation , eq . , by @xmath237 , @xmath238 here in the last expression , @xmath239 denotes the number density of the dominant source for radiation production at a given time . solving eq . finally yields the temperature @xmath70 of the thermal bath , @xmath240 with @xmath241 counting the effective relativistic degrees of freedom contributing to the number density of the thermal bath @xmath242 , in the mssm @xmath243 . * gravitinos * gravitinos are predominantly produced through supersymmetric qcd scattering processes in the thermal bath . the corresponding integrated boltzmann equation is @xmath244 in qcd , up to leading order in the strong gauge coupling @xmath130 , the effective production rate @xmath245 is given by @xcite @xmath246 \ , , \label{eq : gammag}\ ] ] with the energy dependent thermal gluino mass , gluon mass and strong coupling constant @xmath247^{-1/2 } \ , , \end{split}\ ] ] with the typical energy scale during reheating estimated as the average energy per relativistic particle in the thermal bath , @xmath248 . the gravitino mass @xmath62 and the gluino mass at the electroweak scale @xmath249 remain as free parameters combining our initial conditions with the boltzmann equations derived in the previous section poses an initial - value problem . its solution allows us to quantitatively describe the generation of entropy , matter and dark matter due to the production and decay of heavy ( s)neutrinos . we have numerically solved this problem for all values of the input parameters within the ranges specified in eq . . in this section we first illustrate our findings for a representative choice of parameter values . in section [ sec_parameterspace ] we then turn to the investigation of the parameter space . in this paper , we take into account all ( super)particles involved in the reheating process , in particular the gauge degrees of freedom , which were omitted in earlier studies @xcite . this allows us to give a realistic , time - resolved description of the reheating process . furthermore , compared to refs . @xcite , we consider a higher scale of @xmath0@xmath1@xmath2 breaking , @xmath250 , which is compatible with hybrid inflation and cosmic strings , cf . section [ subsec : inflstrs ] . however , many of the techniques employed when solving the boltzmann equations are very similar to those discussed in detail in refs . we hence in the following focus on the physical results , referring the reader to these earlier works for more information on the technical aspects . let us study the evolution of the universe after inflation for @xmath251 as we will see later in section [ subsec : gravitinodm ] , requiring successful leptogenesis as well as the right gravitino abundance to explain dark matter typically forces @xmath53 to be close to @xmath252 . here , we adjust its explicit numerical value such that , given the values for @xmath54 and @xmath8 , the gravitino abundance comes out right in order to account for dark matter . the choice for @xmath54 represents the best - guess estimate in the context of the froggatt - nielsen flavour model employed in this work , a result we recently obtained in a monte - carlo study , cf . @xcite . in scenarios of gauge or gravity mediated supersymmetry breaking the gravitino often acquires a soft mass of @xmath253 , which is why we set @xmath8 to @xmath254 . a gluino mass of @xmath3 is close to the current lower bounds from atlas @xcite and cms @xcite . the values in eq . readily determine several further important model parameters : @xmath255 here , we have chosen opposite signs for the @xmath9 parameters @xmath256 and @xmath257 , so that the sign of the total lepton asymmetry always indicates which contribution from the various ( s)neutrino decays is the dominant one . [ fig : numengden ] presents the comoving number and energy densities of all relevant species as functions of the scale factor @xmath51 . in both panels of this figure some of the displayed curves subsume a number of closely related species . these combined curves are broken down into their respective components in the two panels of fig . [ fig : breakdown ] and in the lower panel of fig . [ fig : tempasymm ] . the upper panel of fig . [ fig : tempasymm ] presents the temperature of the thermal bath as function of @xmath51 . in what follows , we will go through the various stages of the evolution depicted in figs . [ fig : numengden ] , [ fig : breakdown ] and [ fig : tempasymm ] step by step . subsequent to that we will , based on the plots in fig . [ fig : numdenwo ] , discuss the impact of supersymmetry and the particles of the gauge sector on our results . + higgsinos @xmath79 + inflatons @xmath65 ) , ( non)thermally produced ( s)neutrinos of the first generation ( @xmath258 , @xmath259 ) , ( s)neutrinos of the first generation in thermal equilibrium ( @xmath260 , for comparison ) , ( s)neutrinos of the second and third generation ( @xmath261 ) , the mssm radiation ( @xmath262 ) , the lepton asymmetry ( @xmath0@xmath1@xmath2 ) , and gravitinos ( @xmath263 ) as functions of the scale factor @xmath51 . the vertical lines labeled @xmath264 , @xmath265 and @xmath266 mark the beginning , the middle and the end of the reheating process . the corresponding values for the input parameters are given in eq . . , width=425 ] + higgsinos @xmath79 + inflatons @xmath65 ) , ( non)thermally produced ( s)neutrinos of the first generation ( @xmath258 , @xmath259 ) , ( s)neutrinos of the first generation in thermal equilibrium ( @xmath260 , for comparison ) , ( s)neutrinos of the second and third generation ( @xmath261 ) , the mssm radiation ( @xmath262 ) , the lepton asymmetry ( @xmath0@xmath1@xmath2 ) , and gravitinos ( @xmath263 ) as functions of the scale factor @xmath51 . the vertical lines labeled @xmath264 , @xmath265 and @xmath266 mark the beginning , the middle and the end of the reheating process . the corresponding values for the input parameters are given in eq . . , width=425 ] * initial conditions * tachyonic preheating transfers the bulk of the initial vacuum energy into higgs bosons , @xmath267 , and only small fractions of it into nonrelativistic higgsinos , inflatons , gauge degrees of freedom and ( s)neutrinos @xmath268 . the particles in the gauge multiplet decay immediately afterwards around @xmath269 , giving rise to relativistic ( s)neutrinos @xmath270 and an initial abundance of radiation which thermalizes right away . initially , this thermal bath neither exhibits a lepton asymmetry , nor are there any gravitinos present in it . the expansion of the universe between the end of preheating and the decay of the gauge degrees of freedom is practically negligible , @xmath271 . note that technically all plots in figs . [ fig : numengden ] , [ fig : breakdown ] and [ fig : tempasymm ] start at @xmath269 . * decay of the ( s)neutrinos of the second and third generation * among all particles present at @xmath269 , the heavy ( s)neutrinos of the second and third generation have the shortest lifetimes , cf . due to time dilatation , the relativistic ( s)neutrinos stemming from the decay of the gauge particles decay slower than the nonrelativistic ( s)neutrinos produced during preheating . the decay of the ( s)neutrinos of the second and third generation is consequently responsible for an increase in the radiation number and energy densities on two slightly distinct time scales . the gauge particles decay in equal shares into neutrinos and sneutrinos , cf . section [ subsec : ratesratios ] . their number densities thus behave in exactly the same way , explaining the overlapping curves in fig . [ fig : breakdown ] . the production of radiation through the decay of these @xmath272 neutrinos and @xmath273 sneutrinos is efficient as long as the effective rate of radiation production @xmath236 , cf . eq . , exceeds the hubble rate @xmath36 . at @xmath274 it drops below the hubble rate , which roughly coincides with the value of the scale factor at which the comoving energy density of radiation reaches its first local maximum . the period between preheating and this first maximum of the radiation energy density can be regarded as the first stage of the reheating process . in the following we shall refer to it as the stage of @xmath275 reheating . . the ( s)neutrinos of the second and third generation ( @xmath275@xmath276@xmath277 ) * ( upper panel ) * split into ( s)neutrinos that are produced during preheating ( @xmath278 , @xmath279 ) and in the decay of the gauge degrees of freedom ( @xmath272 , @xmath273 ) . in all four cases the sum of the contributions from both generations is shown . the ( s)neutrinos of the first generation ( @xmath280@xmath276@xmath281 , @xmath282@xmath276@xmath283 ) * ( lower panel ) * split into ( s)neutrinos that are produced during preheating ( @xmath284 , @xmath285 ) , in the decay of the gauge degrees of freedom ( @xmath286 , @xmath287 ) , in the decay of the particles from the symmetry breaking sector ( @xmath288 , @xmath289 ) , and from the thermal bath ( @xmath290 , @xmath291 ) . , width=425 ] . the ( s)neutrinos of the second and third generation ( @xmath275@xmath276@xmath277 ) * ( upper panel ) * split into ( s)neutrinos that are produced during preheating ( @xmath278 , @xmath279 ) and in the decay of the gauge degrees of freedom ( @xmath272 , @xmath273 ) . in all four cases the sum of the contributions from both generations is shown . the ( s)neutrinos of the first generation ( @xmath280@xmath276@xmath281 , @xmath282@xmath276@xmath283 ) * ( lower panel ) * split into ( s)neutrinos that are produced during preheating ( @xmath284 , @xmath285 ) , in the decay of the gauge degrees of freedom ( @xmath286 , @xmath287 ) , in the decay of the particles from the symmetry breaking sector ( @xmath288 , @xmath289 ) , and from the thermal bath ( @xmath290 , @xmath291 ) . , width=425 ] * decay of the particles of the symmetry breaking sector * the production of higgsinos and inflatons during preheating is roughly equally efficient , @xmath292 . taking into account kinematic constraints resulting from the mass spectrum described in section [ subsec : fnmodel ] , all particles from the symmetry breaking sector exclusively decay into relativistic ( s)neutrinos of the first generation @xmath293 . the majority of higgs bosons , higgsinos and inflatons survives until @xmath294 , cf . eq . , which corresponds to a scale factor of @xmath295 . roughly up to this time the main part of the total energy is stored in these particles . at later times , i.e. for @xmath296 , the energy budget is dominated by the energy in radiation . higgs bosons that decay earlier than the average lifetime are responsible for the generation of sizeable abundances of @xmath288 neutrinos and @xmath289 sneutrinos . the contributions from higgsino and inflaton decays to this process are essentially negligible . * production and decay of the nonthermal ( s)neutrinos of the first generation * the decay of the particles from the symmetry breaking sector is the most important source for nonthermal ( s)neutrinos . according to our discussion in section [ sec_mssm ] , the ratio between the number densities of @xmath288 neutrinos and @xmath289 sneutrinos is fixed to a constant value at all times , cf . eq . . for our choice of parameters we find @xmath297 . moreover , the large hierarchy between the two decay rates @xmath298 and @xmath299 , cf . eq . , renders the @xmath288 and @xmath289 number densities unable to exceed the number density of the higgs bosons . from the perspective of the rather long - lived higgs bosons the ( s)neutrinos essentially decay right after their production . as long as they are efficiently fueled by higgs decays , the ( s)neutrino number densities continue to rise . but once the supply of higgs bosons is on the decline , they die out as well . the overall timescale of our scenario is hence controlled by the higgs lifetime . however , as we will see below , the characteristic temperature of the reheating process is by contrast associated with the lifetime of the @xmath288 neutrinos . further contributions to the abundances of nonthermal ( s)neutrinos come from preheating as well as the decay of the gauge particles . just as in the case of the second and third ( s)neutrino generation , the nonrelativistic ( s)neutrinos produced during preheating decay at the fastest rate and the number densities of @xmath286 neutrinos and @xmath287 sneutrinos are always the same . * reheating through the decay of @xmath288 neutrinos * the energy transfer from the nonthermal ( s)neutrinos of the first generation to the thermal bath represents the actual reheating process . it is primarily driven by the decay of the @xmath288 neutrinos which soon have the highest abundance among all ( s)neutrino species . in analogy to the notion of @xmath275 reheating , we may now speak of @xmath6 reheating . this stage of reheating lasts as long as @xmath300 , cf . let us denote the two bounding values of the scale factor at which @xmath301 by @xmath264 and @xmath266 . in the case of our parameter example we find @xmath302 and @xmath303 . between these two values of the scale factor the comoving number density of radiation roughly grows like @xmath304 . around @xmath305 the comoving energy density of radiation reaches a local minimum and around @xmath306 a local maximum . similarly , we observe that the end of reheating nearly coincides with the time at which the energy in radiation begins to dominate the total energy budget , @xmath307 . * plateau in the evolution of the temperature * the upper panel of fig . [ fig : tempasymm ] displays the temperature of the thermal bath @xmath70 calculated according to eq . as function of the scale factor @xmath51 . as a key result of our analysis we find that during @xmath6 reheating the temperature stays approximately constant . for @xmath51 between @xmath264 and @xmath266 it varies by less than an order magnitude . we thus conclude that in the first place @xmath264 and @xmath266 represent the limiting values for a plateau in the evolution of the radiation temperature . the origin of this plateau is the continuous production of @xmath288 neutrinos during reheating . as long as these neutrinos are produced much faster than they decay , their comoving number density grows linearly in time , @xmath308 , cf . taking into account that until @xmath309 the expansion of the universe is driven by the energy in the higgs bosons , i.e. nonrelativistic matter , this translates into @xmath310 . the @xmath288 number density in turn controls the scaling behaviour on the right - hand side of the boltzmann equation for radiation during @xmath6 reheating , cf . using @xmath311 , we find @xmath312 and * ( lower panel ) * comoving number densities for the nonthermal ( @xmath313 ) and thermal ( @xmath314 ) contributions to the total lepton asymmetry as well as all ( s)neutrino species ( @xmath259 , @xmath258 , @xmath260 for comparison and @xmath261 ) as functions of the scale factor @xmath51 . the vertical lines in the upper panel labeled @xmath264 , @xmath265 and @xmath266 mark the beginning , the middle and the end of the reheating process . the vertical lines in the lower panel respectively mark the changes in the signs of the two components of the lepton asymmetry . , width=425 ] and * ( lower panel ) * comoving number densities for the nonthermal ( @xmath313 ) and thermal ( @xmath314 ) contributions to the total lepton asymmetry as well as all ( s)neutrino species ( @xmath259 , @xmath258 , @xmath260 for comparison and @xmath261 ) as functions of the scale factor @xmath51 . the vertical lines in the upper panel labeled @xmath264 , @xmath265 and @xmath266 mark the beginning , the middle and the end of the reheating process . the vertical lines in the lower panel respectively mark the changes in the signs of the two components of the lepton asymmetry . , width=425 ] * reheating temperature * the temperature at which the plateau in fig . [ fig : tempasymm ] is located sets the characteristic temperature scale of reheating . in addition , it represents the highest temperature that is ever reached in the thermal bath as long as one restricts oneself to times at which it contains a significant fraction of the total energy budget of the universe , cf . lower panel of fig . [ fig : numengden ] . we define the reheating temperature @xmath315 as the temperature of the thermal bath at @xmath316 , where @xmath265 denotes the value of the scale factor when the decay of the @xmath288 neutrinos into radiation is about to become efficient . this is the case once the hubble rate @xmath36 has dropped to the effective decay rate @xmath317 , @xmath318 this yields a value which is representative for the temperature plateau , cf . [ fig : tempasymm ] . for the chosen set of parameters this equation has the following solution , @xmath319 in figs . [ fig : numengden ] and [ fig : numdenwo ] as well as in the upper panel of fig . [ fig : tempasymm ] the three values of the scale factor marking the initial ( @xmath264 ) , characteristic intermediate ( @xmath265 ) and final ( @xmath266 ) point of the reheating process are indicated by dashed vertical lines . a comparison of our definition of the reheating temperature with other common approaches can be found in appendix [ app : trh ] . * evolution of the temperature away from the plateau * during @xmath275 reheating the temperature first increases up to a maximal value and then decreases like @xmath320 . the initial rise reflects the production of radiation through decays of ( s)neutrinos of the second and third generation while the expansion of the universe is negligible . the subsequent decrease then follows from the boltzmann equation for radiation , cf . eq . , using the fact that its right - hand side stays almost constant up to the end of @xmath275 reheating , @xmath321 finally , we note that between the two stages of reheating and after the end of reheating the temperature drops off like @xmath322 . this is the usual adiabatic behaviour indicating that no radiation , i.e. entropy is being produced , @xmath323 * production and decay of the thermal neutrinos of the first generation * unlike the two heavier ( s)neutrino flavours the ( s)neutrinos of the first generation are also produced thermally @xmath324 . thanks to supersymmetry the evolution of the @xmath325 and @xmath326 number densities is governed by exactly the same boltzmann equation , cf . eq . , so that they are identical at all times . as both species inherit their momentum distribution from the thermal bath , they are always approximately in kinetic equilibrium . simultaneously , the interplay between decays and inverse decays drives them towards thermal equilibrium . initially , there are no thermal ( s)neutrinos present in the thermal bath and inverse decays result in a continuous rise of the thermal ( s)neutrino number densities until @xmath327 . around this time the temperature drops significantly below the mass @xmath53 and the thermal ( s)neutrinos become nonrelativistic . the equilibrium number density @xmath328 begins to decrease due to boltzmann suppression until it almost reaches the actual number density of thermal ( s)neutrinos . the production of thermal ( s)neutrinos can then no longer compete with the expansion of the universe and their comoving number densities do not continue to grow . this picture , however , soon changes because reheating sets in . as the temperature remains almost perfectly constant until @xmath329 , the equilibrium number density @xmath328 is not diminished due to boltzmann suppression any further up to this time . instead it bends over and starts to increase like the volume , @xmath330 . the number densities of the thermal ( s)neutrinos subsequently follow this behaviour of the equilibrium number density . during the second phase of @xmath6 reheating the temperature slightly decreases again , thereby reinforcing the boltzmann factor in @xmath328 . consequently , the equilibrium number density stops growing and shortly afterwards starts declining exponentially . an instant after it has passed its global maximum , the number densities of the thermal ( s)neutrinos overshoot the equilibrium number density . due to their numerical proximity the two values of the scale factor at which @xmath328 and @xmath331 respectively reach their global maxima can not be distinguished from each other in figs . [ fig : numengden ] . both events occur close to @xmath332 . * generation of the baryon asymmetry * the out - of - equilibrium decays of the heavy ( s)neutrinos violate @xmath2 , @xmath81 , and @xmath9 , thereby generating a lepton asymmetry in the thermal bath . a first nonthermal asymmetry is introduced to the thermal bath during @xmath275 reheating . for @xmath333 , the decay of the ( s)neutrinos stemming from preheating leads to an increase of the absolute value of the comoving number density @xmath313 . in the interval @xmath334 the lepton asymmetry is slightly augmented through the decay of the ( s)neutrinos which were produced in the decay of the gauge particles . the main part of the nonthermal asymmetry is , however , generated during @xmath6 reheating , while the scale factor takes values between @xmath335 and @xmath336 . at all other times the effective rate at which the nonthermal asymmetry is produced is at least half an order of magnitude smaller than the hubble rate . among all nonthermal ( s)neutrinos of the first generation only the @xmath288 neutrinos contribute efficiently to the generation of the asymmetry . their decay results in a positive nonthermal asymmetry that gradually overcompensates the negative asymmetry produced during @xmath275 reheating . at @xmath337 the entire initial asymmetry has been erased and @xmath313 changes its sign . washout processes almost do not have any impact on the evolution of the nonthermal asymmetry . the rate @xmath213 at which these processes occur , cf . eq . , is always smaller than the hubble rate @xmath36 by a factor of at least @xmath338 . on top of that , at the time @xmath339 is closest to @xmath36 , which happens around @xmath340 when @xmath341 , the production rate @xmath342 is constantly larger than @xmath213 by a factor of @xmath338 , so that the effect of washout on the nonthermal asymmetry is indeed always negligible . the decays and inverse decays of thermal ( s)neutrinos of the first generation are responsible for the emergence of a thermal , initially negative asymmetry in the bath . as long as the abundance of thermal ( s)neutrinos is far away from the one in thermal equilibrium , the absolute value of this asymmetry increases rapidly . around @xmath327 this is not the case anymore , causing the production of the thermal asymmetry to stall for a short moment . at @xmath343 the washout rate @xmath344 overcomes the production rate @xmath345 of the thermal asymmetry and its absolute value begins to decline . note that at this time the rates @xmath345 and @xmath344 are smaller than @xmath36 by roughly a factor @xmath346 . shortly afterwards , at @xmath347 , the number density of thermal ( s)neutrinos overshoots the equilibrium density which results in the asymmetry being driven even faster towards zero . already at @xmath348 the initial thermal asymmetry is completely erased . meanwhile , washout effects recede in importance . from @xmath349 onwards , @xmath350 permanently dominates over @xmath213 , which is why , once the thermal asymmetry has turned positive , it does not decrease anymore . instead it freezes out at its maximal value around @xmath351 which corresponds to the time when the ratio of @xmath352 and the hubble rate @xmath36 drops below @xmath353 . the final values of @xmath354 and @xmath355 allow us to infer the present baryon asymmetry @xmath356 as well as its composition in terms of a nonthermal ( @xmath357 ) and a thermal ( @xmath358 ) contribution , @xmath359 here , @xmath360 denotes the sphaleron conversion factor , @xmath361 and @xmath362 stand for the effective numbers of relativistic degrees of freedom in the mssm that enter the entropy density @xmath363 of the thermal bath in the high- and low - temperature regime , respectively , and @xmath364 is the comoving number density of photons . as final value for the scale factor we use @xmath365 which is the maximal value depicted in the two plots of fig . [ fig : numengden ] . in our parameter example we find @xmath366 recall that in section [ sec_mssm ] , we set the @xmath9 asymmetry parameter @xmath256 to its maximal value . in this sense , the resulting values for the baryon asymmetry must be interpreted as upper bounds on the actually produced asymmetry and are thus perfectly compatible with the observed value for the baryon asymmetry , @xmath367 @xcite . we also point out that , in fact , the froggatt - nielsen model typically predicts values for @xmath256 that are smaller than the maximal possible value by roughly a factor of @xmath338 , cf . ref . @xcite . using a generic value for @xmath256 according to the froggatt - nielsen model rather than estimating @xmath256 by means of its upper bound , would thus yield an excellent agreement between prediction and observation in the context of this parameter example , @xmath368 . furthermore , we find that in the case under study it is the nonthermal contribution @xmath357 that lifts the total baryon asymmetry @xmath356 above the observational bound . the thermal contribution @xmath358 is smaller than @xmath357 by five orders of magnitude . if we discarded the entire idea of nonthermally produced ( s)neutrinos being the main source of the lepton asymmetry and resorted to standard thermal leptogenesis , we would struggle to reproduce the observed asymmetry . for the chosen value of @xmath54 , standard leptogenesis would result in @xmath369 which is almost an order of magnitude below the observed value , cf . ref . @xcite for details . by contrast , it is still much larger than our result for @xmath358 . [ page : etabthdilu ] this has mainly two reasons . first , in our scenario the decays of the nonthermal neutrinos continuously increase the entropy of the thermal bath , cf . figs . [ fig : numengden ] and [ fig : tempasymm ] , which results in a nonstandard dilution of the thermal asymmetry during and after its production . between , for instance , @xmath370 , which corresponds to the time when the production of the negative asymmetry is reversed and the absolute value of the asymmetry starts to decline , and @xmath371 , the entropy of the thermal bath increases by a factor of @xmath372 . second , in consequence of the specific reheating mechanism at work the generation of the thermal asymmetry is delayed in time , so that it takes place at a lower temperature than in the standard case . this implies a correspondingly smaller abundance of thermal ( s)neutrinos , rendering our thermal mechanism for the generation of an asymmetry less efficient . we will resume this comparison of the thermal asymmetry @xmath358 with the expectation from standard leptogenesis @xmath373 in section [ subsec : baryonasym ] , where we will discuss the respective dependence on the neutrino mass parameters @xmath54 and @xmath53 . * production of gravitino dark matter * inelastic @xmath207 scattering processes in the supersymmetric thermal plasma , mediated predominantly via the strong interaction , are responsible for the production of dark matter in the form of gravitinos . as the right - hand side of the gravitino boltzmann equation , cf . eq . , scales like @xmath374 , the efficiency of gravitino production in the course of reheating is directly controlled by the interplay between the expansion of the universe and the evolution of the temperature . during @xmath275 reheating the temperature roughly declines as @xmath375 , cf . eq . , such that in first approximation @xmath376 once the decay of the ( s)neutrinos of the second and third generation has ceased , the temperature decreases adiabatically , @xmath377 or equivalently @xmath378 , cf . eq . . the rate of gravitino production @xmath379 then begins to decrease much faster than the hubble rate , in fact , initially even slightly faster than @xmath380 , causing the comoving gravitino number density @xmath381 to approach a constant value . the first stage of gravitino production is completed around @xmath382 which corresponds to the time when @xmath379 is half an order of magnitude smaller than @xmath36 . from this time onwards , @xmath379 scales like @xmath380 , the production term in the boltzmann equation is negligibly small and @xmath381 is constant . the decline of @xmath379 is reversed as soon as the temperature plateau characteristic for the phase of @xmath6 reheating is reached such that approximately @xmath383 . while @xmath384 , the gravitino density @xmath381 continues to remain constant and @xmath379 increases almost as fast as @xmath385 . at @xmath386 it has nearly caught up again with the hubble rate , i.e. the ratio @xmath387 reaches again a value of @xmath353 . this time marks the beginning of the second stage of gravitino production . the production term in the boltzmann equation can not be neglected any longer and , assuming for a moment an exactly constant temperature during @xmath6 reheating , we have @xmath388 the gravitino density @xmath381 hence begins to grow again , now even faster than during @xmath275 reheating . this terminates the rise of the rate @xmath379 , turning it into a decline proportional to @xmath389 . we thus obtain the interesting result that , although the temperature evolves differently during @xmath275 and @xmath6 reheating , the rate @xmath379 always runs parallel to the hubble rate during these two stages of the reheating process . at the end of @xmath6 reheating gravitino productions fades away in the same way as at the end of @xmath275 reheating . around @xmath390 , when @xmath387 drops below @xmath353 , the gravitino abundance freezes out . the final value of @xmath381 then allows us to calculate @xmath391 , the present energy density of gravitinos @xmath392 in units of @xmath393 , @xmath394 where @xmath395 denotes the critical energy density of the universe , @xmath396 the hubble rate @xmath36 in the units @xmath397 , @xmath398 the number density of the cmb photons , and @xmath399 , @xmath400 , @xmath401 , and @xmath402 are explained below eq . . recall that after fixing @xmath54 , @xmath8 and @xmath403 we adjusted the heavy neutrino mass , @xmath404 , such that we would obtain the right abundance of gravitinos to account for the observed amount of dark matter @xmath405 @xcite . by construction , we thus now find in our parameter example @xmath406 in conclusion , we would like to emphasize the intriguing simplicity of this mechanism for the generation of dark matter . let us in particular focus on the physical picture behind the second stage of gravitino production . initially , at the onset of @xmath6 reheating , the rate @xmath379 is still very small compared to the hubble rate @xmath36 . but given the constant spacetime density of gravitino production @xmath407 during @xmath6 reheating and the rapid growth of the spatial volume due to the expansion , @xmath379 rapidly grows sufficiently large to get the production of gravitinos going . during the remaining time of @xmath6 reheating this production can then proceed without further hindrance as the universe , although it is expanding , is filled by a thermal bath at a constant temperature . the continuous production of radiation nullifies the expansion and gravitinos are produced as in a static universe . in other words , one key feature of our scenario of reheating is that it turns the universe into a chemistry laboratory in which the temperature is fixed at a certain value so that dark matter can be cooked in it just to the right point . @xmath1@xmath2 vector boson ( * lower panel * ) , to be compared with the result of the full analysis in fig . [ fig : numengden ] . the individual curves show the comoving number densities of the higgs bosons ( @xmath76 ) , nonthermally and thermally produced neutrinos of the first generation ( @xmath408 ) , neutrinos from the first generation in thermal equilibrium ( @xmath409 ) , neutrinos of the second and third generation ( @xmath275 ) , the mssm radiation ( @xmath262 ) , the lepton asymmetry ( @xmath0@xmath1@xmath2 ) , and gravitinos ( @xmath39 ) as functions of the scale factor a. the vertical lines labeled @xmath264 , @xmath265 and @xmath266 mark the beginning , the middle and the end of the reheating process . the corresponding values for the input parameters are given in eq . . , width=425 ] @xmath1@xmath2 vector boson ( * lower panel * ) , to be compared with the result of the full analysis in fig . [ fig : numengden ] . the individual curves show the comoving number densities of the higgs bosons ( @xmath76 ) , nonthermally and thermally produced neutrinos of the first generation ( @xmath408 ) , neutrinos from the first generation in thermal equilibrium ( @xmath409 ) , neutrinos of the second and third generation ( @xmath275 ) , the mssm radiation ( @xmath262 ) , the lepton asymmetry ( @xmath0@xmath1@xmath2 ) , and gravitinos ( @xmath39 ) as functions of the scale factor a. the vertical lines labeled @xmath264 , @xmath265 and @xmath266 mark the beginning , the middle and the end of the reheating process . the corresponding values for the input parameters are given in eq . . , width=425 ] in the previous part of this section we discussed in detail the emergence of the hot thermal universe after inflation . the successful explanation of reheating as well the generation of matter and dark matter by means of our scenario did , however , not rely on any fortunate coincidence between certain particulars but was a direct consequence of the overall setup that we considered . the essential steps in the evolution after symmetry breaking were the following . preheating results in an initial state whose energy density is dominated by nonrelativistic higgs bosons . these decay slowly into nonthermal neutrinos of the first generation which in turn decay into radiation , thereby reheating the universe , generating a lepton asymmetry and setting the stage for the thermal production of gravitinos . at the same time , an additional contribution to the lepton asymmetry is generated by thermally produced ( s)neutrinos . all further details that we took care of are , of course , important for a complete understanding of the physical picture , but merely have a small impact on the final outcome of our calculation . in particular , as we will illustrate in this subsection , the numerical results for the observables of interest , @xmath315 , @xmath356 , and @xmath410 , remain unaffected if one neglects the superpartners of all massive particles or if one excludes the gauge particles from the analysis , cf . [ fig : numdenwo ] , in which we plot the corresponding comoving number densities of all remaining species as functions of the scale factor . this observation renders our scenario of reheating robust against uncertainties in the underlying theoretical framework and opens up the possibility to connect it to other models of inflation and preheating as long as these provide similar initial conditions as spontaneous @xmath0@xmath1@xmath2 breaking after hybrid inflation . in addition to that , the robustness of our scenario justifies to crudely simplify its technical description . if one is interested in the parameter dependence of the observables and less in the exact evolution during reheating , one may simply omit effects due to the gauge degrees of freedom and supersymmetry as it has been done in refs . @xcite and @xcite . * nonsupersymmetric analysis including the gauge multiplet * in a first step , in order to assess the impact of supersymmetry on the reheating process in the abelian higgs model , we neglect the superpartners of all massive particles , i.e. the gauge scalar @xmath81 , the gaugino @xmath411 , the higgsino @xmath79 as well as all heavy sneutrinos @xmath226 . technically , this renders the inflaton @xmath65 stable as it can only decay into a pair of @xmath412 sneutrinos . to avoid overclosure of the universe we thus also omit the inflaton . by contrast , we keep the full particle spectrum of the mssm and the gravitino because we still wish to account for dark matter by thermally produced gravitinos . all in all , these simplifications imply drastically simpler boltzmann equations and induce small changes to the corresponding decay and production rates . again we solve the set of boltzmann equations in combination with the initial conditions set by preheating and the decay of the gauge degrees of freedom . for our key observables we obtain @xmath413 with regard to their first two digits , these results for @xmath315 , @xmath356 , @xmath357 and @xmath410 are the same as in the full analysis . the result for @xmath358 is smaller by a factor @xmath414 reflecting the missing contribution from the thermal sneutrinos of the first generation . in the upper panel of fig . [ fig : numdenwo ] we present the corresponding comoving number densities . they behave very similarly to the original densities in the upper panel of fig . [ fig : numengden ] , the only minor differences being the following . at early times all densities but the one of the higgs bosons are a bit smaller , at most by a factor of @xmath338 . in turn , the density of the higgs bosons is technically a bit larger . but the relative change is of @xmath415 and thus not visible in fig . [ fig : numdenwo ] . the fact that initially more energy remains in the higgs bosons has two reasons . first , there are now simply less particle species present into which the initial vacuum energy could be distributed . second , particles coupling to the gauge sector are produced in smaller numbers after preheating due to the absence of the superpartners of the @xmath0@xmath1@xmath2 vector boson . a direct consequence of the densities being initially slightly smaller is that they become sensitive to the decays of the nonthermal @xmath288 neutrinos a bit earlier . the onset of reheating and the inversion of the lepton asymmetry , for instance , take place at @xmath416 and @xmath417 , respectively , while these events occur later , at @xmath302 and @xmath337 , if supersymmetry is fully included . however , as soon as the @xmath262 and @xmath0@xmath1@xmath2 abundances are dominated by the decay products of the @xmath288 neutrinos , the differences between the two plots in the upper panels of figs . [ fig : numengden ] and [ fig : numdenwo ] begin to vanish . from @xmath418 onwards , they are , up to a factor 2 between the curves for the thermal ( s)neutrinos , at or below the percent level . it is easy to understand why the omission of the heavy superparticles does not have any effect on our final results . according to eq . the initial energy densities of the gauge scalar @xmath81 , the gaugino @xmath411 , the higgsino @xmath79 , the inflaton @xmath65 as well as the heavy sneutrinos @xmath226 are monotonic functions of the higgs - inflaton coupling @xmath23 . setting @xmath23 to its maximal value , @xmath102 , we obtain upper bounds on these densities , @xmath419 we thus conclude that no matter how the dynamics of the above species look like in detail , their influence on the reheating process will always be outweighed sooner or later by the decay of the much more abundant higgs bosons . ignoring these particles does hence not affect the outcome of our calculation . similarly , we can show that only the fermionic decays of the higgs bosons are relevant for reheating . the ratio of @xmath289 sneutrinos to @xmath288 neutrinos increases monotonically with the mass @xmath53 , cf . our upper bound on this mass , @xmath420 , then translates into @xmath421 the nonthermal @xmath289 sneutrinos can hence also be safely neglected . in conclusion , our numerical results in eqs . and substantiate our introductory comment at the beginning of this subsection . the essential feature of our scenario of reheating is the higgs boson decay chain , @xmath422 . from the point of view of the final results for the observables , the inclusion of the full supersymmetric particle spectrum is rather a matter of theoretical consistency than a numerical necessity . * nonsupersymmetric analysis neglecting the gauge multiplet * finally , we wish to demonstrate that one is also free to neglect the decay of the gauge particles if one is only interested in numerical results for the observables . in addition to all massive superparticles we now also exclude the @xmath0@xmath1@xmath2 vector boson from our analysis . consequently , particle production in the decay of gauge particles does not take place any longer , which simplifies our set of boltzmann equations once more . this time we find for our key observables @xmath423 with regard to their first two digits , these results exactly match those in eq . . the lower panel of fig . [ fig : numdenwo ] displays the corresponding comoving number densities , again to be compared with the original densities in the upper panel of fig . [ fig : numengden ] . the absence of ( s)neutrinos of the second and third generation produced through the decay of gauge particles now results in a slightly smaller initial lepton asymmetry and , more importantly , in drastically shorter @xmath275 reheating . while this first stage of reheating still lasted until @xmath274 in our complete analysis , cf . section [ subsec_decay_mp ] , it now comes to an end already at @xmath424 . before the onset of @xmath6 reheating the abundances of radiation , thermal neutrinos and gravitinos are hence significantly reduced . for instance , at @xmath425 the respective comoving number densities are suppressed by factors of the following orders of magnitude , @xmath426 as before , due to this initial suppression these densities are earlier sensitive to the decay of the @xmath288 neutrinos . now the onset of @xmath6 reheating and the inversion of the lepton asymmetry take place at @xmath427 and @xmath428 , which is even earlier than in our nonsupersymmetric analysis including the gauge multiplet . however , during @xmath6 reheating the differences between the two plots in the upper panel of fig . [ fig : numengden ] and the lower panel of fig . [ fig : numdenwo ] again vanish . from @xmath418 onwards , they are at or below the percent level . in conclusion , we find that including the gauge degrees of freedom has a great impact on the dynamics at early times shortly after preheating , but turns out be nonessential when calculating the final numerical results . the value of the boltzmann equations derived in section [ sec_mssm ] is twofold . on the one hand , as we have seen in the last section , they are the basis for a detailed time - resolved description of the dynamics during reheating . on the other hand , as we will demonstrate in this section , solving them in the entire parameter space allows one to study the quantitative dependence of our key quantities , @xmath315 , @xmath356 , and @xmath410 , on the parameters in the lagrangian . in doing so we will mainly focus on the physical aspects of our results , referring the interested reader to ref . @xcite , where we elaborate more comprehensively on the technical details of our approach . the relevant parameters of our model are the scale of @xmath0@xmath1@xmath2 breaking @xmath18 , the heavy neutrino mass @xmath53 , the effective neutrino mass @xmath54 , the gravitino mass @xmath8 , and the gluino mass @xmath403 . requiring consistency with hybrid inflation and the production of cosmic strings fixes the @xmath0@xmath1@xmath2 breaking scale , @xmath429 , and limits the range of possible @xmath53 values , cf . section [ subsec : inflstrs ] . according to the froggatt - nielsen flavour model , @xmath54 should be close to @xmath430 . however , in order to account for the uncertainties of the flavour model , we vary it between @xmath431 and @xmath432 , cf . eq . . for the gravitino mass we consider typical values as they arise in scenarios of gauge or gravity mediated supersymmetry breaking , @xmath433 as for the gluino , we stick without loss of generality to the mass that we used in the parameter example discussed in the previous section , @xmath434 . the generalization to different choices for @xmath403 is straightforward , cf . appendix d in ref . @xcite , and simply amounts to a rescaling of all values for the gravitino mass . notice that gravitino masses as large as @xmath435 are , in fact , inconsistent with unified gaugino masses at the gut scale . if the gluino and the bino had the same mass at the gut scale , the different running of the respective renormalization group equations would then entail a mass ratio of roughly @xmath436 at low energies . the gravitino which we assume to be the lightest superparticle would then have to be lighter than the bino , resulting in an upper bound of @xmath437 . we , however , leave open the question whether gaugino mass unification takes place at the gut scale and work in the following with the full gravitino mass range specified in eq . . at each point of the parameter space defined by the above restrictions we solve the boltzmann equations and record all important numerical results , which we now discuss in turn . in sections [ subsec : trh ] and [ subsec : baryonasym ] we study the parameter dependence of the reheating temperature and the final baryon asymmetry , respectively . in particular , we devote attention to the composition of the asymmetry in terms of a nonthermal and a thermal contribution . by imposing the condition that the maximal possible baryon asymmetry be larger than the observed one , we identify the region in parameter space that is consistent with leptogenesis , cf . the comment below eq . , @xmath438 in section [ subsec : gravitinodm ] we then turn to the generation of dark matter in the form of gravitinos . requiring the final gravitino abundance to match the observed density of dark matter , @xmath439 we are able to derive relations between the neutrino parameters @xmath53 and @xmath54 and the superparticle masses @xmath8 and @xmath403 . combining the two conditions in eqs . and , we are eventually even able to set a lower bound on @xmath8 in terms of @xmath54 . note that in all plots in this section ( figs . [ fig : temprh ] , [ fig : asym ] and [ fig : mgbounds ] ) the position of the parameter point which we investigated in section [ sec : example ] is marked by a small white circle . as a function of the effective neutrino mass @xmath54 and the heavy neutrino mass @xmath53 . the reheating temperature is calculated according to eq . after solving the boltzmann equations , cf . appendix [ app : trh ] for a comparison of our definition of the reheating temperature with other common approaches . the thick horizontal gray lines represent the lower and the upper bound on @xmath53 , respectively , which arise from requiring consistency with hybrid inflation and the production of cosmic strings during the @xmath0@xmath1@xmath2 phase transition , cf . the small white circle marks the position of the parameter point discussed in section [ sec : example ] . [ fig : temprh ] , width=330 ] the process of reheating after the @xmath0@xmath1@xmath2 phase transition is accompanied by an intermediate plateau in the decline of the temperature , which determines the characteristic temperature scale of reheating . in section [ subsec : rhandt ] we concretized this intuitive notion and defined the reheating temperature @xmath315 as the temperature of the thermal bath at the moment when the decay of the @xmath288 neutrinos into radiation is about to become efficient , cf . eq . , @xmath440 in appendix [ app : trh ] we argue that this definition is particularly convenient compared to alternative approaches because it is not only representative for the temperature plateau during reheating , but also associated with a physical feature in the temperature curve . having at hand the solutions of the boltzmann equations for all allowed values of @xmath54 and @xmath53 , eq . enables us to determine the reheating temperature as a function of these two parameters , @xmath441 . as the reheating process is solely controlled by higgs and neutrino decays , @xmath315 obviously does not depend on the gravitino or gluino mass . in fig . [ fig : temprh ] we present the result of our analysis . we find that , within the considered range of neutrino parameters , the reheating temperature varies by almost five orders of magnitude . for @xmath442 and @xmath443 we have , for instance , @xmath444 , while for @xmath445 and @xmath446 we obtain @xmath447 . remarkably , the reheating temperature never exceeds the neutrino mass @xmath53 . instead it is typically smaller than @xmath53 by one or even two orders of magnitude . as the ratio @xmath448 controls the strength of washout process during reheating , we conclude that the effect of washout on the generation of the lepton asymmetry is in most cases negligible , cf . section [ subsec : baryonasym ] where we will come back to this observation . the reheating temperature increases monotonically with both neutrino parameters , @xmath54 and @xmath53 , with the dependence on @xmath53 being much more pronounced than the dependence on @xmath54 . in the following we will derive a simple semianalytical approximation for @xmath315 by means of which this behaviour can be easily understood . a more detailed discussion can be found in appendix c of ref . @xcite . by definition , @xmath315 corresponds to the decay temperature of @xmath6 neutrinos decaying with the effective rate @xmath317 . to first approximation , we may thus write @xmath449 where @xmath450 denotes the average of the relativistic lorentz factor relating @xmath317 to the vacuum decay rate @xmath298 . this first estimate of the reheating temperature fails to accurately reproduce our numerical results because of two imprecisions . first , eq . is based on the assumption that at @xmath316 the dominant contribution to the total energy is contained in radiation . this is , however , never the case . at @xmath316 the decays of the @xmath288 neutrinos have just set in , so that at this time a significant fraction of the total energy is hence always still stored in these neutrinos . on top of that , for @xmath451 , which is the case in almost the entire parameter space , the higgs bosons have not decayed yet at @xmath316 , so that , in the end , they dominate the total energy density at the time of reheating . to remedy this first imprecision , we have to multiply eq . by @xmath452 , where @xmath453 . the second imprecision is related to the fact that we do not explicitly solve the friedmann equation to determine the hubble parameter , but rather calculate it as @xmath454 with the scale factor @xmath51 being constructed as described in section [ sec_mssm ] . as a consequence of this procedure , @xmath36 does not always exactly fulfill the friedmann equation . we account for this technical imprecision by multiplying eq . by @xmath455 , where @xmath456 relates @xmath454 to the exact solution of the friedmann equation at @xmath316 . for appropriate functions @xmath457 , @xmath458 and @xmath459 , the reheating temperature @xmath315 can then be written as @xmath460 the dependence of @xmath457 , @xmath458 and @xmath459 on @xmath54 and @xmath53 follows from the solutions of the boltzmann equations . restricting ourselves to the region in parameter space in which @xmath461 , we find that @xmath458 and @xmath459 are basically constant . we obtain @xmath462 and @xmath463 with deviations around these values of a few percent . the dependence of the correction factor @xmath457 on @xmath54 and @xmath53 is well described by @xmath464 such a behaviour directly follows from the interplay of the decay rates @xmath298 and @xmath299 . for large @xmath298 and small @xmath299 reheating takes place quite early , at a time when most higgs bosons have not decayed yet . for small @xmath298 and large @xmath299 reheating takes place later and not as many higgs bosons are present anymore at @xmath316 . the magnitude of @xmath457 is hence controlled by the ratio @xmath465 which scales like @xmath466 . this explains the parameter dependence in eq . . putting all these results together yields a fitting formula for the reheating temperature that reproduces our numerical results with an error of less than a percent in almost the entire parameter space , @xmath467 as a function of the effective neutrino mass @xmath54 and the heavy neutrino mass @xmath53 . the baryon asymmetry is calculated according to eq . after solving the boltzmann equations . in the bright green ( gray green ) region the nonthermal ( thermal ) asymmetry is consistent with the observed asymmetry . in the red region the total asymmetry falls short of the observational bound . below ( above ) the thin blue line the nonthermal ( thermal ) asymmetry dominates over the thermal ( nonthermal ) asymmetry . the thick horizontal gray lines represent the lower and the upper bound on @xmath53 , respectively , which arise from requiring consistency with hybrid inflation and the production of cosmic strings during the @xmath0@xmath1@xmath2 phase transition , cf . the small white circle marks the position of the parameter point discussed in section [ sec : example ] . [ fig : asym ] , width=445 ] based on the solutions of the boltzmann equations we calculate the nonthermal and thermal contributions to the final baryon asymmetry , cf . eq . , for all values of the neutrino parameters @xmath54 and @xmath53 . we present the result of this analysis in fig . [ fig : asym ] . the parameter regions in fig . [ fig : asym ] where the nonthermal and thermal baryon asymmetries @xmath357 and @xmath358 are consistent with the observational bound @xmath468 are shaded in bright green and gray green , respectively . the overlap of these two regions is coloured in dark green . in the white patch around @xmath469 and @xmath470 the total asymmetry @xmath471 is larger than @xmath468 , but neither of its two contributions is . below the solid blue line in fig . [ fig : asym ] the nonthermal asymmetry dominates over the thermal one . above the solid blue line it is the other way around . we conclude that in the part of parameter space that we are interested in , the thermal asymmetry is almost always outweighed by its nonthermal counterpart . especially in the region in which leptogenesis is consistent with gravitino dark matter , where @xmath53 is typically of @xmath472 , cf . section [ subsec : gravitinodm ] , the thermal asymmetry is negligibly small . in most of the parameter space the nonthermal asymmetry is insensitive to @xmath54 and thus solely controlled by @xmath53 . only for large values of @xmath54 and @xmath53 it depends on both neutrino mass parameters . this behaviour is directly related to the efficiency of the washout processes in the respective parameter regions . let us suppose for a moment that washout does not take place . the final nonthermal asymmetry then only depends on the total number of ( s)neutrinos produced during reheating and the amount of @xmath9 violation per ( s)neutrino decay . neither of these two quantities is , however , affected by changes in @xmath54 , so that the asymmetry , indeed , ends up being a function of @xmath53 only . from this perspective , the insensitivity of @xmath357 to @xmath54 signals that the effect of washout on the generation of the asymmetry is negligible for most values of the neutrino parameters . this result is consistent with our findings for the reheating temperature and in particular the ratio @xmath448 as a function of @xmath54 and @xmath53 , cf . section [ subsec : trh ] . to see this , note that for temperatures @xmath473 the effective washout rate @xmath344 decreases exponentially when raising the ratio @xmath474 , @xmath475 which readily follows from eqs . and . the fact that @xmath476 is of @xmath338 or even larger for most parameter values then explains why the impact of washout is typically vanishingly small . in turn , eq . also illustrates the importance of washout at very large values of @xmath54 and @xmath53 , for which the ratio @xmath476 approaches values of @xmath477 . comparing our results for the reheating temperature and the baryon asymmetry in figs . [ fig : temprh ] and [ fig : asym ] , respectively , we find that washout only plays a significant role if @xmath478 and @xmath479 . interestingly , the parameter region defined by these two conditions covers the entire range of parameters in which the thermal asymmetry exceeds the observed asymmetry . if washout is negligible , the nonthermal asymmetry can be reproduced to good approximation by assuming that all @xmath288 neutrinos decay instantaneously at time @xmath480 into radiation . the resultant baryon asymmetry is then given by @xmath481 where @xmath482 denotes the average energy per @xmath288 neutrino . the ratio @xmath483 is proportional to @xmath484 , the number density of @xmath288 neutrinos at the same time when these decay , normalized to the radiation number density . it directly follows from the solutions of the boltzmann equations and is well described by @xmath485 together with the expression for @xmath256 in eq . this yields the following fitting formula for the nonthermal asymmetry in the case of weak washout , @xmath486 it reproduces our numerical results for @xmath487 within a factor of 2 for most values of @xmath53 . the requirement that the maximal possible asymmetry be larger than the observed one constrains the allowed range of @xmath53 values . [ fig : asym ] implies the following lower bound , @xmath488 where we have averaged out the slight dependence on @xmath54 . if @xmath53 is chosen below this minimal value , the asymmetry falls below the observational bound for two reasons . on the one hand , small @xmath53 implies a small @xmath9 parameter @xmath256 , cf . eq .. on the other hand , according to eq . , a small @xmath53 value also entails a small ratio @xmath483 , i.e. a small abundance of ( s)neutrinos at the time the asymmetry is generated . the combination of both effects then renders the successful generation of the lepton asymmetry impossible . the thermal asymmetry has , to first approximation , the same parameter dependence as the asymmetry generated in standard leptogenesis . it increases monotonically with @xmath53 . if @xmath53 is kept fixed at some value @xmath489 , it is largest for @xmath54 values of @xmath490 . the monotonic behaviour in @xmath53 is a direct consequence of the fact that the @xmath9 parameter @xmath256 scales linearly with @xmath53 . the preference for intermediate values of @xmath54 has the same reason as in the standard case . large @xmath54 corresponds to strong washout , at least for the high values of @xmath53 at which the thermal generation of the asymmetry carries weight . small @xmath54 results in a low temperature and a small neutrino decay rate @xmath298 such that the thermal production of ( s)neutrinos is suppressed . especially in the parameter region in which the thermal asymmetry dominates over the nonthermal asymmetry , the expectation from standard leptogenesis @xmath491 approximates our numerical results reasonably well , @xmath492 here , @xmath493 denotes the final efficiency factor . in the strong washout regime , @xmath494 , it is inversely proportional to @xmath54 and independent of the initial conditions at high temperatures @xcite , @xmath495 combining eqs . and with the expression for @xmath256 in , we obtain @xmath496 in the region in parameter space where @xmath497 this fitting formula reproduces our numerical results within a factor of 2 . despite these similarities it is , however , important to note that our thermal mechanism for the generation of the lepton asymmetry differs from the standard scenario in two important aspects . first , our variant of thermal leptogenesis is accompanied by continuous entropy production , while one assumes an adiabatically expanding thermal bath in the case of standard leptogenesis . consequently , our thermal asymmetry experiences an additional dilution during and after its generation , cf . the comment on page . second , our scenario of reheating implies a particular relation between the temperature at which leptogenesis takes place , which is basically @xmath315 in our case , and the neutrino mass parameters , cf . section [ subsec : trh ] , that differs drastically from the corresponding relation implied by standard leptogenesis . this translates into a different parameter dependence of the ratio @xmath474 as a function of @xmath54 and @xmath53 , which in turn alters the efficiency of washout process and the production of thermal ( s)neutrinos from the bath in the respective regions of parameter space . in the end , our thermal asymmetry therefore rather corresponds to a distorted version of the asymmetry generated by standard leptogenesis . as we have remarked above , in the parameter region where the thermal asymmetry is larger than the nonthermal asymmetry @xmath358 hardly deviates from @xmath373 . but as soon as we go to smaller values of @xmath54 and @xmath53 the difference between the two asymmetries grows . the minimal value of @xmath53 for which the thermal asymmetry is still able to exceed the observational bound , for instance , turns out to be much larger in our scenario than in standard leptogenesis . we find an absolute lower bound on @xmath53 of roughly @xmath498 at an effective neutrino mass @xmath499 , while standard leptogenesis only constrains @xmath53 to values larger than @xmath500 . lowering @xmath53 below @xmath498 either implies a larger ratio @xmath476 or a larger effective neutrino mass @xmath54 , cf . [ fig : temprh ] . in either case the thermal asymmetry is reduced so that it drops below the observed value . in conclusion , we emphasize that the generation of the lepton asymmetry is typically dominated by the decay of the nonthermal ( s)neutrinos . only in the parameter region of strong washout , which is characterized by a small ratio @xmath476 , the nonthermal asymmetry is suppressed and the thermal asymmetry has the chance to dominate . related to that , we find that the viable region in parameter space governed by the nonthermal mechanism is significantly larger than the corresponding region for the thermal mechanism . independently of @xmath54 , the neutrino mass @xmath53 can be as small as @xmath501 , which is an order of magnitude below the bound of @xmath502 , which one obtains in the purely thermal case . * ( upper panel ) * and the reheating temperature @xmath315 * ( lower panel ) * as functions of the effective neutrino mass @xmath54 and the gravitino mass @xmath8 such that the relic density of dark matter is accounted for by gravitinos , cf . eqs . and . in the red region the lepton asymmetry generated by leptogenesis is smaller than the observed one , providing us with a lower bound on the gravitino mass in dependence on @xmath54 . the colour code is the same as in figs . [ fig : temprh ] and [ fig : asym ] . the small white circle marks the position of the parameter point discussed in section [ sec : example ] . [ fig : mgbounds ] , width=442 ] * ( upper panel ) * and the reheating temperature @xmath315 * ( lower panel ) * as functions of the effective neutrino mass @xmath54 and the gravitino mass @xmath8 such that the relic density of dark matter is accounted for by gravitinos , cf . eqs . and . in the red region the lepton asymmetry generated by leptogenesis is smaller than the observed one , providing us with a lower bound on the gravitino mass in dependence on @xmath54 . the colour code is the same as in figs . [ fig : temprh ] and [ fig : asym ] . the small white circle marks the position of the parameter point discussed in section [ sec : example ] . [ fig : mgbounds ] , width=442 ] the final abundance of gravitinos @xmath503 depends on three parameters : the reheating temperature @xmath315 as well as the two superparticle masses @xmath8 and @xmath403 . a key result of our reheating scenario is that @xmath315 is determined by the neutrino mass parameters @xmath54 and @xmath53 . as we keep the gluino mass fixed at @xmath3 , the gravitino abundance thus ends up being a function of @xmath54 , @xmath53 and @xmath8 . based on the solutions of the boltzmann equations we calculate @xmath503 according to eq . for all values of these three masses . by imposing the condition that gravitinos be the constituents of dark matter we can then eliminate one of the free mass parameters , for instance the neutrino mass @xmath53 , @xmath504 the physical picture behind this step is the following . for given @xmath8 , the reheating temperature has to have one specific value so that the abundance of gravitinos comes out right . each choice for @xmath54 then implies one particular value of @xmath53 for which this desired reheating temperature is obtained . solving eq . for @xmath53 yields this value as a function of @xmath54 and @xmath8 . the corresponding reheating temperature follows immediately , @xmath505 in summary , combining the requirement that gravitinos make up the dark matter with the fact that the reheating temperature is determined by neutrino parameters allows us to infer relations between these neutrino parameters and superparticle masses . the lower bound on @xmath53 induced by leptogenesis , cf . eq . , can then be translated into a constraint on the mass parameters @xmath54 and @xmath8 . @xmath506 we present our results for the functions @xmath507 and @xmath508 in the two panels of fig . [ fig : mgbounds ] , respectively . furthermore , we indicate in both plots the constraint arising from the requirement of successful leptogenesis . we observe the following trends in the two plots of fig . [ fig : mgbounds ] . both quantities , @xmath53 and @xmath315 , show a stronger dependence on the gravitino mass than on the effective neutrino mass . for @xmath509 the reheating temperature is almost completely insensitive to @xmath54 . the neutrino mass @xmath53 slightly increases when lowering the value of @xmath54 . for large values of the effective neutrino mass , @xmath510 , exactly the opposite is the case . @xmath53 does not depend on @xmath54 anymore and @xmath315 slightly rises when increasing @xmath54 . in the following we will construct semianalytical approximations for @xmath53 and @xmath315 which will allow us to get some intuition for this behaviour . the final gravitinos abundance @xmath391 can be parametrized in the following way , cf . appendix d of ref . @xcite for details , @xmath511 \ , . \label{eq : omegaanaly}\end{aligned}\ ] ] here , the two coefficient functions @xmath512 subsume all factors contributing to @xmath391 that can be taken care of analytically , @xmath513 \ , , \nonumber \\ c_2 = & \ : \frac{3 g_s^4\left(\mu_0\right)}{100 g_s^4\left(t_{\textrm{rh}}\right)}\,,\end{aligned}\ ] ] they both depend only very weakly on the reheating temperature , so that for our purposes it will suffice to treat them as constants , @xmath514 and @xmath515 . the factor @xmath516 parametrizes all effects that can not be accounted for analytically in the derivation of eq . , i.e. the amount of energy in radiation at @xmath316 , the ratio @xmath387 at @xmath316 as well as the increase in the comoving number densities of gravitinos and radiation after @xmath316 . in principle it depends on all mass parameters , in practice after solving the boltzmann equations we find that it is mainly controlled by @xmath54 , @xmath517 where the exponent @xmath518 is @xmath519 for @xmath510 and @xmath520 for @xmath509 . we insert our results for @xmath521 and @xmath516 into eq . , set @xmath410 to @xmath522 and solve for @xmath315 , @xmath523^{-1 } \ , . \label{eq : trhm1tmgfit}\end{aligned}\ ] ] the corresponding expression for @xmath53 can then be obtained by exploiting eq . , @xmath524^{-4/5 } \ , , \label{eq : m1m1tmgfit}\end{aligned}\ ] ] where the exponent @xmath52 is given as @xmath525@xmath1@xmath526 so that @xmath527 for @xmath510 and @xmath528 for @xmath509 . these two fitting formulae reproduce our numerical results with deviations of @xmath529 and nicely illustrate the different dependence of @xmath315 and @xmath53 on @xmath54 for small and large values of @xmath54 , respectively . as expected , they show that the dependence on @xmath54 is always very mild and solely stems from the factor @xmath516 , i.e. corrections beyond the purely analytical result for @xmath410 . if we were to omit these corrections and set @xmath516 to @xmath530 , the reheating temperature required for gravitino dark matter would be a function of @xmath8 only , @xmath531 , in accordance with the fact that the only parameters entering the gravitino production rate @xmath532 are the masses of the gravitino and the gluino . the relation between the gravitino mass and the neutrino parameters @xmath54 and @xmath53 translates the lower bound on @xmath53 imposed by the requirement of successful leptogenesis , cf . eq . , into a lower bound on @xmath8 . as we can read off from fig . [ fig : mgbounds ] , @xmath8 must be at least of @xmath533 to obtain consistency between leptogenesis and gravitino dark matter . in fact , the bound on @xmath8 slightly varies with @xmath54 . for @xmath54 values between @xmath431 and @xmath534 it monotonically increases from roughly @xmath535 to @xmath536 , from @xmath537 onwards it remains at @xmath538 . for such low gravitino masses the first term in the brackets on the right - hand side of eq . is negligibly small , and @xmath315 . ] so that the fitting formula for @xmath53 can be easily solved for @xmath8 , @xmath539 imposing the condition that @xmath53 be larger than @xmath540 , cf . eq . , provides us with an analytical expression for the lower bound on @xmath8 , @xmath541 this estimate reproduces our numerical results with a precision at the level of @xmath542 . physically , the connection between the bounds on @xmath8 and @xmath53 is the following . for gravitino masses below @xmath543 , a reheating temperature @xmath544 is required to avoid overproduction of gravitinos . according to our reheating mechanism such low reheating temperatures are associated with comparatively small values of the neutrino mass , @xmath545 . the low temperature and low mass then entail a small abundance of ( s)neutrinos at the time the asymmetry is generated and a small @xmath9 parameter @xmath256 , cf . eqs . and , respectively . both effects combine and result in an insufficient lepton asymmetry , rendering dark matter made of gravitinos with a mass below @xmath543 inconsistent with leptogenesis . in conclusion , we find that our scenario of reheating can be easily realized in a large fraction of parameter space . the two conditions of successful leptogenesis and gravitino dark matter , in combination with constraints from hybrid inflation , allow us to interconnect parameters of the neutrino and supergravity sector . in particular , we are able to determine the neutrino mass @xmath53 and the reheating temperature @xmath315 as functions of the the effective neutrino mass @xmath54 and the gravitino mass @xmath8 . furthermore , the consistency between all ingredients of our scenario indicates preferences for @xmath53 and @xmath315 , namely @xmath53 values close to @xmath252 and @xmath315 values close to @xmath546 . finally , we obtain a lower bound on the gravitino mass of roughly @xmath547 . a phase of false vacuum of unbroken @xmath0@xmath1@xmath2 symmetry at the gut scale can account for the observed acoustic peaks in the cosmic microwave background via hybrid inflation . subsequent tachyonic preheating , followed by the decay of heavy gauge and higgs particles and heavy neutrinos sets the initial conditions of the hot early universe . we have studied the @xmath0@xmath1@xmath2 breaking phase transition for the full supersymmetric abelian higgs model and given a detailed time - resolved description of the reheating process taking all ( super)particles into account . the competition of cosmic expansion and entropy production leads to an intermediate plateau of constant temperature , during which baryon asymmetry and gravitino dark matter are produced . the initial conditions of the thermal phase of the universe are determined by the parameters of the fundamental lagrangian , i.e. the masses and couplings of elementary particles . likewise , the constant plateau temperature is fixed by neutrino parameters . the temperature scale of reheating is hence no longer an unknown cosmological parameter , but rather an effective quantity that is determined by mass parameters that can in principle be measured in experiments . the consistency of hybrid inflation , leptogenesis and gravitino dark matter restricts the parameter space . for a gluino mass of @xmath548 we find a lower bound on the gravitino mass of about @xmath4 . the order of magnitude of @xmath53 , the mass of the lightest of the heavy neutrinos , is @xmath549 . for a wide range of light neutrino masses this results in a reheating temperature of order @xmath550 . we point out that lowering the scale of @xmath0@xmath1@xmath2 breaking would significantly weaken the bound on the gravitino mass . if @xmath0@xmath1@xmath2 breaking is unrelated to hybrid inflation and takes place at a scale @xmath551 , the gravitino could have a mass of @xmath552 @xcite . similarly , for a lower @xmath0@xmath1@xmath2 scale reheating would occur at a higher temperature because of faster higgs decays . this would result in a stronger washout of the lepton asymmetry generated in ( s)neutrino decays . small @xmath18 hence implies an upper bound on the effective neutrino mass @xmath54 of about @xmath553 @xcite . in this paper we have demonstrated that , if @xmath0@xmath1@xmath2 is broken at the gut scale , this restriction does no longer apply , rendering the proposed reheating mechanism viable for all reasonable masses of the light neutrinos . tachyonic preheating is a complicated nonequilibrium process , which requires further theoretical investigations . a remarkable result of this work is that the final baryon asymmetry and dark matter density are rather insensitive to many of the related theoretical uncertainties , such as the details of the production and relaxation of cosmic strings . for instance , even if 50% of the false vacuum energy density is initially stored in strings , they quickly loose most of their energy and the effect on the final baryon asymmetry and dark matter abundance is negligible . this robustness is due to the fact that after all most of the vacuum energy density is transferred to heavy higgs bosons whose slow decays , via heavy neutrinos , dominate the reheating process . throughout our analysis we have assumed that the gravitino is the lightest superparticle . however , the proposed mechanism for the ignition of the hot early universe also works if the gravitino is very heavy with a neutralino as lsp . in this case ordinary wimp dark matter can be nonthermally produced from gravitino decays . consistency of hybrid inflation , leptogenesis and dark matter density then leads to constraints on gravitino and neutralino masses . in ref . @xcite we give a detailed description of this alternative scenario . further important questions concern the effect of the inflaton on tachyonic preheating @xcite and possible modifications of superpotential and khler potential of the symmetry breaking sector in connection with the detailed description of the cosmic microwave background , which will be discussed elsewhere . * acknowledgements * the authors thank g. vertongen for collaboration at the initial stage of this work and k. kamada and f. takahashi for helpful discussions and comments . this work has been supported by the german science foundation ( dfg ) within the collaborative research center 676 `` particles , strings and the early universe '' . in this section , we present the full supersymmetric lagrangian of the abelian higgs model in unitary gauge , following the notation of ref . @xcite . our starting point is an arbitrary superpotential @xmath554 , given in terms of chiral fields @xmath555 , whose scalar and fermionic components are denoted by @xmath556 and @xmath557 , and the canonical khler potential @xmath558 where @xmath559 is the gauge coupling and @xmath560 is the @xmath12 gauge charge of @xmath555 . @xmath37 denotes the @xmath12 vector superfield , @xmath561 containing the scalar degree of freedom @xmath81 , the fermionic components @xmath562 and @xmath563 , the vector @xmath564 as well as the auxiliary fields @xmath565 , @xmath566 and @xmath85 . in the wess - zumino gauge , which we will not use , one has @xmath567 , @xmath568 and @xmath569 . the supersymmetric lagrangian can be derived in the standard manner by calculating d- and f - terms of khler potential and superpotential and eliminating all auxiliary fields . in order to obtain fields with canonical mass dimension we perform the rescalings @xmath570 where @xmath571 corresponds to one specific , conveniently chosen @xmath572 and @xmath573 is an arbitrary nonvanishing mass scale . in the following , we will promote @xmath573 to a time - dependent function . note , however , that our discussion also applies to the even more general case of a fully spacetime - dependent scalar field @xmath574 . after some calculations , including several integrations by part , one finds the lagrangian , @xmath575 with @xmath576\ , \nonumber\\ \mathcal{l}_{\textrm{wz}}^{\textrm{ferm } } = & \ : \sum_i \exp\left(\frac { p_i \sqrt{2}c}{pv}\right ) \frac{i p_i}{\sqrt{2}}\phi_i^ * \psi_i \xi -\frac{1}{2 } \sum_{i , j } w_{ij } \psi_i\psi_j + \textrm{h.c.}\ , \nonumber \\ v_f = & \ : \sum_i \exp\left(-\frac { p_i \sqrt{2}c}{pv}\right ) w_i^ * w_i\ , \nonumber\\ v_d = & \ : \frac{1}{8 } \sum_{ij } p_i p_j \exp\left(\frac { ( p_i + p_j ) \sqrt{2}c}{pv}\right ) \phi_i^\phi_i \phi_j^ \phi_j\ , \nonumber\end{aligned}\ ] ] @xmath577 \nonumber \\ - & \ : \sum_i \left\{w_i \left(\frac{p_i^2}{2(pv)^2}\phi_i\chi^2 + \frac{i p_i}{p v}\psi_i\chi\right ) + \textrm{h.c . } \right\ } \ , \nonumber\end{aligned}\ ] ] and @xmath578 evaluating the exponential functions in @xmath579 to leading order in @xmath580 yields the familiar lagrangian in wess - zumino gauge ; the remaining terms , collected in @xmath581 , represent additional terms involving the gauge degrees of freedom @xmath81 and @xmath562 . the symmetry breaking sector defined in section [ sec_2 ] , cf . eq . , contains the superfields @xmath14 , @xmath15 and @xmath16 with @xmath0@xmath1@xmath2 charges @xmath582 and @xmath583 . in unitary gauge , cf . , one has @xmath584 with @xmath585 , @xmath586 , and @xmath587 , one now obtains . ] @xmath588 \nonumber \\ - & \ : \cosh\left(\frac{\sqrt{2}c}{v}\right ) \frac{p_s^2}{4 } s^ * s a_\mu a^\mu \ , , \label{eq : lgauge}\\ \mathcal{l}_{\textrm{wz}}^{\textrm{ferm } } = & \:\sinh\left(\frac{\sqrt{2}c}{v}\right ) \frac{i p_s}{\sqrt{2}}s^ * \tilde s \xi + \frac{1}{2 } \sqrt{\lambda } \phi \tilde s \tilde s + \sqrt{\lambda } s \tilde \phi \tilde s + \textrm{h.c.}\ , , \label{eq_lferm } \\ v_f = & \ : \frac{\lambda}{4 } \|v^2_{b - l } - s^2\|^2 + \cosh\left(\frac{\sqrt{2}c}{v}\right ) \lambda \phi^ * \phi \ , s^ * s \ , \\ v_d = & \ : \frac{1}{8 } p_s^2 \sinh^2\left(\frac{\sqrt{2}c}{v}\right ) ( s^ * s)^2 \ , \label{eq_vd } \end{aligned}\ ] ] and @xmath589 \nonumber \\ + & \ : \cosh\left(\frac{\sqrt{2}c}{v}\right ) \bigg[- \frac{i } { v } s^\bar{\chi}\bar{\sigma}^\mu\partial_\mu \frac{s \ , \chi}{v } + \frac { p_s } { \sqrt{2 } v } s^ s \left(\chi\xi + \bar{\chi}\bar{\xi}\right ) \label{eq_nonwz}\\ + & \ : \left\{\frac{1}{\sqrt{2 } v } s^\bar{\chi}\bar{\sigma}^\mu \tilde s \partial_\mu \frac{c}{v } + \frac{i p_s}{2v } s^ \bar{\chi}\bar{\sigma}^\mu \tilde s a_\mu + \textrm{h.c . } \right\ } \bigg ] + \ : \left\{\sqrt{\lambda } \phi s \frac{1}{2 v^2 } s \chi^2 + \textrm{h.c . } \right\ } \nonumber \ .\end{aligned}\ ] ] the ground state of the theory corresponds to @xmath590 . identifying the mass scale @xmath573 with the time - dependent vacuum expectation value of the higgs field in the broken phase , @xmath591 , which approaches @xmath18 at large times , the lagrangian @xmath581 yields kinetic terms for @xmath81 and @xmath562 and a mass term for @xmath562 and @xmath563 . the mass terms for @xmath564 and @xmath81 are contained in eqs . and , respectively . as expected , in unitary gauge the vector field @xmath37 describes a massive vector multiplet @xcite . shifting @xmath592 around its expectation value , @xmath593 , one reads off the masses given in eq . . note that due to the time - dependence of @xmath573 , the kinetic term for @xmath81 in eq . yields a contribution to the mass @xmath594 . in the main part of this paper , we omit this term for two reasons . first , it is much smaller than the contribution to @xmath594 obtained from eq . throughout the preheating process and hence the latter governs the production during tachyonic preheating . second , as we show in section [ sec : robust ] , our final results prove insensitive to the dynamics of the gauge sector and we can hence ignore this technically rather complicated contribution . to calculate the lepton asymmetry consistently to first order in the @xmath9 violation parameter @xmath595 , @xmath208 scattering processes involving an ( anti-)(s)lepton in the initial and final state must be considered . scatterings with an on - shell neutrino in the intermediate state are already included in decay and inverse decay processes . we are hence left with the task to calculate the off - shell contribution of these processes . for the nonsupersymmetric case , this was discussed in refs . @xcite and @xcite . here we explain the supersymmetric case . we first study the @xmath9-violating contribution of the full @xmath208 scattering processes and will see that this vanishes to @xmath596 . hence to this order in the yukawa coupling , the @xmath9-violating off - shell contributions can be added by subtracting the corresponding on - shell contributions . the right - hand side of the integrated boltzmann equation is given by the interaction density @xmath597 , cf . eqs and . for distinct final and initial states , this is related to the corresponding @xmath598-matrix elements @xmath599 where the summation over the lower case letters on the left - hand side runs over different particle species and the summation over capital letters on the right - hand side additionally includes the summation over all internal degrees of freedom as well as phase space integrals for all initial and final state particles . considering the case of @xmath208 scatterings in the boltzmann equation for the lepton asymmetry , the initial and final states of interest are @xmath600 . the internal degrees of freedom are helicity , weak isospin and flavour . @xmath601 denotes the phase space distribution function of particle species @xmath602 . using this notation , we now consider the @xmath9-violating contributions of the full @xmath208 scattering processes , @xmath603 & = \sum_{i , f } \left[\|s_{\bar{f}i}\|^2 f_i - \|s_{f \bar{i } } \|^2 f_{\bar i } \right]\\ & = \sum_{i , f } \left[\|s_{\bar f i}\|^2 + \|s _ { f i}\|^2 - \|s_{f \bar{i } } \|^2 - \|s _ { \bar f \bar i}\|^2 \right ] f_i \\ & = \sum_i \left[1 - 1\right ] f_i + { \cal o}((h^{\nu})^4 ) = { \cal o}((h^{\nu})^4 ) \ , . \end{split}\ ] ] the bar indicates @xmath9 conjugation and @xmath604 are the phase space distributions of the light mssm ( anti-)particles in thermal equilibrium . here in the second line of eq . , we extended the summation over the final states to include the lepton number conserving processes . these can be grouped in pairs of @xmath209 conjugates and hence , due to @xmath209 invariance , yield a vanishing contribution in total . in the third line , we exploit the unitarity of the @xmath598 matrix , i.e. that the summation over all possible final states yields 1 . since however in eq . the sum runs only over all possible two - particle final states , we obtain corrections caused by neglecting multi - particle final states . for off - shell intermediate states these corrections are of @xmath605 @xcite , however close to the resonance pole they are enhanced to @xmath606 @xcite . concluding , we find that the @xmath9-violating contributions of the @xmath208 scattering processes involved in the production of the lepton asymmetry vanish , with corrections of @xmath596 . hence the on- and off - shell contributions cancel each other and we can use the usual ` recipe ' of replacing the @xmath9-violating contributions of the off - shell ( s)neutrino decays by the negative of the respective on - shell contributions , i.e.@xmath607 where @xmath457 is a flavour index . note that looking at this line of argument closely , this argument holds separately for neutrinos and sneutrinos because of distinct sets of initial and final states , but the summation over flavour and lepton / slepton is unavoidable . apart from the definition of the reheating temperature employed in this work , cf . , there are alternative ways to define the reheating temperature . for instance , we could use the temperature at the beginning ( @xmath305 ) or the end of reheating ( @xmath306 ) or the temperature when half of the total available energy has been transferred to radiation ( @xmath309 for the parameter example discussed in section [ sec : example ] ) . in either case , although the respective value for @xmath265 may significantly vary , thanks to the temperature plateau during reheating the resulting reheating temperature would not change much . for the parameter point investigated in section [ sec : example ] , we find @xmath608 our definition of the reheating temperature may hence be regarded as a compromise between several more extreme approaches . but more important than that , it picks up on a physical feature that other definitions would miss . in fig . [ fig : tempasymm ] we observe that the temperature declines less during the first part of reheating , @xmath609 , than during the second part , @xmath610 . the stage of @xmath6 reheating evidently splits up into two phases , during the first of which the temperature is basically constant , whereas during the second one the temperature slightly decreases . the reason for this substructure in the temperature plateau is the following . as soon as the @xmath288 neutrinos decay more efficiently their comoving number density starts to grow slower than @xmath611 . this diminishes the production rate of radiation . according to eq . , a constant temperature can then no longer be maintained . the advantage of our definition for @xmath315 now is that we read it off the curve in fig . [ fig : tempasymm ] at exactly that value of the scale factor at which the transition between these two phases of @xmath6 reheating takes place . our definition thus yields a temperature that is both representative as it mediates between several more extreme values and especially singled out as it is associated with a prominent feature in the temperature curve . for completeness , we should however mention that for other parameter choices this picture may change . if the higgs decay rate @xmath299 is , for instance , larger than the neutrino decay rate @xmath317 , which can for example be achieved by going to lower values of the @xmath0@xmath1@xmath2 scale , the scaling behaviour of the @xmath288 number density changes when the neutrino production efficiency begins to cease and not when the decays of the neutrinos themselves set in . the slight kink in the temperature plateau is then located at @xmath309 which is in this case before the decay of the @xmath288 has become fully efficient . but the definition of the reheating temperature in eq . remains reasonable nonetheless . after all , if @xmath612 , the bulk of the total energy is first almost entirely accumulated in @xmath288 neutrinos before it is passed on to radiation . the energy in radiation thus receives its major contribution just when these neutrinos decay with a sufficient efficiency . the characteristic temperature at the time when this happens is then again obtained from eq . . further details on the reheating temperature in regions in parameter space in which @xmath612 can be found in ref . @xcite . s. raby , eur . j. c * 59 * , 223 ( 2009 ) , arxiv:0807.4921 [ hep - 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ex ] . w. buchmuller , v. domcke and k. schmitz , arxiv:1203.0285 [ hep - ph ] . j. martin and v. vennin , arxiv:1110.2070 [ astro-ph.co ] . w. buchmuller , v. domcke , k. schmitz and f. takahashi , in preparation .	the decay of a false vacuum of unbroken @xmath0@xmath1@xmath2 symmetry is an intriguing and testable mechanism to generate the initial conditions of the hot early universe . if @xmath0@xmath1@xmath2 is broken at the grand unification scale , the false vacuum phase yields hybrid inflation , ending in tachyonic preheating . the dynamics of the @xmath0@xmath1@xmath2 breaking higgs field and thermal processes produce an abundance of heavy neutrinos whose decays generate entropy , baryon asymmetry and gravitino dark matter . we study the phase transition for the full supersymmetric abelian higgs model . for the subsequent reheating process we give a detailed time - resolved description of all particle abundances . the competition of cosmic expansion and entropy production leads to an intermediate period of constant ` reheating ' temperature , during which baryon asymmetry and dark matter are produced . consistency of hybrid inflation , leptogenesis and gravitino dark matter implies relations between neutrino parameters and superparticle masses . in particular , for a gluino mass of @xmath3 , we find a lower bound on the gravitino mass of @xmath4 . desy 11 - 174 + march 2012 * spontaneous @xmath0@xmath1@xmath2 breaking as + the origin of the hot early universe * w. buchmller , v. domcke , k. schmitz + _ deutsches elektronen - synchrotron desy , 22607 hamburg , germany _ .	41,602	410
because many natural systems are organized as networks , in which the nodes ( be they cells , individuals , populations or web servers ) interact in a time - dependent fashion the study of networks has been an important focus in recent research . one of the particular points of interest has been the question of how the hardwired _ structure _ of a network ( its underlying graph ) affects its _ function _ , for example in the context of optimal information storage or transmission between nodes along time . it has been hypothesized that there are two key conditions for optimal function in such networks : a well - balanced adjacency matrix ( the underlying graph should appropriately combine robust features and random edges ) and well - balanced connection strengths , driving optimal dynamics in the system . however , only recently has mathematics started to study rigorously ( through a combined graph theoretical and dynamic approach ) the effects of configuration patterns on the efficiency of network function , by applying graph theoretical measures of segregation ( clustering coefficient , motifs , modularity , rich clubs ) , integration ( path length , efficiency ) and influence ( node degree , centrality ) . various studies have been investigating the sensitivity of a system s temporal behavior to removing / adding nodes or edges at different places in the network structure , and have tried to relate these patterns to applications to natural networks . brain functioning is one of the most intensely studied contexts which requires our understanding of the tight inter - connections between system architecture and dynamics . the brain is organized as a `` dynamic network , '' self - interacting in a time - dependent fashion at multiple spacial and temporal scales , to deliver an optimal range for biological functioning . the way in which these modules are wired together in large networks that control complex cognition and behavior is one of the great scientific challenges of the 21st century , currently being addressed by large - scale research collaborations , such as the human connectome project . graph theoretical studies of empirical empirical data support certain generic topological properties of brain architecture , such as modularity , small - worldness , the existence of hubs and `` rich clubs '' @xcite . in order to explain how connectivity patterns may affect the system s dynamics ( e.g. , in the context of stability and synchronization in networks of coupled neural populations ) , and thus the observed behavior , a lot of effort has been thus invested towards formal modeling approaches , using a combination of analytical and numerical methods from nonlinear dynamics and graph theory , in both biophysical models @xcite and simplified systems @xcite . these analyses revealed a rich range of potential dynamic regimes and transitions @xcite , shown to depend as much on the coupling parameters of the network as on the arrangement of the excitatory and inhibitory connections @xcite . the construction of a realistic , data - compatible computational model has been subsequently found to present many difficulties that transcend the existing methods from nonlinear dynamics , and may in fact require : ( 1 ) new analysis and book - keeping methods and ( 2 ) a new framework that would naturally encompass the rich phenomena intrinsic to these systems both of which aspects are central to our proposed work . in a paper with dr . verduzco - flores @xcite , one of the authors of this paper first explored the idea of having network connectivity as a bifurcation parameter for the ensemble dynamics in a continuous time system of coupled differential equations . we used configuration dependent phase spaces and our probabilistic extension of bifurcation diagrams in the parameter space to investigate the relationship between classes of system architectures and classes of their possible dynamics , and we observed the robustness of the coupled dynamics to certain changes in the network architecture and its vulnerability to others . as expected , when translating connectivity patterns to network dynamics , the main difficulties were raised by the combination of graph complexity and the system s intractable dynamic richness . in order to break down and better understand this dependence , we started to investigate it in simpler theoretical models , where one may more easily identify and pair specific structural patterns to their effects on dynamics . the logistic family is historically perhaps the most - studied family of maps in nonlinear dynamics , whose behavior is by now relatively well understood . therefore , we started by looking in particular at how dynamic behavior depends on connectivity in networks with simple logistic nodes . this paper focuses on definitions , concepts and observations in low - dimensional networks . future work will address large networks , and different classes of maps . dynamic networks with discrete nodes and the dependence of their behavior on connectivity parameters have been previously described in several contexts over the past two decades . for example , in an early paper , wang considered a simple neural network of only two excitatory / inhibitory neurons , and analyzed it as a parameterized family of two - dimensional maps , proving existence of period - doubling to chaos and strange attractors in the network @xcite . masolle , attay et al . have found that , in networks of delay - coupled logistic maps , synchronization regimes and formation of anti - phase clusters depend on coupling strength @xcite and on the edge topology ( characterized by the spectrum of the graph laplacian ) @xcite . yu has constructed and studied a network wherein the undirected edges symbolize the nodes relation of adjacency in an integer sequence obtained from the logistic mapping and the top integral function @xcite . in our present work , we focus on investigating , in the context of networked maps , extensions of the julia and mandelbrot sets traditionally defined for single map iterations . for three different model networks , we use a combination of analytical and numerical tools to illustrate how the system behavior ( measured via topological properties of the _ julia sets _ ) changes when perturbing the underlying adjacency graph . we differentiate between the effects on dynamics of different perturbations that directly modulate network connectivity : increasing / decreasing edge weights , and altering edge configuration by adding , deleting or moving edges . we discuss the implications of considering a rigorous extension of fatou - julia theory known to apply for iterations of single maps , to iterations of ensembles of maps coupled as nodes in a network . the logistic map is historically perhaps the best - known family of maps in nonlinear dynamics . iterations of one single quadratic function have been studied starting in the early 19th century , with the work of fatou and julia . the prisoner set of a map @xmath3 is defined as the set of all points in the complex dynamic plane , whose orbits are bounded . the escape set of a complex map is the set of all points whose orbits are unbounded . the julia set of @xmath3 is defined as their common boundary @xmath4 . the filled julia set is the union of prisoner points with their boundary @xmath4 . for polynomial maps , it has been shown that the connectivity of a map s julia set is tightly related to the structure of its critical orbits ( i.e. , the orbits of the map s critical points ) . due to extensive work spanning almost one century , from julia @xcite and fatou @xcite until recent developments @xcite , we now have the following : + * fatou - julia theorem . * _ for a polynomial with at least one critical orbit unbounded , the julia set is totally disconnected if and only if all the bounded critical orbits are aperiodic . _ + for a single iterated logistic map@xcite , the fatou - julia theorem implies that the julia set is either totally connected , for values of @xmath5 in the mandelbrot set ( i.e. , if the orbit of the critical point 0 is bounded ) , or totally disconnected , for values of @xmath5 outside of the mandelbrot set ( i.e. , if the orbit of the critical point 0 is unbounded ) . in previous work , the authors showed that this dichotomy breaks in the case of random iterations of two maps @xcite . in our current work , we focus on extensions for networked logistic maps . although julia and mandelbrot sets have been studied somewhat in connection with coupled systems @xcite , none of the existing work seems to address the basic problems of how these sets can be defined for networks of maps , how different aspects of the network hardwiring affect the topology of these sets and whether there is any fatou - julia type result in this context . these are some of the questions addressed in this paper , which is organized as follows : in section [ logistic_maps ] , we introduce definitions of our network setup , as well as of the extensions of mandelbrot and julia sets that we will be studying . in order to illustrate some basic ideas and concepts , we concentrate on three examples of 3-dimensional networks , which differ from each other in edge distribution , and whose connectivity strengths are allow to vary . in section [ complex_maps ] , we focus on the behavior of these 3-dimensional models when we consider the nodes as complex iterated variables . we analyze the similarities and differences between node - wise behavior in each case , and we investigate the topological perturbations in one - dimensional complex slices of the mandelbrot and julia sets , as the connectivity changes from one model to the next , through intermediate stages . in section [ real_maps ] , we address the same questions for real logistic nodes , with the advantage of being able to visualize the entire network mandelbrot and julia sets , as 3-dimensional real objects . in both sections , we conjecture weaker versions of the fatou - julia theorem , connecting points in the mandelbrot set with connectivity properties of the corresponding julia sets . finally , in section [ discussion ] , we interpret our results both mathematically and in the larger context of network sciences . we also briefly preview future work on high - dimensional networks and on networks with adaptable nodes and edges . we consider a set of @xmath6 nodes coupled according to the edges of an oriented graph , with adjacency matrix @xmath7 ( on which one may impose additional structural conditions , related to edge density or distribution ) . in isolation , each node @xmath8 , @xmath9 , functions as a discrete nonlinear map @xmath10 , changing at each iteration @xmath11 as @xmath12 . when coupled as a network with adjacency @xmath13 , each node will also receive contributions through the incoming edges from the adjacent nodes . throughout this paper , we will consider an additive rule of combining these contributions , for a couple of reasons : first , summing weighted incoming inputs is one simple , yet mathematically nontrivial way to introduce the cross talk between nodes ; second , summing weighted inputs inside a nonlinear integrating function is reminiscent of certain mechanisms studied in the natural sciences ( such as the integrate and fire neural mechanism studied in our previous work in the context of coupled dynamics ) . the coupled system will then have the following general form : @xmath14 where @xmath15 are the weights along the adjacency edges . one may view this system simply as an iteration of an @xmath6-dimensional map @xmath16 , with @xmath17 ( in the case of real - valued nodes ) , or respectively @xmath18 ( in the case of complex - valued nodes ) . the new and exciting aspect that we are proposing in our work is to study the dependence of the coupled dynamics on the parameters , in particular on the coupling scheme ( adjacency matrix ) viewed itself as a system parameter . to fix these ideas , we focused first on defining these questions and proposing hypotheses for the case of quadratic node - dynamics . the logistic family is one of the most studied family of maps in the context of both real and complex dynamics of a single variable . it was also the subject of our previous modeling work on random iterations . in this paper in particular , we will work with quadratic node - maps , with their traditional parametrization @xmath19 , with @xmath20 and @xmath21 for the complex case and @xmath22 and @xmath23 for the real case . the network variable will be called respectively @xmath24 in the case of complex nodes , and @xmath25 in the case of real nodes . we consider both the particular case of identical quadratic maps ( equal @xmath5 values ) , and the general case of different maps attached to the nodes throughout the network . in both cases , we aim to study the asymptotic behavior of iterated node - wise orbits , as well as of the @xmath6-dimensional orbits ( which we will call multi - orbits ) . as in the classical theory of fatou and julia , we will investigate when orbits escape to infinity or remain bounded , and how much of this information is encoded in the critical multi - orbit of the system . for the following definitions , fix the network ( i.e. , fix the adjacency @xmath13 and the edge weights @xmath26 ) . to avoid redundancy , we give definitions for the complex case , but they can be formulated similarly for real maps : for a fixed parameter @xmath27 , we call the * filled multi - julia set * of the network , the locus of @xmath28 which produce a bounded multi - orbit in @xmath29 . we call the * filled uni - julia set * the locus of @xmath30 so that @xmath31 produces a bounded multi - orbit . the * multi - julia set ( or the multi - j set ) * of the network is defined as the boundary in @xmath29 of the filled multi - julia set . similarly , one defines the * uni - julia set ( or uni - j set ) * of the network as the boundary in @xmath32 of its filled counterparts . we define the * multi - mandelbrot set ( or the multi - m set ) * of the network the parameter locus of @xmath27 for which the multi - orbit of the critical point @xmath33 is bounded in @xmath29 . we call the * equi - mandelbrot set ( or the equi - m set ) * of the network , the locus of @xmath21 for which the critical multi - orbit is bounded for * equi - parameter * @xmath34 . we call the * @xmath35th node equi - m set * the locus @xmath21 such that the component of the multi - orbit of @xmath33 corresponding to the @xmath35th node remains bounded in @xmath32 . we study , using a combination of analytical and numerical methods , how the structure of the julia and mandelbrot sets varies under perturbations of the node - wise dynamics ( i.e. , under changes of the quadratic multi- parameter @xmath36 ) and under perturbations of the coupling scheme ( i.e. , of the adjacency matrix @xmath13 and of the coupling weights @xmath26 ) . in this paper , we start with investigating these questions in small ( 3-dimensional ) networks , with specific adjacency configurations . in a subsequent paper , we will move to investigate how similar phenomena may be quantified and studied analytically and numerically in high - dimensional networks . in both cases , we are interested in particular in observing differences in the effects on dynamics of three different aspects of the network architecture : ( 1 ) increasing / decreasing edge weights , ( 2 ) increasing / decreasing edge density , ( 3 ) altering edge configuration by adding , deleting or moving edges . while a desired objective would be to obtain general results for all network sizes ( since many natural networks are large ) , we start by studying simple , low dimensional systems . in this study , we focus on simple networks formed of three nodes , connected by different network geometries and edge weights . to fix our ideas , we will follow and illustrate three structures in particular ( also see figure [ 3d_net ] ) : ( 1 ) two input nodes @xmath37 and @xmath38 are self driven by quadratic maps , and the single output node @xmath39 is driven symmetrically by the two input nodes ; @xmath37 additionally communicates with @xmath38 via an edge of variable weight @xmath40 , which can take both positive and negative values . we will call this the * _ simple dual model_. ( 2 ) in addition to the simple dual scheme , the output node @xmath39 is also self - driven , i.e. there is a self - loop on @xmath39 of weight @xmath41 ( which can be positive or negative ) . we will call this the _ self - drive model_. ( 3 ) in addition to the self - driven model , there is also feedback from the output node @xmath39 into the node @xmath38 , via a new edge of variable weight @xmath3 . we will call this the _ feedback model_. unless specified , edges have positive unit weight . notice that the same effect as negative feed - forward edges from @xmath37 and @xmath38 into @xmath39 can be obtained by changing the sign of @xmath41 , etc . the three connectivity models we chose to study and compare are described by the equations below : + _ simple dual model : _ * @xmath42 * _ self - drive model : _ * @xmath43 * _ feedback model : _ * @xmath44 for a fixed multi - parameter @xmath45 for example , one can see all three systems as generated by a network map @xmath46 , @xmath47 , @xmath48 , defined as @xmath49_1),f_{c_2}([az]_2),f_{c_3}([az]_3))$ ] , for any @xmath50 . we try to classify and understand the effects that coupling changes have on the topology of multi - j and multi - m sets for both complex and real networked maps . we do nt expect all classical topology results on the julia and mandelbrot sets for single maps ( e.g. , fatou - julia theorem , or connectivity of the mandelbrot set ) to carry out for networks of coupled maps . however , since the topology of the full sets in @xmath51 is somewhat harder to inspect , we study as a first step their equi - slices and node - wise equi - slices , which are objects in @xmath32 . we will track and compare in particular the differences between the three models , but also the geometric and topological changes produced on the equi - slices within each one model for different values of the parameters @xmath40 , @xmath41 and @xmath3 . none of these results , however , can be directly extrapolated to similar conclusions on the full sets . to offer some insight into the latter , we study the multi - m and multi - j sets in the context of real maps , for which there objects can be visualized in @xmath52 . a first intuitive question is when the nodes of the network have similar behavior , and whether if one node - wise orbit is bounded , the others will remain bounded . this relationship is trivial to establish in some cases , such as for example in the simple dual model with independent input nodes ( i.e. , @xmath53 ) . indeed , in this model , for any fixed @xmath21 , the origin s orbit in @xmath51 under @xmath54 can be described as : @xmath55 the projection of the orbit in any of the three components only depends on the previous states of @xmath37 and @xmath38 , and these three sequences are simultaneously bounded in @xmath32 , hence the node - specific equi - mandelbrot sets are all identical with the traditional mandelbrot set . some basic connections between node - wise equi - m sets in each of the three models are stated below . we will prove these incrementally ( recall that the dual model is a particular case of self - drive for @xmath56 , and the self - drive is a particular case of feedback model with @xmath57 ) . , the equi - mandelbrot set for the nodes @xmath38 and @xmath39 are identical ( red ) , but different from the set for the node @xmath37 ( blue ) * b. * for the self - drive model with negative feedback , @xmath58 and @xmath59 , the equi - mandelbrot sets for the three nodes @xmath37 , @xmath38 and @xmath39 ( shown respectively in blue , green and red ) are all different . * c. * for the feedback model with with negative feedback , @xmath58 and @xmath59 , @xmath60 , the equi - mandelbrot set for the nodes @xmath38 and @xmath39 are identical ( red ) , but different from the set for the node @xmath37 ( blue ) . in all panels , the computations were generated based on @xmath61 iterations , and for a test radius of @xmath62 . _ _ ] in the simple dual model , the node - wise equi - m sets for the nodes @xmath38 and @xmath39 are identical subsets of the traditional mandelbrot set ( which is the equi - m set for node @xmath37 ) . [ prop_simple_dual ] an additional self - drive @xmath63 applied to the output node changes the balance of inputs to @xmath39 , in the following sense : in the self - drive model , the node - wise equi - m sets of @xmath38 and @xmath39 remain subsets of the standard mandelbrot set , but the equi - m set of @xmath39 is strictly contained in the equi - m set of @xmath38 ( figure [ node_differences]b ) . [ prop_self_drive ] finally , introducing any arbitrary feedback @xmath64 re - couples the behavior of nodes @xmath38 and @xmath39 , producing a common equi - mandelbrod set , largely shrunk from the simple dual version : in the feedback model with @xmath63 and @xmath64 , the node - wise equi - m sets for the nodes @xmath38 and @xmath39 are again identical subsets of the traditional mandelbrot set ( figure [ node_differences]c ) . [ prop_feedback ] for the rest of the section , the term of `` equi - m set '' will be referring to the equi - mandelbrot set of the network , which is the intersection of the three node - specific sets . we illustrate the equi - m set for the three models and for different levels of cross - talk @xmath40 , @xmath41 and @xmath3 between nodes . starting with the simple dual input version of the model , we show in figure [ dual_input ] the effects of changing the level @xmath40 of talk between the input nodes , on the shape of the equi - m set . it is not surprising that , in both positive and negative @xmath40 ranges , increasing @xmath65 gradually shrinks the equi - mandelbrot set . this can be motivated intuitively by the fact that an additional contribution to the node @xmath38 may cause the critical orbit to increase faster in the @xmath38 , and subsequently the @xmath39 components , hence points in the traditional set will no longer be included in the mutants for @xmath66 . as @xmath40 increases in the positive range , we noticed that the network m sets form nested subsets ( which is not true for the negative range ) , that they remain connected for all values of @xmath40 , and that the hausdorff dimension of the boundary increases with @xmath40 ( in figure [ dual_input ] , notice an increased wrinkling of the boundary as @xmath40 takes larger positive values , and an increase smoothing as @xmath40 takes negative values with increasing absolute value ) . perturbations of @xmath40 in the positive range seem to have a much more substantial contribution to the size of the equi - m set , while perturbations of @xmath40 in the negative range have a lesser influence on the size , and affect mostly the region close tot the boundary of the equi - m set , and the boundary topological details . we will track the same changes in @xmath40 in the other network models , and investigate if this trend is consistent . increases : * a. * @xmath58 ; * b. * @xmath67 ; * c. * @xmath53 ( traditional mandelbrot set ) ; * d. * @xmath68 ; * e. * @xmath69 . _ _ ] figure [ equimand_selfdrive ] illustrates the evolution of the equi - m set in the case of the model with self - drive , for a grid of positive and negative values of the input connectivity @xmath40 and of the self - drive @xmath41 . below are some simple visual observations based on our numerical computations , to be addressed analytically in future work . decreasing @xmath41 in the negative range produces no alteration of the m sets when @xmath70 . however , it induces dramatic changes in shape and connectivity when @xmath71 . if for @xmath72 relatively large , increasing @xmath40 only slightly alters the shape of the set , for small @xmath72 the size of the set is also altered with increasing @xmath40 ( generating smaller and smaller subsets ) , and the complexity of its boundary also seems to increase . the effects of varying @xmath71 for a fixed value of @xmath41 become more dramatic with decreasing @xmath41 in the negative range . these effects include changes in shape and topology , the region @xmath71 and @xmath73 allowing the m - set to break into multiple connected components . and @xmath41 . * the rows show , from bottom to top , increasing values of the self - drive : @xmath74 , @xmath75 ; @xmath76 ; @xmath56 ( this row representing the simple dual model , as shown in figure [ dual_input ] ) ; @xmath59 ; @xmath77 ; @xmath78 . the columns show , from left to right , increasing values of cross - talk between the two input nodes : @xmath58 , @xmath67 , @xmath53 , @xmath68 and @xmath69 . all the equi - m sets were generated from @xmath61 iterations , and plotted at the same scale , in the complex square @xmath79 \times [ -1.5,1.5]$].__,scaledwidth=90.0% ] in this section , we will track the changes in the uni - julia set when the parameters of the system change . one of our goals is to test , first in the case of equi - parameters @xmath21 , then for general parameters in @xmath51 , if a fatou - julia type theorem applies in the case of our three networks . first , we try to establish a hypothesis for connectedness of uni - j sets , by addressing numerically and visually questions such as : `` is it true that if @xmath5 is in the equi - m set of a network , then the uni - julia set is connected ? '' `` is it true that , if @xmath5 is not in the equi - m set of the network , then the uni - julia set is totally disconnected ? '' clearly , this is not simply a @xmath51 version of the traditional fatou - julia theorem , but rather a slightly different result involving the projection of the julia set onto a uni - slice . notice that a connected uni - j set in @xmath32 may be obtained from a disconnected @xmath51 network julia set , and conversely , that a disconnected uni - j projection may come from a connected julia in @xmath51 . we will further discuss @xmath51 versions of these objects in the context of iterations of real variables , where one can visualize the full mandelbrot and julia sets for the network as subsets of @xmath52 . here , we will first investigate uni - j sets for equi - parameters @xmath80 , with a particular focus on tracking the topological changes of the uni - j set as the system approaches the boundary of the equi - m set and leaves the equi - m set . and @xmath74 , for different values of the equi - parameter @xmath5 ( marked with colored dots on the equi - m template in upper left ) : @xmath81 ( red ) ; @xmath82 ( green ) ; @xmath83 ( blue ) ; @xmath84 ( orange ) ; @xmath85 ( purple ) . all sets were based on @xmath86 iterations . both equi - m and uni - j sets coincide in this case with the traditional mandebrot and julia sets for single map iterations.__,scaledwidth=65.0% ] and @xmath76 , for different values of the equi - parameter @xmath5 ( marked with colored dots on the equi - m template in upper left ) : @xmath87 ( red ) ; @xmath88 ( green ) ; @xmath84 ( orange ) ; @xmath89 ( blue ) ; @xmath90 ( dark purple ) ; @xmath91 ( cyan ) ; @xmath92 ( magenta ) . for the first four panels , @xmath5 is in the equi - m set ; for the last two , @xmath5 is outside of the equi - m set . all sets were based on 100 iterations.__,scaledwidth=90.0% ] , as the network profile is changed , from * a. * simple dual with @xmath93 , to * b. * self - drive with additional @xmath74 , to * c. * feedback with additional @xmath94.__,scaledwidth=60.0% ] first , we fix the network type and the connectivity profile ( i.e. , the parameters @xmath40 , @xmath41 and @xmath3 ) , and we observe how the uni - j sets evolves as the equi - parameter @xmath5 changes . in figures [ unijulia1 ] and [ unijulia2 ] we illustrate this for two examples of self - driven models : one with @xmath74 and @xmath53 , the other with @xmath76 and @xmath58 . as the parameter point @xmath95 approaches the boundary of the equi - m set , the topology of the uni - j set if affected , with its connectivity braking down `` around '' the boundary . second , we look at the dependence of uni - julia sets on the coupling profile ( network type ) . as an example , we fixed the equi - parameter @xmath96 , and we first considered a simple dual network with negative feed - forward and small cross - talk @xmath97 . we then added self - drive @xmath74 to the output node , then additionally introduced a small negative feedback @xmath98 . the three resulting uni - julia sets are shown in figure [ unijulia_model_comparison ] . notice that a very small degree of feedback @xmath3 produces a more substantial difference than a significant change in the self - drive @xmath41 . third , one can study the dependence of uni - julia sets on the strength of specific connections within the network . as a simple illustration of how complex this dependence may be , we show in figures [ c2_julia ] and [ c3_julia ] the effects on the uni - j sets of slight increases in the cross - talk parameter @xmath40 , for two different values of the equi - parameter @xmath5 . an immediate observation is that uni - j sets no not exhibit the dichotomy from traditional single - map iterations no longer stands : uni - j sets can be connected , totally disconnected , but also disconnected into a ( finite or infinite ) number of connected components , without being totally disconnected . based on our illustrations , we can further conjecture , in the context of our three models , a description of connectedness for uni - j sets , as follows : and equi - parameter @xmath99 , as the input cross - talk @xmath40 is increased . the panels show , left to right : @xmath100 , @xmath93 , @xmath53 , @xmath101 and @xmath102.__,scaledwidth=90.0% ] and equi - parameter @xmath96 , as the input cross - talk @xmath40 is increased . the panels show , left to right : @xmath103 , @xmath104 , @xmath105 , @xmath97 , @xmath53 , @xmath106 and @xmath107 , @xmath108.__,scaledwidth=90.0% ] for any of the three models described , and for any equi - parameter @xmath21 , the uni - j set is connected only if @xmath5 is in the equi - m set of the network , and it is totally disconnected only if @xmath5 is not in the equi - m set of the network . * the conjecture implies a looser dichotomy regarding connectivity of uni - j sets than that delivered by the traditional fatou - julia result for single maps : if @xmath5 is in the equi - m set of the network , then the uni - j set is either connected or disconnected , without being totally disconnected . if @xmath5 is not in the equi - m set of the network , then the uni - j set is disconnected ( allowing in particular the case of totally disconnected ) . + finally , we want to remind the reader that uni - julia sets can be defined for general parameters @xmath45 , as shown in figure [ general_unijulia ] . , with @xmath109 , @xmath110 and @xmath111 . the panels represent uni - j sets for a self - drive network with @xmath74 , as the cross - talk @xmath40 changes from * a. * @xmath53 , to * b. * @xmath112 , to * c. * @xmath113.__,scaledwidth=70.0% ] * center . * self - drive network with @xmath114 and @xmath78 . * right . * self - drive network with @xmath115 and @xmath78 . plots were generated with @xmath116 iterations and in resolution @xmath117.__,scaledwidth=90.0% ] the same definitions apply for iterations of real quadratic maps , with the real case presenting the advantage of easy visualization of full julia and mandelbrot sets , rather than having to consider equi - slices , as we did in the complex case . in figures [ realmand ] and [ realjulia ] , we illustrate a few multi - m and multi - j sets respectively , for some of the same networks considered in our complex case . moving to illustrate the _ relationship _ between the multi - m and the multi - j set in this case , consider for example the self - drive real network with @xmath115 and @xmath78 , for different parameters @xmath118 . while more computationally intensive , higher - resolution figures would be necessary to establish the geometry and fractality of these sets , one may already notice basic properties . for example , figure [ realjulia ] shows that , if one were to consider complex equi - parameters , the multi - julia set may not only be connected ( figure [ realjulia]a ) , or totally disconnected ( not shown ) , but may also be broken into a number of connected components ( figures [ realjulia]b anc c ) . and @xmath78 , with equi - parameters respectively : * a. * @xmath119 ; * b. * @xmath120 ; * c. * @xmath96 . plots were generated with @xmath116 iterations and in resolution @xmath117.__,scaledwidth=90.0% ] this remained true if we returned to our restriction of having real parameters , once we allow arbitrary ( that is , not necessarily equi ) parameters . the panels of figure [ real_comp ] show the multi - j sets for two different , but close parameters , @xmath121 and @xmath122 respectively , both of which are not in the multi - m set . the figures suggest a disconnected ( although not totally disconnected ) multi - j set in the first case , and a connected multi - j set in the second case . this implies that in this case the fatou - julia dichotomy fails in its traditional form and that the statement relating boundedness of the critical orbit with connectedness of the multi - j set does not hold for real networks . more precisely , we found parameters for which the multi - julia set appears to be connected , although the critical multi - orbit is unbounded . on the other hand , the counterpart of the theorem may still hold , in the following form : `` the multi - j set is connected if the parameter belongs to the multi - m set . '' part of our current work consists in optimizing the numerical algorithm for multi - m and j sets in real networks , with high enough resolution to allow ( 1 ) observation of possible fractal properties of multi - j sets and of multi - m sets boundaries and ( 2 ) computation of the genus of the filled multi - j sets , in attempt to phrase a topological extension of the theorem that takes into account the number of handles and tunnels that open up in these sets as their connectivity breaks down when leaving the mandelbrot set . and @xmath78 . the two multi - parameters @xmath121 ( left panels ) and @xmath122 ( right panels ) are not in the mandelbrot set for the network . the top row shows the 3-dimensional julia sets , the bottom panels show top views of the same sets . plots were generated with @xmath116 iterations and in resolution @xmath117.__,scaledwidth=60.0% ] in this paper , we used a combination of analytical and numerical approaches to propose possible extensions of fatou - julia theory to networked complex maps . we started by showing that , even in networks where all nodes are identical maps , their behavior may not be `` synchronized : '' the node - wise mandelbrot sets may be identical in some cases , which in others they may differ substantially , depending on the coupling pattern . we then investigated how specific changes in the network hard - wiring trigger different effects on the structure of the network mandelbrot and julia sets , focusing in particular on observing topological properties ( connectivity ) and fractal behavior ( haussdorff dimension ) . we found instances in which small perturbations in the strength of one single connection may lead to dramatic topological changes in the asymptotic sets , and instances in which these sets are robust to much more significant changes . more generally , our paper suggests a new direction of study , with potentially tractable , although complex mathematical questions . while existing results do not apply to networks in their traditional form , it appears that connectivity of the newly defined uni - julia sets may still be determined by the behavior of the critical orbit . we conjectured a weaker extension of the fatou - julia theorem , which was based only on numerical inspection , and which remains subject to a rigorous study that would support or refute it . there are a few interesting aspects which we aim to address in our future work on iterated networks . for example , we are interested in studying the structure of equi - m and uni - j sets for larger networks , and in understanding the connection between the network architecture and its asymptotic dynamics . this direction can lead to ties and applications to understanding functional networks that appear in the natural sciences , which are typically large . the authors previous work has addressed some of these aspects in the context of continuous dynamics and coupled differential equations . however , when translating network architectural patters into network dynamics , the great difficulty arises from a combination of the graph complexity and the system s intractable dynamic richness . addressing the question at the level of low - dimensional networks can help us more easily identify and pair specific structural patterns to their effects on dynamics , and thus better understand this dependence . the next natural step is to return to the search for a similar classification in high - dimensional networks , where specific graph measures or patters ( e.g. node - degree distribution , graph laplacian , presence of strong components , cycles or motifs ) may help us , independently or in combination , classify the network s dynamic behavior . * described schematically on the left , together with their adjacency matrices . both systems have connectivity parameters @xmath123 , @xmath124 . _ _ ] @xmath125 = @xmath126 $ ] , @xmath127 + + @xmath128 = @xmath129 , @xmath130 * described schematically on the left , together with their adjacency matrices . both systems have connectivity parameters @xmath123 , @xmath124.__,scaledwidth=90.0% ] + * described schematically on the left , together with their adjacency matrices . both systems have connectivity parameters @xmath123 , @xmath124 . _ _ ] @xmath125 = @xmath126 $ ] , @xmath127 + + @xmath128 = @xmath131 , @xmath130 * described schematically on the left , together with their adjacency matrices . both systems have connectivity parameters @xmath123 , @xmath124.__,scaledwidth=90.0% ] of high interest are methods that can identify robust versus vulnerable features of the graph from a dynamics standpoint . as figures [ 4dim_mand ] and [ 10dim_mand ] show , it is clear that a small perturbation of the graph ( e.g. , adding a single edge ) have the potential , even for higher dimensional networks , to produce dramatic changes in the asymptotic dynamics of the network , and readily lead to substantially different m and j sets . however , this is not consistently true . we would like to understand whether a network may have a priori knowledge of which structural changes are likely to produce large dynamic effects . this is a real possibility in large natural learning networks , including the brain where such knowledge probably affects decisions of synaptic restructuring and temporal evolution of the connectivity profile . nodes * , formed of two cliques @xmath132 and @xmath133 , with @xmath134 nodes in each . th adjacency matrix is therefore similar to those in figure [ 4dim_mand ] , with square blocks @xmath135 , @xmath136 and @xmath137 of size @xmath134 . the densities ( number of ones in each block , i.e. number of @xmath132-to-@xmath133 and respectively @xmath133-to-@xmath132 connecting edges ) were takes in each panel to be ( out of the total of @xmath138 : * a. * @xmath139 ; * b. * @xmath140 , @xmath141 ; * c. * @xmath142 ; * d. * @xmath143 . in all cases , the connectivity parameters ( i.e. , edge weights ) were @xmath144 and @xmath145.__,title="fig:",scaledwidth=24.0% ] nodes * , formed of two cliques @xmath132 and @xmath133 , with @xmath134 nodes in each . th adjacency matrix is therefore similar to those in figure [ 4dim_mand ] , with square blocks @xmath135 , @xmath136 and @xmath137 of size @xmath134 . the densities ( number of ones in each block , i.e. number of @xmath132-to-@xmath133 and respectively @xmath133-to-@xmath132 connecting edges ) were takes in each panel to be ( out of the total of @xmath138 : * a. * @xmath139 ; * b. * @xmath140 , @xmath141 ; * c. * @xmath142 ; * d. * @xmath143 . in all cases , the connectivity parameters ( i.e. , edge weights ) were @xmath144 and @xmath145.__,title="fig:",scaledwidth=24.0% ] nodes * , formed of two cliques @xmath132 and @xmath133 , with @xmath134 nodes in each . th adjacency matrix is therefore similar to those in figure [ 4dim_mand ] , with square blocks @xmath135 , @xmath136 and @xmath137 of size @xmath134 . the densities ( number of ones in each block , i.e. number of @xmath132-to-@xmath133 and respectively @xmath133-to-@xmath132 connecting edges ) were takes in each panel to be ( out of the total of @xmath138 : * a. * @xmath139 ; * b. * @xmath140 , @xmath141 ; * c. * @xmath142 ; * d. * @xmath143 . in all cases , the connectivity parameters ( i.e. , edge weights ) were @xmath144 and @xmath145.__,title="fig:",scaledwidth=24.0% ] nodes * , formed of two cliques @xmath132 and @xmath133 , with @xmath134 nodes in each . th adjacency matrix is therefore similar to those in figure [ 4dim_mand ] , with square blocks @xmath135 , @xmath136 and @xmath137 of size @xmath134 . the densities ( number of ones in each block , i.e. number of @xmath132-to-@xmath133 and respectively @xmath133-to-@xmath132 connecting edges ) were takes in each panel to be ( out of the total of @xmath138 : * a. * @xmath139 ; * b. * @xmath140 , @xmath141 ; * c. * @xmath142 ; * d. * @xmath143 . in all cases , the connectivity parameters ( i.e. , edge weights ) were @xmath144 and @xmath145.__,title="fig:",scaledwidth=24.0% ] our future work includes understanding and interpreting the importance of this type of results in the context of networks from natural sciences . one potential view , proposed by the authors in their previous joint work , is to interpret iterated orbits as describing the temporal evolution of an evolving system ( e.g. , copying and proofreading dna sequences , or learning in a neural network ) . along these lines , an initial @xmath146 which escapes to @xmath147 under iterations may represent a feature of the system which becomes in time unsustainable , while an initial @xmath146 which is attracted to a simple periodic orbit may represent a feature which is too simple to be relevant or efficient for the system . then the points on the boundary between these two behaviors ( i.e. , the julia set ) may be viewed as the optimal features , allowing the system to perform its complex function . we study how this `` optimal set of features '' changes when perturbing its architecture . once we gain enough knowledge of networked maps for fixed nodes and edges , and we formulate which applications this framework may be appropriate to address symbolically , we will allow the nodes dynamics , as well as the edge weights and distribution , to evolve in time together with the iterations . this process may account for phenomena such as learning , or adaptation a crucial aspect that needs to be understood about systems . this represents a natural direction in which to extend existing work by the authors on random iterations in the one - dimensional case . the work on this project was supported by the suny new paltz research scholarship and creative activities program . we additionally want to thank sergio verduzco - flores , for his programing suggestions , and mark comerford , for the useful mathematical discussions . 10 rdulescu a , verduzco - flores s , 2015 . nonlinear network dynamics under perturbations of the 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network : an analytical proof . , 5(4 ) : 425444 , 1991 . the figures show four uni - j sets , for @xmath148 and @xmath149 nodes . the equi - parameters , adjacency matrices , and connectivity parameters of each network are given below , from left to right : +	many natural systems are organized as networks , in which the nodes interact in a time - dependent fashion . the object of our study is to relate connectivity to the temporal behavior of a network in which the nodes are ( real or complex ) logistic maps , coupled according to a connectivity scheme that obeys certain constrains , but also incorporates random aspects . we investigate in particular the relationship between the system architecture and possible dynamics . in the current paper we focus on establishing the framework , terminology and pertinent questions for low - dimensional networks . a subsequent paper will further address the relationship between hardwiring and dynamics in high - dimensional networks . + for networks of both complex and real node - maps , we define extensions of the julia and mandelbrot sets traditionally defined in the context of single map iterations . for three different model networks , we use a combination of analytical and numerical tools to illustrate how the system behavior ( measured via topological properties of the _ julia sets _ ) changes when perturbing the underlying adjacency graph . we differentiate between the effects on dynamics of different perturbations that directly modulate network connectivity : increasing / decreasing edge weights , and altering edge configuration by adding , deleting or moving edges . we discuss the implications of considering a rigorous extension of fatou - julia theory known to apply for iterations of single maps , to iterations of ensembles of maps coupled as nodes in a network . * real and complex behavior for networks of coupled logistic maps * + anca rdulescu@xmath0 , ariel pignatelli@xmath1 + @xmath2 department of mathematics , suny new paltz , ny 12561 @xmath1 department of mechanical engineering , suny new paltz , ny 12561 +	13,912	435
hadrons are the bound states of the strong interaction which is described by the quantum chromodynamics ( qcd ) in the framework of yang - mills gauge theory . one of the main goals of the hadron physics is to understand the composition of hadrons in terms of quarks and gluons . the quark model is proved successful in classifying the mesons and baryons as @xmath5 and @xmath6 composite systems . almost all the well established mesons can be described as a quark - antiquark state except some mesons with exotic quantum numbers which are impossible for a @xmath5 system , but no experimental evidence is reported for exotic baryons which are inconsistent with the @xmath6 configuration until the beginning of this century . theoretically , the qcd does not forbid the existence of the hadrons with the other configurations , such as the glueballs , the hybrids and the multiquarks . in this review , we focus on the pentaquarks . if the pentaquark really exists , it will provide a new stage to test the qcd in the nonperturbative region and to investigate the properties of the strong interaction . in the quark model language , the pentaquark is a hadron state with four valence quarks and one valence antiquark as @xmath7 @xcite . because the pentaquark can decay to a three - quark baryon and a quark - antiquark meson , its width was suggested to be wide @xcite , but it was predicted to have a narrow width due to its particular quark structure @xcite . in principle , any baryon may have the five - quark contents , and experiments have revealed the important role of the intrinsic sea quarks in understanding the structure of the proton . on the other hand , the pentaquark state may also mix with the corresponding three - quark state or hybrid state , so the situation is much more complicated . the pentaquark is easier to be identified if it has no admixture with any three - quark state , i.e. , if the flavor of the anti - quark @xmath8 in the @xmath7 state is different from any of the other four quarks @xcite . early experiments in 1960 s and 1970 s were performed to search for a baryon with positive strangeness as the pentaquark candidate referred to as the @xmath9 @xcite , but no enhancements were found . this field developed rapidly on both the experimental and the theoretical aspects in the last decade since the first report for a positive strangeness pentaquark - like baryon , referred to as the @xmath0 , by the leps collaboration @xcite . its mass and width are closed to the prediction of the chiral soliton model @xcite . this particle was quickly reported in subsequent experiments by some groups , and many theoretical models were applied to understanding this particle and to predicting other pentaquarks , such as the diquark cluster model @xcite , the diquark - triquark model @xcite , the constituent quark model @xcite , the chiral quark model @xcite , the bag model @xcite , the meson - baryon binding @xcite , the qcd sum rules @xcite , the lattice qcd @xcite and the chiral soliton model in new versions @xcite . unfortunately , many experiments found no signals for this particle . what is worse , the signals observed in the early experiments by some groups disappeared when including the new data with high statistics . however , some groups reconfirmed their observations for this particle with very high statistical significance in their updated experiments . so even the existence of the pentaquark is a mystery . the production mechanism and the analysis method should be investigated in details . recently , a charged charmonium - like meson @xmath10 was observed by bes @xcite and belle @xcite . it is a suggestive evidence for the existence of the multiquark meson . this arouses much interest on the study of the multiquark states . in this paper , we review the experimental search for the pentaquark states . in sect . ii and iii , we concentrate on the searches for the @xmath0 with positive or negative results . in sect . iv , we focus on a non - strange pentaquark candidate . in sect . v , the other observed pentaquark candidates are presented . then we discuss the results in sect . vi and a brief summary is contained in the last section . the pentaquark candidate @xmath0 was widely discussed and searched for since the first report on the experimental observation by the leps collaboration @xcite . the skyrme s idea that baryons are solitons @xcite arouses interesting , and the soliton picture consists with the qcd in the large @xmath11 limit @xcite . the @xmath0 , if exists , is the lightest member in the predicted antidecuplet @xcite . its mass and width were predicted in the chiral soliton model @xcite . in quark model language , the @xmath0 is described as a pentaquark state @xmath12 . unlike the other pentaquark @xmath7 states that the antiquark may have the same flavor with at least one quark , the lowest fock state of the @xmath0 composes of five quarks with the anti - quark being @xmath13 , which is of different flavor from the other four @xmath14 quarks . therefore it is easy to be identified from other ordinary baryons with minimal @xmath15 configurations @xcite . for the pentaquark states that the antiquark has the same flavor of some quark , the mixing of the pentaquark state and the corresponding three - quark state and hybrid state makes the situation complicated , because any three - quark baryon may have five - quark components from both perturbative and nonperturbative aspects , such as the baryon - meson fluctuation picture @xcite and the light - cone fock state expansion @xcite . since the @xmath0 has the same quark constituents as the combination of @xmath16 and @xmath17 , these two modes are expected as the primary decay channel , and thus are usually used in reconstructions in the experiments . after the first report for the @xmath0 , the signals were observed by many groups @xcite , and some groups confirmed their results with new data @xcite . all these results are briefly listed in table [ thetay ] . negative results were also reported by many groups , and some early positive results in the photoproduction experiment by clas @xcite and in the proton - proton collision by cosy - tof @xcite were rejected by their high - statistics experiments later @xcite . those are discussed in the next section . .experiments with positive signals for the @xmath0 . [ cols="<,^,^,^,^,^,^,>",options="header " , ] the @xmath3 baryon , if exists , is the lightest charmed pentaquark state like the @xmath0 but the @xmath13 quark replaced with the @xmath18 quark . its lowest fock state is @xmath19 which has the same constituent quarks with the combination of @xmath20 or @xmath21 . thus these are estimated as the dominant decay channels . the signal for the @xmath3 was only observed in the dis experiment by the h1 collaboration @xcite . the analysis was based on the data at hera in 1996 - 2000 . the @xmath22 was reconstructed via the decay channal @xmath23 . in the distribution of @xmath24 with opposite - charge combinations , a peak was observed at @xmath25 mev/@xmath26 with a gaussian width of @xmath27 mev/@xmath26 . the background was estimated from the monte carlo simulation . the number of the signal events is @xmath28 , corresponding to a statistical significance of 5.4@xmath29 . however this resonance was not observed in any other experiment as listed in the table [ thetac ] . the @xmath30 baryon , if exists , is a double strangeness @xmath31 pentaquark candidate , having the lowest fock state @xmath32 . it is recognized as an isospin quartet together with its partners @xmath33 , @xmath34 and @xmath35 . the primary decay channel of @xmath2 is estimated to be @xmath36 . the signal for the @xmath2 was only reported by the na49 collaboration at cern @xcite . the analysis was base on the data of 158 gev/@xmath37 proton beam colliding with the lh@xmath38 target . the @xmath39 was reconstructed via the decay channel @xmath40 . then 1640 @xmath39 and 551 @xmath41 were selected . a peak was observed in the @xmath36 mass spectrum . combined with the @xmath42 data as the antiparticle , the fitted peak is at @xmath43 mev/@xmath26 with 69 signals over the background of 75 events corresponding to a statistical significance of 5.8@xmath29 estimated as @xmath44 . one of its isospin partners @xmath45 was also observed , and the fitted mass is @xmath46 mev/@xmath26 . unfortunately this resonance was not observed in any other experiment as listed in the table [ xi1860 ] . among all the experiments in which the @xmath0 was observed , the width was claimed to be narrow . however , the mass position of the signals in different experiments spreads in a large region from 1520 to 1560 mev/@xmath26 as shown in fig . [ mass ] . this does not consist with a narrow resonance . the mass values in leps early experiment @xcite , diana experiments @xcite , saphir experiment @xcite and jinr experiments @xcite are around 1540 mev/@xmath26 , while the values in hermes experiment @xcite , svd early experiment @xcite , itep s analysis on the old neutrino experiments @xcite and e522 experiment @xcite are near 1530 mev/@xmath26 . besides , the leps recent experiment @xcite , the svd updated experiment @xcite and the zeus experiment @xcite provide even lower mass values close to 1520 mev/@xmath26 , and the obelix experiment gives a much higher value . therefore , it is possible that even the observed signals do not correspond to the same particle . the width of the @xmath0 are not measured as accurate as the mass , and thus the values in almost all the experiments are consistent as shown in fig . [ width ] . mass values for the @xmath0 observed in various experiments . the error bars represent the statistical uncertainties.,scaledwidth=70.0% ] width values for the @xmath0 measured in various experiments . the error bars represent the statistical uncertainties . the experiments with only the upper limit on the width are listed in the table [ thetay ] . the dashed line is the upper limit of the intrinsic width of @xmath0 at 90% c.l . by belle @xcite.,scaledwidth=70.0% ] the statistical significance of the signals was usually estimated as @xmath47 in the early experiments . this estimator neglects the uncertainty of the background , and thus the significance may be overestimated . the estimator @xmath44 which assumes a smooth background with a well defined shape and @xmath48 which assumes a statistical independent background with uncertainties are more proper since the production mechanism is still unknown . if so , the significance of the signals in the svd updated analysis @xcite reduces to 5.6@xmath29 and @xmath49 estimated as @xmath48 for the two samples respectively , but is still large enough as an evidence . in this case , however , the hermes result decreases to 2.7 - 3.9@xmath29 , the jinr @xmath50 result decreases to 4.1@xmath29 and the jinr @xmath51 result decreases to 3.5@xmath29 . these are not enough to be claimed as an evidence . besides , the log - likelihood difference is also a suitable alternative and was used by the leps @xcite and the diana @xcite . the @xmath52 produced in the inclusive experiments may affect the result of the @xmath0 reconstructed via the @xmath53 mode . as pointed out by m. zavertyaev @xcite , the decay @xmath54 could lead to a spurious sharp peak at 1540 mev/@xmath26 when the momentum of the @xmath52 is greater than 2 gev/@xmath37 . on the other hand , the @xmath55 decays of the @xmath52s could enhance the background when the @xmath56 or the proton was paired with a @xmath57 or a @xmath58 . up to now , no positive result is reported for the @xmath0 production in the @xmath59 experiments . a possible way to understand these null results with no contradict with the positive ones is to assume that the @xmath0 production cross section is highly suppressed in @xmath59 annihilations . using the quark constituent counting rules , a.i . titov et al . estimated the ratio of @xmath0 to @xmath60 production in the fragmentation region and showed the ratio decreases very fast with energy @xcite . this ratio is often applied to estimate the yield of the @xmath0 , because @xmath60 is a narrow resonance with similar mass to the @xmath0 and is easily reconstructed . the low value of this ratio implies a very different production mechanism for @xmath0 if it really exists . for the dis experiments performed at hera @xcite , the zeus and the h1 provided opposite conclusions . these two experiments were almost in the same conditions and with the data collected during the same period , but even using the same cuts the h1 could not produce the @xmath0 signal observed by the zeus . it is very confused , and probably the signal observed by the zeus is a statistical fluctuation . for the photoproduction experiments , there is a contradiction in the @xmath61 experiments between the upper limit on the cross section given by the clas @xcite and the result reported by the saphir @xcite and a contradiction in the @xmath62 experiments between the upper limit on the cross section given by the clas @xcite and the result reported by the leps @xcite . as claimed by the leps , the contradiction between the leps and clas results is due to the different measurements . if the @xmath0 is preferably produced at the forward angles , the clas would possibly not detect the @xmath63 meson associated with the @xmath0 , because the most forward angle for the @xmath63 detection is about @xmath64 for the clas while most acceptance is of forward @xmath64 for the leps @xcite . it may be a similar case for the contradiction between the clas and the saphir . if this is true , it will be a suggestion on the angular distribution for the @xmath0 production . in the experiments at higher energy the @xmath0 productions will be boosted to a much more forward direction and thus escape the coverage of the detectors in most high energy experiments . this may be a possible explanation for the null results in most high energy experiments . in the improved analysis by the diana @xcite , a very narrow intrinsic width for the @xmath0 was estimated as @xmath65 mev/@xmath26 . this result passed the upper limits given by the belle @xcite , the e559 @xcite and the j - parc @xcite . therefore there is no contradiction between these experiments , although opposite conclusions were reported for the existence of the @xmath0 . among all the experimental search for the @xmath0 pentaquark candidate , those with negative results have higher statistics and are consequently more reliable in usual , but it is hard to prove that all the observed peaks are fakes or fluctuations . especially , the updated results by the leps @xcite , the diana @xcite and the svd @xcite can be claimed as strong evidence for the @xmath0 . so the existence of the @xmath0 is still debatable . for the other pentaquark candidates such as @xmath1 , @xmath2 and @xmath3 , the signals are much less reliable since none of them is confirmed in any other experiment . we reviewed the experimental search for the pentaquark states during the last decade . both the most widely studied candidate @xmath0 and the other candidates like @xmath1 , @xmath2 and @xmath3 as well as the non - strange pentaquark candidates are included . since the first observation of the pentaquark - like baryon state @xmath0 , this field has aroused much interest , but even the existence of the pentaquark is debatable up to now . if the pentaquark really exists , it will open a new world of the qcd and hadron physics . in particular , if the @xmath0 exists , its production mechanism is almost unknown and needs to be investigate , whether it is a pentaquark or not . besides it will imply the existence of a flavor multiplet . if the pentaquark does not exist , the peaks observed in the experiments with positive results need a reasonable explanation . in addition , the contradictions between the experiments should be examined in details . this will improve the analysis method and raise the reliability of the result in future experiments . in order to confirm or rule out the existence of the pentaquark , particularly the @xmath0 , comparisons between experiments in similar conditions are required . among the experiments , the updated results by the leps @xcite , the diana @xcite and the svd @xcite provide the best positive evidence for the @xmath0 . therefore more experiments at medium energy may lead to a clear conclusion . 132ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1016/s0031 - 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10029-y [ * * , ( ) ] , @noop ( ) , link:\doibase 10.1016/j.physletb.2007.01.041 [ * * , ( ) ] , link:\doibase 10.1143/ptps.168.90 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.100.252002 [ * * , ( ) ] , link:\doibase 10.1140/epja / i2011 - 11089 - 0 [ * * , ( ) ] , link:\doibase 10.1088/1674 - 1137/33/12/051 [ * * , ( ) ] , link:\doibase 10.1103/physrevc.83.022201 [ * * , ( ) ] , @noop ( ) , @noop * * , ( ) , link:\doibase 10.1134/s002136400818001x [ * * , ( ) ] , link:\doibase 10.1103/physrevd.69.077501 [ * * , ( ) ] , link:\doibase 10.1016/j.physletb.2004.02.029 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.70.094042 [ * * , ( ) ] , link:\doibase 10.1142/s0217751x06032101 [ * * , ( ) ] , link:\doibase 10.1016/j.nuclphysa.2005.02.082 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.97.102001 [ * * , ( ) ] , @noop , ( ) , link:\doibase 10.1103/physrevd.72.051101 [ * * , ( ) ] , link:\doibase 10.1103/physrevc.75.055208 [ * * , ( ) ] , link:\doibase 10.1016/j.physletb.2004.03.012 [ * * , ( ) ] , link:\doibase 10.1140/epjc / s2004 - 02042 - 9 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.73.091101 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.74.051101 [ * * , ( ) ] , link:\doibase 10.1016/j.physletb.2005.07.023 [ * * , ( ) ] , link:\doibase 10.1016/j.nuclphysa.2005.02.075 [ * * , ( ) ] , link:\doibase 10.1103/physrevc.85.015205 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.71.032004 [ * * , ( ) ] , link:\doibase 10.1140/epjc / s10052 - 007 - 0407 - 3 [ * * , ( ) ] , link:\doibase 10.1016/j.physletb.2005.02.016 [ * * , ( ) ] , link:\doibase 10.1103/physrevc.70.022201 [ * * , ( ) ] , link:\doibase 10.1103/physrevd.75.032003 [ * * , ( ) ] , link:\doibase 10.1016/j.physletb.2008.01.063 [ * * , ( ) ] , link:\doibase 10.1140/epjc / s2005 - 02281 - 2 [ * * , ( ) ] , link:\doibase 10.1103/physrevlett.95.152001 [ * * , ( ) ] , @noop ( ) , link:\doibase 10.1103/physrevc.70.042202 [ * * , ( ) ] ,	it has been ten years since the first report for a positive strangeness pentaquark - like baryon state . however the existence of the pentaquark state is still controversial . some contradictions between the experiments are unsolved . in this paper we review the experimental search for the pentaquark candidates @xmath0 , @xmath1 , @xmath2 , @xmath3 and @xmath4 in details . we review the experiments with positive results and compare the experiments with similar conditions but opposite results .	8,691	131
frequently it is the case in the study of real - world complex networks that we observe essentially a sample from a larger network . there are many reasons why sampling in networks is often unavoidable and , in some cases , even desirable . sampling , for example , has long been a necessary part of studying internet topology @xcite . similarly , its role has been long - recognized in the context of biological networks , e.g. , protein - protein interaction @xcite , gene regulation @xcite and metabolic networks @xcite . finally , in recent years , there has been intense interest in the use of sampling for monitoring online social media networks . see @xcite , for example , for a representative list of articles in this latter domain . given a sample from a network , a fundamental statistical question is how the sampled network statistics be used to make inferences about the parameters of the underlying global network . parameters of interest in the literature include ( but are by no means limited to ) degree distribution , density , diameter , clustering coefficient , and number of connected components . for seminal work in this direction , see @xcite . in this paper , we propose potential solutions to an estimation problem that appears to have received significantly less attention in the literature to date the estimation of the degrees of individual sampled nodes . degree is one of the most fundamental of network metrics , and is a basic notion of node - centrality . deriving a good estimate of the node degree , in turn , can be helpful in estimating other global parameters , as many such parameters can be viewed as functions that include degree as an argument . while a number of methods are available to estimate the full degree distribution under network sampling ( e.g. , @xcite ) , little work appears to have been done on estimating the individual node degrees . our work addresses this gap . formally , our interest lies in estimation of the degree of a vertex , provided that vertex is selected in a sample of the underlying graph . there are many sampling designs for graphs . see ( * ? ? ? * ch 5 ) for a review of the classical literature , and @xcite for a recent survey . canonical examples include ego - centric sampling@xcite , snowball sampling , induced / incident subgraph sampling , link - tracing and random walk based methods@xcite . under certain sampling designs where one observes the true degree of the sampled node ( e.g. ego - centric and one - wave snowball sampling ) , degree estimation is unnecessary . in this paper , we focus on _ induced subgraph sampling _ , which is structurally representative of a number of other sampling strategies@xcite . formally , in induced subgraph sampling , a set of nodes is selected according to independent bernoulli(@xmath0 ) trials at each node . then , the subgraph induced by the selected nodes , i.e. , the graph generated by selecting edges between selected nodes , is observed . this method of sampling shares stochastic properties with incident subgraph sampling ( wherein the role of nodes and edges is reversed ) and with certain types of random walk sampling @xcite . the problem of estimating degrees of sampled nodes has been given a formal statistical treatment in @xcite , for the specific case of traceroute sampling as a special case of the so - called _ species problem _ @xcite . to the best of our knowledge , a similarly formal treatment has not been applied more generally for other , more canonical sampling strategies . however , a similar problem would be estimating personal network size for a group of people in a survey . some prior works in this direction @xcite consider estimators obtained by scaling up the observed degree in the sampled network , in the spirit of what we term a method of moments estimator below . but no specific graph sampling designs are discussed in these studies . we focus on formulating the problem using the induced subgraph sampling design and exploit network information beyond sampled degree to propose estimators that are better than naive scale - up estimators . key to our formulation is a risk theoretic framework used to derive our estimators of the node degrees , through minimizing frequentist or bayes risks . this contribution is accompanied by a comparative analysis of our proposed estimators and naive scale - up estimators , both theoretical and empirical , in several network regimes . we note that when sampling is coupled with false positive and false negative edges , e.g. , in certain biological networks , our methods are not immediately applicable . sampling designs that result in the selection of a fraction of edges from the underlying global network ( induced and incident subgraph sampling , random walks etc . ) are our primary objects of study . we use induced subgraph sampling as a rudimentary but representative model for this class and aim to simultaneously estimate the true degrees of all the observed nodes with a precision better than that obtained by trivial scale - up estimators with no network information used . let us denote by @xmath1 a true underlying network , where @xmath2 . this network is assumed static and , without loss of generality , undirected . the true degree vector is @xmath3 . the sampled network is denoted by @xmath4 where , again without loss of generality , we assume that @xmath5 . write the sampled degree vector as @xmath6 . throughout the paper , we assume that we have an induced subgraph sample , with ( known ) sampling proportion @xmath0 . it is easy to see from the sampling scheme that @xmath7 . therefore , the method of moments estimator ( mme ) for @xmath8 is @xmath9 . thus , @xmath10 is a natural scale - up estimator of the degree sequence of the sampled nodes . in this section , we propose a class of estimators that minimize the unweighted @xmath11-risk of the sampled degree vector and discuss their theoretical properties . we aim to demonstrate , under several conditions , that the risk minimizers are superior to the regular scale - up estimators , the former taking into account the inherent relationships inside the network . we note that although a maximum likelihood approach to estimation is perhaps intuitively appealing , a closed form derivation of the mle in this setting is probitive . another option is to look at marginal likelihoods . but the mle based on univariate marginal likelihoods are essentially equivalent to the mme for this sampling scheme . we will frequently use the the first and second moments of the sampled degree vector in our estimation methods . the following lemma will be useful . [ lem : meancovariancedegree ] under induced subgraph sampling , the mean and covariance matrix of the observed degree vector are @xmath12 where the diagonals of @xmath13 are @xmath14 and the @xmath15-th off - diagonal is denoted by @xmath16 , which denotes the number of common neighbors of node @xmath17 and node @xmath18 in the network @xmath19 . adopting the standard definition of ( unweighted ) frequentist @xmath11 risk of an estimator @xmath20 of a parameter @xmath21 , i.e. , @xmath22 , the frequentist risks are calculated for a general class of estimators . we also define @xmath23 , a _ restricted risk function _ assuming the sampled graph @xmath24 is restricted to some class @xmath25 . our proposed candidates are the elements in the class of linear functions of the observed degree vector that minimize the risk or the restricted risk w.r.t . some class . it is expected that the optimal estimator will be a function of the parameter and hence another ( naive ) estimator will need to be plugged in . our final estimate will then be a plug - in risk minimizer . here we estimate the node degrees individually , assuming that the estimate for the @xmath26 node is of the form @xmath27 , where @xmath28 is a scalar and @xmath29 is the observed degree in the sample . since @xmath7 , where @xmath8 is the true degree of the @xmath26 node , @xmath30 differentiating w.r.t . @xmath28 and equating to 0 , we get the optimal @xmath31 . plugging in the mme of @xmath32 , we get the plug - in univariate risk minimizer @xmath33 taylor expanding the above formula ( during taylor expansions of functions of @xmath29 , we will assume that @xmath29 is concentrated around its mean , so that the taylor expanded approximation is close ) and taking expectation , we see that @xmath34 = \frac{1}{p } \mathbb{e}\left [ d^_i\left(1 + \frac{1-p}{d^_i}\right)^{-1 } \right ] \approx \frac{1}{p } \mathbb{e}\left [ d^_i\left(1 - \frac{1-p}{d^_i}\right ) \right ] = d^0_i - \frac{1-p}{p } \enskip . $ ] + the above calculation suggests that an adjustment needs to be made to @xmath35 by bias - correction , so that its risk becomes comparable to that of @xmath36 . in fact , we will show in proposition [ unirisk ] that our bias - corrected plug - in estimator has a lower risk than mme when the true degree is bigger than a lower bound , which can be expressed as a closed form function of the sampling proportion . ultimately , our proposed univariate risk minimizer is given by @xmath37 note that the proposed estimator @xmath38 can be written as @xmath39 this is almost an additive adjustment to the multiplicative scale - up estimate . we also have , @xmath40 . + from the above discussion , it is obvious to see that when @xmath36 overestimates @xmath41 , @xmath38 can not be any better than @xmath36 . we first argue that this happens with small probability . to demonstrate this , we use the erds - rnyi model . + for any node @xmath17 selected in induced subgraph sampling from an erds - rnyi model @xmath42 ( for @xmath43 and @xmath44 denoting the probability of edge formation such that @xmath45 . ) and for any @xmath46 , @xmath47\\ \mathbb{p}\left(\hat{d}^{\rm mme}_i < ( 1-\epsilon)d^0_i\right ) & \leq \exp\left[-np_e\left(1 - e^{-\frac{\epsilon^2p}{2}}\right)\right]\end{aligned}\ ] ] @xmath48 \\ & \hspace{-1.5cm}\leq \mathbb{e}_{d^0_i}\left[\exp\left(- \frac{\epsilon^2 p}{2+\epsilon } d^0_i\right)\right ] \intertext{in the above inequality , we use a version of chernoff inequality . if $ x_1 , x_2 , \cdots , x_n$ are independent with $ 0\leq x_i\leq 1 $ for all $ i$ , $ x = \sum_i x_i$ and $ \mu = \mathbb{e}x$ , then $ $ \mathbb{p}\left[x > ( 1+\epsilon)\mu\right ] \leq \exp\left(-\frac{\epsilon^2}{2+\epsilon}\mu\right)$$ and $ $ \mathbb{p}\left[x < ( 1-\epsilon)\mu\right ] \leq \exp\left(-\frac{\epsilon^2}{2}\mu\right).$$ since $ d^0_i$ approximately follows a poisson$\left(np_e\right)$ , we have } \mathbb{e}_{d^0_i}\left[\exp\left(- \frac{\epsilon^2 p}{2+\epsilon } d^0_i\right)\right ] & = \psi\left(-\frac{\epsilon^2 p}{2+\epsilon}\right ) \\ & \hspace{-1cm}= \exp\left[-np_e\left(1 - e^{-\frac{\epsilon^2p}{2+\epsilon}}\right)\right ] \intertext{where $ \psi$ denotes the moment generating function of poisson distribution . the other inequality follows similarly when the lower tail chernoff bound is used . } \end{aligned}\ ] ] we extend the idea presented in the previous section to the multivariate case , in order to minimize the overall @xmath11 sum over all sampled nodes . the rationale for this extension is to exploit the covariance structure we derived in lemma [ lem : meancovariancedegree ] in estimating the degree vector . accordingly , we consider all estimates of the form @xmath49 , where @xmath50 is an @xmath51 matrix . using lemma [ lem : meancovariancedegree ] , we get the @xmath11 risk + @xmath52 . + the multivariate risk minimizer is defined as + @xmath53 . + differentiating the objective function w.r.t . @xmath50 and equating it to @xmath54 , we get + @xmath55 + plugging in the mme of @xmath56 and @xmath57 , we get the plug - in multivariate risk minimizer @xmath58 where @xmath59 denotes the number of common neighbors of node @xmath17 and node @xmath18 in the sample , and @xmath60 is given by a matrix whose diagonals are @xmath29 and whose off - diagonals are @xmath59 , @xmath61 . in this section , we propose a bayesian solution to our estimation problem , by putting a prior on the degree distribution . the principal motivation behind this approach is the desire to incorporate additional information on global network structure , where the natural candidate in this context is the degree distribution . in case such a subjective prior is not available , an estimate of the degree distribution may be used . we propose and analyze estimators based on both known ( subjective ) and estimated degree distributions below . first , let us assume that we know the degree distribution @xmath62 of the underlying network . under the assumption that the true degree of node @xmath17 follows @xmath62 , and under induced subgraph sampling of @xmath63 , the conditional distribution of @xmath64 is @xmath65 . then it can be easily shown that the bayes estimator under square error loss is @xmath66 if the true degree distribution is not known , then it needs to be estimated , for example using techniques described in or similar to @xcite . let @xmath67 be a reasonable " estimator for @xmath62 . then an empirical bayes estimator is given by @xmath68 generally speaking , if @xmath69 denotes the distribution of @xmath29 given @xmath70 , then this empirical bayes estimate can be expressed as @xmath71 these estimators take the form of a weighted mean , as expected for bayes estimates under quadratic loss . the weights are functionals of both sampling design and the degree distribution . for the latter estimator , only the estimated degree distribution comes into play , and thus the proposed empirical bayes estimator incorporates the sampling and sampled network information . in this section , we present results on the relative performance of our proposed estimators from a risk - theoretic perspective , and we discuss several conditions under which one outperforms the other . all these estimates will be benchmarked against the regular scale - up estimate @xmath72 . proofs may be found in the supplementary materials . in the first part of our risk analysis , we look at the @xmath11 frequentist risk of our proposed univariate and multivariate estimators . our main results in this section will compare the risk incurred by our proposed estimators to the scale up estimator and discuss conditions under which our proposed estimators perform better . [ unirisk ] assuming @xmath73 , we have @xmath74 . in other words , the univariate risk minimizer @xmath75 will outperform the mme when the true degree @xmath8 is sufficiently large . [ multirisk ] let us denote the class of all sampled graphs of size @xmath76 ( where @xmath77 for all @xmath17 , i.e. , there is no isolated node ) as @xmath78 . also assume that there exists an @xmath79 such that @xmath80 are nonempty . then we have @xmath81 over sampled graphs belonging to @xmath82 . scrutiny of the conditions in proposition [ multirisk ] , along with definition of the set @xmath83 , reveals a general characterization of the graphs where the proposed multivariate estimator performs better . it is to be noticed that @xmath84 shrinks @xmath85 by some factor . the term on the right side of the inequality in the definition of @xmath86 provides a lower bound on the shrinkage factor and the term on the left decreases as the cardinality of @xmath87 increases , i.e. , the graph becomes less sparse . hence , the proposed estimator can be expected to work better than the standard scale - up estimator under the assumption of sparsity of the sampled graph . this will also be demonstrated in the simulation section . the eigenvector condition imposes a geometric constraint on the sample degree - degree matrix @xmath60 . what it essentially means is that the angle between the eigenvectors of @xmath88 and @xmath89 should be smaller than @xmath90 . or , in other words , by selecting an @xmath91 sufficiently small but positive , our class of sampled graphs are restricted where the associated matrix @xmath88 has eigenvectors at least @xmath92 angle away from any orthogonal direction to @xmath89 . thus , our estimator performs better for sparse graph satisfying a mild geometric condition . the performance of the bayes estimators is evaluated here under several conditions and network paradigms . note that these estimators are compared to the regular scale - up estimator with respect to their frequentist risk functions . we start with our estimator in its most general form and state conditions on the prior degree distribution that will ensure lower risk . from that , we assess its risk when the prior degree distribution is replaced with an appropriate estimate . we also explicitly derive the bayes estimator for the erds - rnyi class of random graphs and state conditions under which the bayes estimator yields lower risk than the scale - up estimator . [ suffrisk ] let @xmath93 be the true degree of sample node @xmath17 , and @xmath29 , the observed degree . denote by @xmath94 the class of sampled graphs where the following two conditions hold : @xmath95 where @xmath96 . then @xmath97 under induced subgraph sampling . the conditions ( [ suffrisk : cond1 ] ) and ( [ suffrisk : cond2 ] ) essentially constrain the tail behavior of the prior degree disbution . the first condition ensures that the tail decays at a rate such that it is not too `` thick '' and the second condition ensures that it is not too `` thin '' . as @xmath8 becomes bigger , the rhs in condition ( [ suffrisk : cond1 ] ) becomes smaller and that is reminiscent of the sparsity property of the underlying graph , meaning that not a lot of nodes can have very high degree , an observation consistent with sparse graphs . on the other hand , the lhs in the condition ( [ suffrisk : cond2 ] ) can be interpreted as the mean of the tail probabilities weighted by the posterior distribution . this has to be bounded away from zero in order for the bayes estimate to have lower risk than the mme . in real problems , where the true degree distribution is unknown , one either has to choose @xmath98 subjectively or use the data to come up with a reasonable estimate . estimating @xmath98 for a general case is beyond the scope of this paper and will not be discussed here . for our analysis , we will just assume that we have an estimate of the degree distribution at our disposal ( e.g. , @xcite ) , denoted by @xmath99 . using @xmath99 will give us our proposed empirical bayes estimate @xmath100 , the behavior of which can be described as follows . [ pluginapprox ] let @xmath67 be an estimate of @xmath62 such that @xmath101 then under assumption ( [ suffrisk : cond2 ] ) , with @xmath98 replaced by @xmath99 , we have @xmath102 it is easily seen that with the assumption , the upper bound in can be simplified to @xmath103 assuming a large network , the sum in the denominator can be approximated by @xmath104 . then the upper bound is @xmath105 from the above discussion , it is evident that if @xmath106 , @xmath107 for all @xmath17 and hence their risk functions will also be close . thus , using proposition [ suffrisk ] , it is expected that @xmath108 it is well known that the asymptotic degrees in erds - rnyi graph models follow a poisson distribution , under standard conditions . in this section , we study the effects of using a poisson prior degree distribution for large erds - rnyi graphs . the goal is to demonstrate the efficacy of the bayesian approach compared to scale - up estimates as in the last section . however , studying specific models like erds - rnyi will give us more insight about the performance of the proposed bayes estimate . in this scenario , the prior @xmath62 is given by @xmath109 where @xmath110 is the prior mean . for a large erds - rnyi graph with number of nodes @xmath111 and edge probability @xmath44 , @xmath112 . we denote , by @xmath113 , the shifted poisson distribution on @xmath114 whose p.m.f . is given by @xmath115 it is easy to check that with a @xmath116 prior on @xmath70 , the posterior distribution is @xmath117 . hence the bayes estimate with respect to the quadratic loss function is @xmath118 [ erdosbayesrisk ] assuming @xmath119 the quadratic risk of the bayes estimator using a @xmath116 prior is smaller than that of the mme . the above result shows that if the sampled node is such that its true degree belongs to a neighborhood around the mean of the underlying degree distribution , then the bayes estimator is uniformly better than the mme . in case the underlying mean is unknown , it can easily be estimated from the sample . ( e.g. , for known @xmath111 , @xmath120 ) if @xmath121 is a consistent estimator of @xmath110 in the sense that @xmath122 when @xmath123 , @xmath124 and @xmath125 , then the empirical bayes estimator @xmath126 will converge in probability to the bayes estimator in the sense that @xmath127 . hence , the result of prop . ( [ erdosbayesrisk ] ) is expected to hold . this will also be demonstrated in the simulations . for our simulation study , we look at two different regimes of network erds - rnyi random graphs and heavy tailed degree distributions . we compare four methods of estimation - the regular mme , univariate risk minimizer , multivariate risk minimizer and the bayes estimate . as priors in bayes estimation , we use both exponentially decaying ( poisson ) and polynomially decaying degree distribution as priors . table [ ertable ] records the euclidean distance between the true and estimated degree vectors across some combinations of graph size @xmath111 , edge strength @xmath44 and sampling proportion @xmath0 . the errors are averaged over 50 different samples from each given graph @xmath63 . from the output , it is clear that the bayes estimators with true @xmath110 and estimated @xmath110 outperform other estimators by a very wide margin in terms of @xmath11 risk . also , our theoretical prediction in the discussion following proposition [ multirisk ] was that the multivariate risk minimizer ( mrm ) works better than the mme for sparse graphs . this is experimentally verified in this simulation , since we see that the relative risk of mrm compared to mme decreases as the sparsity of the underlying graph increases , i.e. , as @xmath44 decreases . the method with lowest total quadratic loss is shown in red for each condition . we compared four methods of estimation in simulated scale free networks which follow a power law degree distribution . as priors in bayes estimation , we compared the true polynomial prior and quadratic prior . we computed the @xmath128 distances across some combinations of sparsity ( denoted by @xmath129 , given by the ratio of total edges to all possible edges ) , sampling proportion @xmath0 and heaviness of the tail of the degree distribution , controled by @xmath130 . the results are shown in table [ sftable ] . the bayes estimators or the multivariate risk minimizers work better than the other estimators . one important thing to observe here is that for the most sparse graph , the bayes estimator with true prior works the best and as @xmath129 increases , multivariate risk minimizers work better than the rest , but there is hardly any improvement over mme . again , the method with lowest total quadratic loss is shown in red for each condition . [ table : sim1 ] [ table : sim2 ] in february 2015 , the defense advanced research projects agency ( darpa ) , an agency of the u.s . department of defense , announced the _ memex _ program in response to the use of the internet in human trafficking , especially chat forums , advertisements and job services sections . darpa - funded research determined the trafficking industry spent $ 250 m to post more than 60 m advertisements over a two - year time frame@xcite . indexing and cross - referencing the ads with the same contact number , similar address or zip codes help identify and track the illegal trafficking activities . this leads to a massive background network structure where each node represents an advertisement and an edge between two nodes are created if they share certain features . it is not unreasonable to expect that , in surveillance of networks like this , sampling may well arise , either by choice or by circumstance . we mimic this situation by pretending that this underlying network generated by the _ memex _ program is unknown to us and sampling it using induced subgraph sampling . the nodes associated with trafficking activities are flagged in the data . there are 31,248 nodes , of which 12,387 are flagged and there are 10,200,838 edges . our goal was to estimate the true degrees of flagged nodes that we saw in our sample . we compared the @xmath11 distance of regular scale - up estimators , and our proposed univariate , multivariate and bayes estimators . for the bayes estimator , a number of polynomial priors were taken into consideration with varying degree of decay , denoted by @xmath131 . the results are shown in table [ humantraffic ] . almost everything works better than the naive scale - up estimator in terms of total @xmath11 loss , although the relative improvement is more modest than in simulation . [ humantraffic ] in this paper , we addressed the problem of estimation of true degrees of sampled nodes from an unknown graph . we proposed a class of estimators from a risk - theory perspective where the goal was to minimize the overall @xmath11 risk of the degree estimates for the sampled nodes . we considered estimators that minimize both frequentist and bayes risk functions and compared the frequentist @xmath11 risks of our proposed estimator to the naive scale - up estimator . the basic objective of proposing these estimators was to exploit the additional network information inherent in the sampled graph , beyond the observed degrees . our theoretical analyses , simulation studies and real data show clear evidence of superior performance of our estimators compared to mme , especially when the graph is sparse and the sampling ratio is low , mimicking the real - world examples . there are a number of ways our current work could be extended . firstly , a theoretical analysis of the bayes estimators under priors for random graph models beyond erds - rnyi is desirable , although likely more involved . secondly , although induced subgraph sampling serves as a representative structural model for a certain class of adaptive sampling designs , the specific details of the sufficiency conditions discussed in this paper can be expected to vary slightly with the other sampling designs ( e.g. , incident subgraph or random walk designs ) . finally , the success of the bayesian method appears to rely heavily upon appropriate choice of prior distribution , as observed in our theoretical analysis and computational experiments . it would be of interest to explore the performance of the empirical bayes estimate in conjunction with the nonparametric method of degree distribution estimation proposed in @xcite . more generally , the method in @xcite can in principle be extended to estimate individual vertex degrees . but the computational challenge of implementation and the corresponding risk analysis can be expected to be nontrivial . d. killworth , c. mccarty , h. r. bernard , g.a . shelley , and e.c . estimation of seroprevalence , rape , and homelessness in the united states using a social network approach . , 22(2):289308 , 1998 . j. leskovec and c. faloutsos . sampling from large graphs . in _ proceedings of the 12th acm sigkdd international conference on knowledge discovery and data mining _ , kdd 06 , pages 631636 , new york , ny , usa , 2006 . acm . b. ribeiro and d. towsley . estimating and sampling graphs with multidimensional random walks . in _ proceedings of the 10th acm sigcomm conference on internet measurement _ , imc 10 , pages 390403 , new york , ny , usa , 2010 .	the need to produce accurate estimates of vertex degree in a large network , based on observation of a subnetwork , arises in a number of practical settings . we study a formalized version of this problem , wherein the goal is , given a randomly sampled subnetwork from a large parent network , to estimate the actual degree of the sampled nodes . depending on the sampling scheme , trivial method of moments estimators ( mmes ) can be used . however , the mme is not expected , in general , to use all relevant network information . in this study , we propose a handful of novel estimators derived from a risk - theoretic perspective , which make more sophisticated use of the information in the sampled network . theoretical assessment of the new estimators characterizes under what conditions they can offer improvement over the mme , while numerical comparisons show that when such improvement obtains , it can be substantial . illustration is provided on a human trafficking network .	7,972	237
in a graph @xmath14 , an independent set @xmath15 is a subset of the vertices of @xmath0 such that no two vertices in @xmath15 are adjacent . the independence number @xmath2 is the cardinality of a largest set of independent vertices and an independent set of size @xmath2 is called an @xmath16-set . the maximum independent set problem is to find an independent set with the largest number of vertices in a given graph . it is well - known that this problem is np - hard @xcite . therefore , many attempts are made to find upper and lower bounds , or exact values of @xmath2 for special classes of graphs . this paper is aimed toward studying this problem for the generalized petersen graphs . for each @xmath7 and @xmath8 @xmath17 , a generalized petersen graph @xmath18 , is defined by vertex set @xmath19 and edge set @xmath20 ; where @xmath21 and subscripts are reduced modulo @xmath7 . an induced subgraph on @xmath22-vertices is called the inner subgraph , and an induced subgraph on @xmath23-vertices is called the outer cycle . + in addition , we call two vertices @xmath24 and @xmath25 as twin of each other and the edge between them as a spoke . in @xcite , coxeter introduced this class of graphs . later watkins @xcite called these graphs `` generalized petersen graphs '' , @xmath18 , and conjectured that they admit a tait coloring , except @xmath26 . this conjecture later was proved in @xcite . since 1969 this class of graphs has been studied widely . recently vertex domination and minimum vertex cover of @xmath18 have been studied . for more details see for instance @xcite , @xcite , @xcite and @xcite . a set @xmath27 of vertices of a graph @xmath28 is called a vertex cover of @xmath0 if every edge of @xmath0 has at least one endpoint in @xmath27 . a vertex cover of a graph @xmath0 with the minimum possible cardinality is called a minimum vertex cover of @xmath0 and its size is denoted by @xmath29 . in @xcite and @xcite @xmath30 , has been studied . since for every simple graph @xmath0 , @xmath31 @xcite , their results imply the following results for @xmath3 , and @xmath9 : * @xmath32 * for all @xmath33 , @xmath34 . * for all @xmath35 , @xmath36 * if both @xmath7 and @xmath8 are odd , then @xmath37 . + also , if @xmath38 , then @xmath39 . * @xmath40 @xmath41 if and only if @xmath7 is even and @xmath8 is odd . * for all even @xmath8 , we have * * if @xmath42 then @xmath43 . * * if @xmath44 then @xmath45 . * for all odd @xmath7 , we have @xmath46 , where @xmath47 . recently , fox et al . proved the following results in @xcite : * for all @xmath48 , @xmath49 * for any integer @xmath50 , we have that @xmath51 * if @xmath52 are integers with @xmath7 odd and @xmath8 even , then @xmath40 @xmath53 , where @xmath47 . * if @xmath52 are even , then @xmath40 @xmath54 , where @xmath47 . notice that the problem of finding the size of a maximum independent set in the graph @xmath18 is trivial for even @xmath7 and odd @xmath8 , since @xmath18 is a bipartite graph . for odd @xmath7 and @xmath8 , @xmath18 is not bipartite but we can remove a set of @xmath55 edges from @xmath18 to obtain a bipartite graph . thus in this case we have @xmath56 . so , for odd @xmath8 , we have upper and lower bounds for @xmath40 that are at most @xmath55 away from @xmath7 . in contrast , for even @xmath8 , @xmath18 has a lot of odd cycles . in fact , the number of odd cycles in @xmath18 is at least as large as @xmath13 . this observation shows that for even @xmath8 , the graph @xmath18 is far from being a bipartite graph and as we see in continuation , we need more complicated arguments for finding lower and upper bounds for @xmath40 compared to the case that @xmath8 is an odd number . this paper is organized as follows . in section @xmath57 , we provide an upper bound for @xmath40 for even @xmath58 . in section @xmath59 , we present some lower bounds when @xmath8 is even . in both section @xmath57 and section @xmath59 we compare our bounds with the previously existing bounds . some exact values for @xmath40 are given in section @xmath60 by applying results presented in sections @xmath57 and section @xmath59 . finally , in section @xmath61 we prove behsaz - hatami - mahmoodian s conjecture for some cases by using known lower bounds . we checked the conjecture with our table for @xmath11 , and it had no inconsistency . in this section we present an upper bound for @xmath40 when @xmath62 is even and we will show that the presented upper bound is equal to @xmath40 in some cases . our upper bound is better than the upper bound given by behsaz et.al . in @xcite . + for @xmath63 , we call the set @xmath64 a @xmath65-segment and we denote it by @xmath66 . let @xmath67 $ ] be the subgraph of @xmath18 induced by @xmath66 . let @xmath68 be the set of all maximum independent sets of @xmath18 . for every @xmath69 we denote by @xmath70 the number of @xmath65-segments @xmath66 for which @xmath71 . + define @xmath72 . since @xmath68 is nonempty , @xmath73 is also nonempty . let @xmath74 [ note ] for any @xmath75 , @xmath76 if and only if @xmath77 . [ type ] for any @xmath69 , we say @xmath66 is of type @xmath78 with respect to @xmath79 if @xmath80 , of type @xmath57 with respect to @xmath79 if @xmath81 , and of type @xmath59 with respect to @xmath79 if @xmath82 . + let @xmath83 . for a given @xmath84 , we say @xmath66 is of special type @xmath57 with respect to @xmath79 , if @xmath85 and @xmath86 becomes an independent set for @xmath67 $ ] . since @xmath67 $ ] has a perfect matching of spokes @xmath87 , @xmath88 . so every @xmath66 is one of the above types . + note that @xmath89 . [ lemma type 1 ] if @xmath8 is an even number then @xmath90 ) = 2k$ ] and @xmath67 $ ] has a unique @xmath16-set shown in figure [ type 1 ] . + as a type @xmath78 segment.,scaledwidth=80.0% ] @xmath67 $ ] has a perfect matching @xmath91 . so @xmath90 ) \leq \frac{\|v(g[i_t])\|}{2 } = 2k.$ ] on the other hand , figure [ type 1 ] is an example of an independent set of @xmath67 $ ] of size @xmath65 . so @xmath90)= 2k$ ] . + to show that @xmath67 $ ] has a unique @xmath16-set , let @xmath79 be an @xmath16-set of @xmath92)$ ] . since @xmath90)= 2k$ ] , @xmath93 and @xmath79 must contain precisely one vertex from each edge @xmath94 where @xmath95 . notice that the set of @xmath23-vertices of @xmath92)$ ] induces a path of length @xmath65 . therefore @xmath96 the set of @xmath22-vertices of @xmath92)$ ] induces a matching of size @xmath8 . this means that @xmath97 these two observations show that any @xmath16-set @xmath79 of @xmath67 $ ] has @xmath8 vertices from @xmath23-vertices and @xmath8 other vertices from @xmath22-vertices of @xmath98).$ ] moreover , every such @xmath79 contains precisely one vertex from each edge @xmath99 where @xmath100 and @xmath101 where @xmath102 . now , consider two cases : + case 1 : @xmath103 . + in this case , @xmath104 and @xmath105 are forced not to be in @xmath79 . so @xmath106 is forced to be in @xmath79 . then @xmath107 and @xmath108 are forced not to be in @xmath79 and this forces @xmath109 and @xmath110 to be in @xmath79 . since @xmath110 is in @xmath79 , @xmath111 . therefore @xmath112 , so @xmath113 and thus @xmath114 so , we showed that if @xmath103 then @xmath115 too . now , if we repeat the same argument for @xmath116 instead of @xmath117 , we can deduce that @xmath118 and by a simple induction , it follows that @xmath119 for any @xmath120 . particularly , @xmath121 therefore @xmath122 this shows that @xmath123 . hence @xmath124 and @xmath125 . so @xmath126 but we already showed that @xmath109 is forced to be in @xmath79 . this contradiction shows that there is no type @xmath78 @xmath66 for which @xmath127 + case 2 : @xmath128 . + in this case , similar to the argument in case 1 , each vertex is either forced to be in @xmath79 or it is forced not to be in @xmath79 . so , there is a unique pattern for @xmath129 when @xmath130 . since the pattern shown in figure [ type 1 ] is an instance of an independent set of size @xmath65 for @xmath67 $ ] , it is the unique pattern for such an independent set . lemma [ lemma type 1 ] guarantees that there is a unique pattern for @xmath131 , if @xmath66 is of special type @xmath57 with respect to @xmath79 . [ lemma uv ] for every @xmath69 , if @xmath132 then @xmath133 , and @xmath134 also , if @xmath135 is a special type @xmath57 segment with respect to @xmath79 then @xmath136 and @xmath137 . if @xmath132 then @xmath80 . so by lemma [ lemma type 1 ] , there is a unique pattern for @xmath131 . based on this pattern , @xmath138 and @xmath139 . therefore @xmath136 and @xmath137 , since @xmath79 is an independent set of vertices of @xmath18 . a similar argument shows that @xmath140 and @xmath141 . the proof of the second part of the lemma is similar . [ corollary 1 ] if @xmath132 then @xmath142 notice that if @xmath143 then for any edge @xmath144)$ ] either @xmath145 or @xmath146 . since @xmath132 , lemma [ lemma uv ] implies that @xmath136 and @xmath147 . on the other hand , @xmath148)$ ] for @xmath149 . thus @xmath150 [ maintheorem ] @xmath151 for any even number @xmath62 and any integer @xmath152 . let @xmath153 . we consider two cases . + case @xmath78 : @xmath154 . + in this case @xmath155 . so @xmath156 for any @xmath157 . if we add all of these @xmath7 inequalities we get : + @xmath158 on the other hand @xmath159 , since every element of @xmath160 is contained in precisely @xmath65 of the sets @xmath66 . thus : @xmath161 case @xmath57 : @xmath162 + in this case @xmath163 . similar to the inequality [ e : 1 ] we have : + @xmath164 so , to prove the theorem , it suffices to show that there exists @xmath165 such that @xmath166 . + if we can show that for any @xmath167 , there exists an @xmath168 so that @xmath169 , then it follows that @xmath166 . + on the contrary , suppose that there exists @xmath170 in such a way that in the sequence @xmath171 before we see an element of @xmath172 , we see an element of @xmath173 . without loss of generality we can assume that @xmath174 by lemma [ lemma type 1 ] , @xmath175 is of the form depicted in figure [ type 11 ] . + as a type @xmath78 segment.,scaledwidth=80.0% ] since @xmath176 , by corollary [ corollary 1 ] , @xmath177 based on our assumption , @xmath178 in particular , @xmath179 . since @xmath180 , by lemma [ lemma uv ] we have @xmath181 . on the other hand , we know that @xmath160 must have one vertex from each edge @xmath182 where @xmath183 since @xmath184 , either @xmath185 or @xmath186 . but notice that @xmath187 is adjacent to @xmath188 which is in @xmath160 , for @xmath189 . thus , @xmath190 and @xmath185 must be in @xmath160 . this means that @xmath191 . now , define @xmath192 . one can easily see that @xmath193 . based on the choice of @xmath153 , @xmath194 . therefore , there must be an index @xmath195 so that @xmath196 . since @xmath160 and @xmath197 agree on every element except @xmath198 and @xmath199 , the only candidate for @xmath200 is @xmath201 . so @xmath202 and @xmath203 . moreover @xmath204 and @xmath205 . thus @xmath206 . by proposition [ note ] , @xmath207 . notice that if any of @xmath208 are of type @xmath209 with respect to @xmath197 , they are of type @xmath209 with respect to @xmath160 , as well . so , in the sequence @xmath208 , any type @xmath59 segment with respect to @xmath197 appears after an element of type @xmath78 with respect to @xmath197 . since @xmath202 by corollary [ corollary 1 ] , @xmath210 . then from our assumption @xmath211 . + this means that the same argument can be applied to @xmath197 and if we define @xmath212 , then @xmath213 . if we consecutively repeat this argument for @xmath214 where @xmath215 and @xmath216 , then we observe that @xmath217 and @xmath218 for @xmath219 , and none of @xmath220 are of type @xmath78 with respect to @xmath160 . also , @xmath221 for @xmath222 are of special type @xmath57 with respect to @xmath223 . since @xmath223 and @xmath160 agree on the @xmath224 , then @xmath221 for @xmath222 are of special type @xmath57 with respect to @xmath160 . + in other words , if @xmath225 belongs to @xmath226 and the next element of @xmath226 appears before the first element of @xmath227 in the sequence @xmath228 then all of @xmath229 are special type @xmath57 with respect to @xmath160 . in particular , @xmath230 is of special type @xmath57 with respect to @xmath160 . + as @xmath231 , by lemma [ lemma uv ] , @xmath232 and since @xmath233 and @xmath160 agree on the @xmath234 we conclude that @xmath235 . + now consider three cases : * @xmath236 : + since @xmath230 is of special type @xmath57 with respect to @xmath160 , by lemma [ lemma uv ] , we have @xmath237 . this is a contradiction as we assumed @xmath180 and therefore @xmath238 . * @xmath239 : + since @xmath180 , by lemma [ lemma uv ] , @xmath240 . also we know that @xmath241 . thus , @xmath242 is of type @xmath59 with respect to @xmath160 and none of @xmath243 are of type @xmath78 with respect to @xmath160 which is a contradiction . * @xmath244 : + @xmath225 is of type @xmath78 with respect to @xmath160 and for every @xmath245 , @xmath221 is of special type @xmath57 with respect to @xmath160 . in particular , @xmath230 is of special type @xmath57 with respect to @xmath160 , and therefore @xmath246 , ( see figure [ type 2 ] ) . on the other hand , @xmath247 as @xmath180 , and since @xmath248 , @xmath249 is adjacent to @xmath250 . this is a contradiction . so in all the cases , we get a contradiction which means , after any type @xmath78 segment @xmath251 , a type @xmath59 segment @xmath252 will appear before we see another type @xmath78 segment . this means that @xmath253 and the theorem follows , as we argued earlier . in this section we introduce some lower bounds for @xmath40 where @xmath8 is even and @xmath189 . + here we explain a construction for an independent set in @xmath18 for even numbers @xmath7 and @xmath8 . it happens that for every even @xmath254 , our lower bound is equal to the actual value , using a computer program for finding the independence number in @xmath18 . [ nk even ] if @xmath7 and @xmath8 are even and @xmath62 then : @xmath255 where @xmath200 is the remainder of @xmath7 modulo @xmath65 . we partition the vertices of @xmath18 into @xmath256 @xmath65-segments and one @xmath200-segment . since @xmath7 and @xmath8 are even numbers , @xmath200 is also an even number and it is straightforward to see that if we choose a subset of the form shown in figure [ type 2 ] , from each @xmath65-segment they form an independent set @xmath160 of size @xmath257 . then we try to extend this independent set by adding more vertices from the remaining @xmath200-segment . without loss of generality , we may assume that the @xmath200-segment consists of the vertices @xmath258 . consider two cases : * @xmath259 : + in this case the set @xmath260 is an independent set of size @xmath261 . * @xmath262 : + in this case the set : + @xmath263 is an independent set of size @xmath264 as a special type @xmath57 segment.,scaledwidth=80.0% ] in the next theorem we establish a lower bound for @xmath12 for odd @xmath7 and even @xmath8 . + [ n odd k even ] if @xmath7 is odd and @xmath62 is even then we have : @xmath265 where @xmath200 is the remainder of @xmath7 modulo @xmath65 . we construct an independent set for the graph @xmath18 . similar to the proof of theorem [ nk even ] , first we partition the vertices of the graph into @xmath266 @xmath65-segments and a remaining segment of size @xmath200 . without loss of generality , we can assume that the last @xmath65-segment starts from the first spoke and the remaining segment starts from the @xmath267-st spoke and finishes at the @xmath268-th spoke . we also label the @xmath65-segments with indices @xmath269 . from each of @xmath65-segments @xmath270 , we choose @xmath271 vertices as shown in figure [ type 2 ] . we also choose the following vertices from the last @xmath65-segment and the remaining @xmath200-segment : * @xmath272 : + @xmath273 * @xmath274 + @xmath275 * @xmath276 + @xmath277 one can easily check that in each case , the given set is an independent set of size specified in the theorem . notice that the upper bound given in theorem [ maintheorem ] and the lower bound in theorem [ nk even ] , and theorem [ n odd k even ] are very close to each other for every fixed even @xmath189 . more precisely , we have the following corollary : if @xmath278 is an even number then @xmath279 notice that our lower bounds are considerably better than the lower bounds obtained in @xcite and @xcite . in this section , we will find the exact value of @xmath40 for some pairs of @xmath52 . + [ k=4 ] if @xmath280 , then : + @xmath281 @xmath282 this result is straight consequence of theorems [ maintheorem ] , [ nk even ] , and [ n odd k even ] . notice that for @xmath283 and @xmath284 or @xmath285 , the upper bound and lower bound differ by @xmath78 . in fact , for @xmath286 the exact of @xmath3 is the same as our lower bound as we checked by computer . if @xmath278 is an even number and @xmath287 or @xmath288 then @xmath289 . this assertion is trivial consequence of theorems [ maintheorem],[nk even ] and [ n odd k even ] . in fact the upper bound and lower bounds we have for @xmath3 are identical in these cases . [ conj 1 ] ( @xcite ) . for all @xmath7 , @xmath8 we have @xmath290 . + notice that , since @xmath291 , this conjecture is equivalent to @xmath292 . the above conjecture is valid in the following cases : * @xmath293 . * @xmath7 is even and @xmath8 is odd . * @xmath52 are odd and @xmath294 . * @xmath52 are even . * @xmath7 is odd , @xmath8 is even and @xmath295 . * this case is a straight consequence of ( i ) , ( ii ) , ( iii ) , proposition [ k=4 ] and ( viii ) . * in this case @xmath18 is a bipartite graph and @xmath296 . * @xmath297 ( @xcite ) . for @xmath298 this lower bound is greater than @xmath299 . * let @xmath300 where @xmath301 and @xmath302 . we consider the following subcases : * * if @xmath259 and @xmath303 then by theorem [ nk even ] , @xmath304 for any @xmath305 . for @xmath306 conjecture holds based on the information provided in table @xmath78 . * * if @xmath307 and @xmath308 then by theorem [ nk even ] , @xmath309 for any @xmath310 . for @xmath311 , conjecture follows from @xmath312 . * * if @xmath262 and @xmath303 then by theorem [ nk even ] , @xmath313 for any @xmath314 . for @xmath315 conjecture holds based on the information provided in table @xmath78 . * * if @xmath316 and @xmath308 then by theorem [ nk even ] , @xmath317 for any @xmath318 . for @xmath319 , conjecture follows from @xmath312 . * similar to the previous part , let @xmath300 where @xmath301 and @xmath302 . we consider the following subcases : * * if @xmath320 and @xmath303 then @xmath321 which is isomorphic to @xmath322 . ( for more information about isomorphic generalized petersen graphs see @xcite ) . * * if @xmath320 and @xmath323 then by theorem [ n odd k even ] , @xmath324 for every @xmath325 . * * if @xmath326 then @xmath327 has to be larger than @xmath78 . in fact if @xmath326 and @xmath303 then @xmath328 . for @xmath326 and @xmath329 then by theorem [ n odd k even ] , @xmath330 for every @xmath331 . ( note that since @xmath7 is odd and @xmath8 is even , @xmath332 implies that @xmath333 ) . * * if @xmath334 then by theorem [ n odd k even ] , @xmath335 for @xmath336 . for @xmath337 then the assertion is concluded from part @xmath312 . if @xmath295 then @xmath338 , and behsaz - hatami - mahmoodian s conjecture holds . in this section we will prove that the independence number of generalized petersen graphs with fixed @xmath8 can be found in linear time , @xmath13 . this result is a special case of a deep theorem stating that the problem of finding the independence number of graphs with bounded treewidth can be solved in linear time of the number of vertices of the graph . in the continuation we will show that for every fixed @xmath8 and any integer @xmath9 the treewidth of @xmath18 is bounded . first , we need to formally define the concepts of tree decomposition and treewidth of a graph . let @xmath14 be a graph . a tree decomposition of @xmath0 is a pair @xmath339 , where @xmath340 is a family of subsets of @xmath341 , and @xmath342 is a tree whose nodes are the subsets @xmath343 , satisfying the following properties : * the union of all sets @xmath343 equals @xmath341 . * for every edge @xmath344 in the graph , there is a subset @xmath343 that contains both @xmath22 and @xmath345 . * if @xmath343 is on the path from @xmath346 to @xmath347 in @xmath342 then @xmath348 . in other words , for all vertices @xmath349 all nodes @xmath346 which contain @xmath22 induce a connected subtree of @xmath342 . the width of @xmath350 is defined to be the size of the largest @xmath346 minus one . the treewidth , @xmath351 , of the graph @xmath0 is defined to be the minimum width of all its tree decompositions . the treewidth will be taken as a measure of how much a graph resembles a tree . ( @xcite ) [ alghorithm ] the problem of finding a maximum independent set of a graph @xmath0 with bounded treewidth , @xmath352 can be solved in @xmath353 by dynamic programming techniques , where @xmath7 is the number of vertices of graph . for more details see for instance @xcite , @xcite , @xcite , and @xcite . for any fixed @xmath8 , the problem of finding independence number of the graphs @xmath18 can be solved by an algorithm with running time @xmath13 . by theorem [ alghorithm ] , we only need to show that for a given number @xmath8 , the treewidth of @xmath18 is bounded . consider the following tree decomposition of @xmath18 of width @xmath354 . without loss of generality we can only consider the case where @xmath355 . let @xmath342 be the path of order @xmath356 and define @xmath357 as follows : + @xmath358 + @xmath359 + @xmath360 + @xmath361 + @xmath362 and so on . + notice that in each step we remove two elements and add two other elements . therefore @xmath363 for all @xmath209 . one can easily see that @xmath350 is a tree decomposition for @xmath18 where @xmath364 . thus , @xmath365 and by theorem [ alghorithm ] , the proof is complete . table @xmath78 : independence number of @xmath366 . the authors would like to thank the referee for careful reading of this paper and very helpful comments . and the authors like to thank professor e. s. mahmoodian for suggesting the problem and very useful comments . we also thank nima aghdaei , and hadi moshaiedi for their computer program and algorithm to create presented table of @xmath3 . hans l. bodlaender . dynamic programming on graphs with bounded treewidth . in _ automata , languages and programming ( tampere , 1988 ) _ , volume 317 of _ lecture notes in comput . _ , pages 105118 . springer , berlin , 1988 .	determining the size of a maximum independent set of a graph @xmath0 , denoted by @xmath1 , is an np - hard problem . therefore many attempts are made to find upper and lower bounds , or exact values of @xmath2 for special classes of graphs . this paper is aimed toward studying this problem for the class of generalized petersen graphs . we find new upper and lower bounds and some exact values for @xmath3 . with a computer program we have obtained exact values for each @xmath4 . in @xcite it is conjectured that the size of the minimum vertex cover , @xmath5 , is less than or equal to @xmath6 , for all @xmath7 and @xmath8 with @xmath9 . we prove this conjecture for some cases . in particular , we show that if @xmath10 , the conjecture is valid . we checked the conjecture with our table for @xmath11 and it had no inconsistency . finally , we show that for every fixed @xmath8 , @xmath12 can be computed using an algorithm with running time @xmath13 . * keywords * : generalized petersen graphs , independent set , tree decomposition	8,497	318
macroscopic quantum systems such as superconductors and superfluids are the remarkable consequence of many of bosonic particles occupying the same lowest energy state , and thus forming a bose - einstein condensate ( bec ) . the design of closely spaced two dimensional electron systems ( 2des ) which can be contacted independently is the foundation to create a bec of excitons in semiconductors @xcite . exposed to a strong perpendicular magnetic field @xmath1 , the density of states of each of the 2des will condense into a discrete set of sub - bands , the landau levels . the total number of occupied states is then parameterized by the filling factor @xmath2 . if the electron densities @xmath3 are tuned to be identical in both layers , the filling factors will simultaneously be at @xmath4 at a particular @xmath1 . governed by coulomb interactions , the bilayer system can then be viewed as a bose condensate of interlayer quasi - excitons by coupling an electron from layer 1 to a vacant state from layer 2 and vice versa . since these excitons have an infinite life time , their properties can be investigated via electrical transport experiments . transport experiments in the counter - flow configuration @xcite , where constant currents of equal magnitude but opposite direction are imposed on the two layers have indeed shown that exclusively if @xmath5 ( denoted as total filling factor 1 , or simply @xmath6 ) , the hall and longitudinal voltages across both layers ( nearly ) vanish . while this by itself can be interpreted as the result of a dissipationless flow of charge - neutral electron - hole pairs in one direction , interlayer tunneling experiments @xcite have shown an i / v characteristic that has an astonishing resemblance to the one of the josephson effect . however , the bilayer at @xmath6 is only partially analogous to a josephson junction @xcite , and it is important to recognize the experiment as tunneling between two electron systems that _ only as a whole _ form the correlated state @xcite . this fact might also explain why no true dc supercurrent at zero bias has been observed so far . suitable bilayer samples are required to be weakly tunneling @xcite , however , they only possess a very small single electron tunnel splitting @xmath7 of up to approximately 100 @xmath8k . even though interlayer phase coherence is completely _ spontaneous _ only for @xmath9 , it has been demonstrated @xcite that single electron tunneling can co - exists with this correlated state which is still dominated by coulomb interactions . our interlayer tunneling experiments indicate that the bose condensation strongly changes the nature of the tunneling process . more specifically , we exploit a pure dc tunneling configuration which reveals the existence of critical tunneling currents @xmath0 . these critical currents terminate the regime of interlayer phase coherence , i.e. , when the total current @xmath10 exceeds the threshold value of @xmath0 , the 4-terminal interlayer resistance abruptly increases by many orders of magnitude . our data originate from three different samples from the same wafer . the double quantum well structure consists of two 19 nm gaas quantum wells , separated by a 9.9 nm superlattice barrier composed of alternating layers of alas ( 1.70 nm ) and gaas ( 0.28 nm ) . the quantum wells have an intrinsic electron density of about @xmath11 m@xmath12 and a low - temperature mobility which exceeds 40 m@xmath13/vs . while sample a is a standard hall bar geometry with a length of 880 @xmath8 m and a width of 80 @xmath8 m , samples b and c are patterned into a quasi - corbino ring @xcite , both with an outer diameter of 860 @xmath8 m and a ring width of 270 @xmath8 m . a commonly used selective depletion technique @xcite was used to provide separate contacts to the layers . the densities in the two layers are balanced with a front and back gate which cover the entire region of the structures including the edges . the modulation of a tunable dc bias @xmath14 with a low amplitude ac sine wave @xmath15 which is applied between the two layers ( i.e. , the interlayer bias ) is a convenient and commonly used method to determine the differential conductance @xmath16 . while a @xmath17 counter - shifts the fermi energies of both systems , @xmath15 is used to induce an ac ( tunneling ) current which can be detected via a sensitive lock - in technique . in the zero magnetic field case , if both layers have identical densities and @xmath18 , the fermi energies of both layers align , and owing to momentum and energy conservation , electron tunneling becomes possible . under the application of a magnetic field , however , it generally requires a finite energy e@xmath14 to add / extract an electron to / from one of the correlated 2des @xcite . this means that no peak in @xmath19 centered around @xmath14=0 is expected under application of a ( strong ) perpendicular magnetic field . figure [ fig:1 ] shows the results of the common tunneling experiment as previously described . the tunable dc bias was modulated with a small ( @xmath20v ) ac voltage . the current was detected by measuring the voltage drop across a 10 k@xmath21 resistor connected towards common ground . these measurements were performed on sample a ( hall bar ) at @xmath22 mk and @xmath6 with balanced carrier densities in the two layers leading to three different @xmath23 . this ratio of the center - to - center distance @xmath24 between the layers ( here @xmath25 nm ) and the magnetic length @xmath26 characterizes the strength of the @xmath6 state due to coulomb interactions . for figure [ fig:1 ] we use the common notation where we plot the 2-point ( 2pt ) differential conductance @xmath19 versus the 2pt voltage @xmath14 , i.e. , the curve illustrates the measured @xmath27 induced by the ac modulation of 7 @xmath8v versus the variable dc interlayer bias . the peaks centering @xmath28 can be identified as the familiar enhanced tunneling anomaly @xcite of the @xmath6 state . from high to low values of @xmath29 , the full width at half maximum ( fwhm ) in these three cases is about 60 @xmath8v , 160 @xmath8v and 200 @xmath8v . while an increase of the tunneling amplitude upon decreasing @xmath29 was to be expected based on earlier reports , it is yet remarkable that its fwhm appears to increase as well . as we will show next , the answer to this apparent inconsistency is hidden in the modality of a two - terminal tunneling experiment where the interlayer resistance becomes much smaller than other series resistances . ( dotted line ) , @xmath30 ( dashed line ) , @xmath31 ( solid line ) . in addition to its amplitude , also the width increases with decreasing @xmath29 . these data were produced on sample a at @xmath22 mk.,scaledwidth=80.0% ] using a sufficiently sensitive dc measurement setup , the tunneling experiment can be simplified by measuring the dc current directly . in addition , with a separate pair of contacts , a 4-point ( 4pt ) setup is possible to probe the dc voltage that drops across the barrier as well . figure [ fig:2 ] illustrates these 4pt measurements , performed again on sample a at @xmath6 for a single @xmath29 of 1.44 . the current was again detected by measuring the voltage drop across a resistor connected towards the common ground . the top panel thus illustrates this dc current as a function of the 2pt dc voltage . consistent with the prior observation of an enhanced tunneling conductance at small bias voltages , the dc current displays a relatively steep slope around @xmath28 which abruptly terminates when the current exceeds values of approximately -1.5 na or + 1.25 na.v in figure [ fig:2 ] can not be directly compared to the results in figure [ fig:1 ] which are measured at a different value of @xmath29 . in addition , we observed a hysteretic behavior which may lead to a smearing of the curves in ac modulated measurement . ] the existence of such a critical current had already been predicted @xcite but had not been clearly demonstrated . most strikingly , the 4pt measurements on the bottom panel reveal a plateau in the probed dc voltage close to zero which accompanies the region where the current flow is enhanced . . clearly visible are critical currents below which the characteristic has a steeper slope . bottom panel : 4pt voltage @xmath32 which was measured simultaneously versus @xmath33 . a plateau exists around zero bias where the 4pt voltages is nearly zero . the plateau terminates at exactly the same 2pt voltages where the critical currents occur . the inset shows a schematics of the experiment with the source ( s ) and drain ( d ) contacts and the location of the voltage probes @xmath34 . shaded contacts connect to the lower layer . these data were produced on sample a at a @xmath29 of 1.44 and @xmath22 mk.,scaledwidth=85.0% ] we emphasize that the unexpected increase in width of the differential tunneling conductance curves in figure [ fig:1 ] is now explainable in terms of a strongly reduced 4pt dc voltage . in view of this reduction , which affects the 4pt ac modulation as well , the @xmath19 curves would rescale to a very narrow peak with a very high amplitude if plotted versus @xmath32 and with @xmath35 . please note that at no other ( total ) filling factor is such a behavior observable , i.e. , the strong reduction of the 4pt interlayer voltages is a peculiarity of the @xmath6 state . we associate the range in which @xmath32 is small with a state in which interlayer coherence is established . its width apparently depends on the sum of all series resistances @xmath36 in the system , such as contact arms and the series resistance for the current measurement . if @xmath36 is large compared to the 4pt interlayer resistance , the experiment is essentially performed by controlling the current so that the existence of critical currents can be resolved . in a voltage - controlled experiment on the other hand , critical currents are concealed in the limit of vanishing ( interlayer ) resistances . for a set of six different @xmath29=\{1.99 , 1.92 , 1.85 , 1.78 , 1.70 , 1.44}. the mid panel shows the probed 4pt voltage @xmath32 which was not measured simultaneously , and the bottom panel illustrates the calculated 4pt interlayer resistance . the enhanced noise around @xmath37 originates from the noise in detecting small voltages.,scaledwidth=80.0% ] figure [ fig:3 ] demonstrates how the tunneling process evolves upon reducing the ratio @xmath29 from high to low values , i.e. , upon reducing the electron densities in both layers simultaneously and adjusting the magnetic field . these data were produced on sample b ( corbino geometry ) where the voltage @xmath33 was applied between the two outer circumferences of the upper and lower layer . moving from high to low values of @xmath29 , plateaus in the 4pt voltage appear which progressively take on lower values . at the same time the critical currents grow . the resulting 4pt interlayer resistance has , at the lowest @xmath29 , a value of only about 200 ohms at @xmath38 mk . once the critical current is exceeded , the 4pt interlayer resistance is nearly of the same magnitude for all @xmath29 , which suggests that the condensate is destroyed and the current is maintained by bare electron tunneling . the observed asymmetry which is particularly pronounced in sample b for low @xmath29 is owing to a strong hysteresis . the question of the lowest obtainable 4pt resistance and/or its accuracy is directly related to the question which factors influence the 4pt voltage . in addition to a temperature - activated behavior , it is relevant where exactly the potential is probed ( see insets in figures [ fig:2 ] and [ fig:4 ] ) because residual resistances come into play . more precisely , _ any _ current @xmath10 that crosses the boundary of a two - dimensional electron system under quantum hall conditions will produce a voltage difference across the contact of the order of the hall voltage h / e@xmath39 ( @xmath4025 @xmath8v at @xmath41 a ) . since the sign of the hall voltage depends on the sign of the magnetic field @xmath1 , it should be possible to account for its influence by inverting the magnetic field . and indeed , the inversion from @xmath42 to @xmath43 also inverted the slope of @xmath32 around @xmath37 in figure [ fig:2 ] . the mean value calculated from the curves at @xmath44 and @xmath45 , however , did not completely cancel out @xmath32 within the plateau region . this might be caused by longitudinal resistance components , if the current flows through dissipative regions @xcite . nevertheless , as we have shown , this ( residual ) voltage and the resulting 4pt interlayer resistance was a lot smaller for sample b ( corbino ) . for this sample , the voltage was probed in a `` longitudinal '' configuration , i.e. , the voltage was probed ( across the barrier ) at contacts that lie between the source and drain and at the same side of the current flow . for @xmath46 ( sample b ) . only red dots are actual data points , the black dashed lines are used to guide the eye . the inset shows a simplified schematics of the experiment where the contacts to probe the voltage and the source and drain contacts are marked . shaded contacts connect to the lower layer . unused ohmic contacts are disregarded.,scaledwidth=80.0% ] figure [ fig:4 ] finally shows the current plotted versus the 4pt voltage for @xmath46 . in this representation our data resemble earlier reports @xcite where however the maximal current was of order 20 pa , or about 1000 times smaller . note that in @xcite the i / v characteristic was deduced from integrating the differential tunneling conductance data which may have masked the critical current behavior reported here . even though the sample characteristics differ only marginally ( qw / barrier / qw width in @xcite is 18 nm/9.9 nm/18 nm ) which yields a comparable value of @xmath7 , the effective single particle tunneling amplitude in our samples appears to be larger . hence , we assume that the different magnitudes of the maximal currents can be attributed to a different bare interlayer tunneling which strongly influences the tunneling anomaly at @xmath6 @xcite . for reasons of completeness , we would like to elaborate on an experimental detail . generally , the application of an interlayer bias will imbalance the electron densities of both layers , while the total density @xmath47 remains constant . this has the consequence that the regular quantum hall ( qh ) states will shift to lower / higher fields , owing to a higher / lower density in the respective single layer . the @xmath6 qh state on the other hand depends only on @xmath48 and thus does not shift to a different magnetic field . using the shubnikov - de haas oscillations in transport experiments in the low field regime , we were able to adjust front and back gate voltages while sweeping the interlayer bias to keep the density in each of the two layers constant . however , interlayer tunneling experiments did not significantly differ from unadjusted measurements . the bias - induced imbalance for the electron bilayer system in question is @xmath49 for @xmath50=500 @xmath8v , but might become irrelevant as the ( effective ) 4pt interlayer bias nearly vanishes for @xmath51 . in a different experiment , sample c ( corbino ring ) was set up in a drag experiment as described in @xcite , where a voltage is applied across only one layer ( drive layer ) , while the other ( drag ) layer is kept as an open circuit . only at a total filling factor of one has this been shown to produce a voltage drop of equal sign and magnitude across the adjacent drag layer . at the same time , the conductance through the drive layer vanishes , i.e. , the bulk is in a gapped state . in this situation we applied a variable resistor @xmath52 between the inner and outer circumference of the drag layer as shown in the inset of figure [ fig:5 ] . for @xmath53 the system behaves as before . however , upon decreasing @xmath52 from @xmath54 to @xmath55 ohms , a current begins to flow through the bridge connecting inner and outer circumference of the drag layer . simultaneously , a current of equal magnitude in the circuit of the drive layer can be measured . in the light of the strongly reduced 4pt interlayer resistance we demonstrated with figure [ fig:2 ] through [ fig:4 ] , these results can be explained in terms of parallel - tunneling at both sample edges carried by quasiparticles @xcite . . as soon as @xmath52 is sufficiently reduced , the drive current @xmath10 increases , even though the bulk is in a gapped state . meanwhile , a current @xmath56 with the same sign and magnitude through the load resistance can be measured . the inset shows a schematics of the experiment in a pseudo-3d view . bottom panel : 4pt drive voltage @xmath33 as a function of @xmath52 . these data were produced on sample c at a @xmath57 and @xmath58 mk.,scaledwidth=90.0% ] the observables in any of these transport geometries are the currents and voltages _ in the leads _ @xcite , so we have no direct access to what is happening within the bulk . the ground state of our bilayer system in the correlated regime can be described by the halperin ( 111 ) state @xcite , as the laughlin wave function describes the ground state of the fractional quantum hall effect ( fqhe ) . and like in the fqhe , it is convenient to introduce the _ quasiparticle _ concept . these quasiparticles experience enhanced interlayer tunneling , just as the quasiparticle hamiltonian of a superconductor has pair creation and annihilation terms . the quasiparticles in our system arise at the interface where the single particle electron current from the leads meets the correlated @xmath6 phase . since for @xmath6 the bulk of the drive layer is in a gapped state , it prohibits any regular single electron current flow across the annulus . however , in a process which is analogous to andreev reflection @xcite , the injection of a single electron leads to a condensate current , or an excited state from the condensate ground state , respectively . to put it simply , every incident single electron in the top layer excites an exciton in the bulk . to conserve total charge in both layers , or to counter for this sudden net flow of excitons in the bulk , respectively , an electron must exist into the leads in the bottom layer . the bose condensate thus changes the single electrons into quasiparticles which are easily transferred . this constant flow of quasiparticles is the process we would like to refer to as _ quasiparticle tunneling _ @xcite . if the inner and outer circumference of the bottom layer are physically connected over a sufficiently small resistance @xmath52 , it offers a short cut path across the gapped bulk for reflected single electrons . once having passed that bridge , each electron will by itself undergo the same process of triggering condensate currents and quasiparticle tunneling at the other edge . while this model is able to account for our data , we can not definitively say whether this configuration really allows us to trigger such an excitonic current through the bulk of the @xmath6 qh state or not . it is also possible that some still unknown ( tunneling ) process is taking place . we have presented dc tunneling experiments on electron double layer systems at a total filling factor of one which clearly show the existence of critical tunneling currents @xmath0 . when the total current @xmath10 exceeds the critical value , the 4pt interlayer resistance increases by many orders of magnitude . the results can be explained in terms of quasiparticle tunneling which is possible due to the bose condensation . these observations could have grave consequences for the interpretation of the @xmath6 qh state and the transport experiment performed within this regime . this could be of particular relevance if the currents that are imposed in regular transport are smaller than @xmath0 . we thank j. g. s. lok for the design of the corbino geometry and j. h. smet for giving us access to some of his equipment . also , we would like to acknowledge the german ministry of research and education ( bmbf ) for its financial support and gratefully thank both allan h. macdonald and ady stern for discussions . 10 fertig h a 1989 _ phys . b _ * 40 * 1087 macdonald a h and rezayi e h 1990 _ phys . rev . b _ * 42 * 3224 tutuc e and shayegan m and huse d a 2004 _ phys . 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and fischer a and dietsche w and von klitzing k and eberl k 1997 _ phys . * 78 * 1763 eisenstein j p and pfeiffer l n and west k w 1992 _ phys . lett . _ * 69 * 3804 fertig h a and murthy g 2005 _ phys . rev . lett . _ * 95 * 156802 su j j and macdonald a h 2008 _ arxiv:0801.3694 _ , to be published macdonald a h , private communication . halperin b i 1983 _ helv	we have investigated the tunneling properties of an electron double quantum well system where the lowest landau level of each quantum well is half filled . this system is expected to be a bose condensate of excitons . our four - terminal dc measurements reveal a nearly vanishing interlayer voltage and the existence of critical tunneling currents @xmath0 which depend on the strength of the condensate state .	6,301	98
in recent years , renewable energy has gained much popularity and attention because of it s potential in economic and environmental advantages . some of the benefits include- high stainability , low carbon emission , reduction of environmental impact , saving fuel cost and so on . other advantages include economical benefits to remote communities and supporting the microgrids during the operation in islanded mode . although renewable energy , e.g. , wind and solar , offers huge benefits @xcite , their practical use is limited due to their intermittent nature which makes it very challenging to ensure a steady power supply in the grid @xcite . because of the variable nature of the renewable energy based power generation sources , transmission and distribution system operators need advanced monitoring and control . wind power generation relies on wind speed which varies depending on location and time . for economic and stable operation of the wind power plant , accurate forecasting of wind power is critical . there are two main wind power forecasting approaches , physical method and statistical method . in the first approach , the physical system and power translation processes are modelled in detail . therefore , physical approaches not only need the information of historical wind speed data but also other information , i.e. , meteorological output , hub height of the turbine and physical modelling of power conversion process from wind speed are essential @xcite . on the other hand , in a statistical approach , wind power output is modelled as a time - series where the power output at any time instant depends on its previous observation values . the physical approach provides good accuracy for long term forecasting but not so good for short term forecasting as it is computationally very demanding . on the contrary , statistical approaches are well suited for short therm forecasting . for short term wind power forecasting , different approaches are well studied @xcite . in a conventional statistical approach , wind power output behaviour is modelled as a time - series . autoregressive ( ar ) model has been used for wind energy forecasting in @xcite and autoregressive moving average ( arma ) model has been used in @xcite . the artificial neural network ( ann ) is also widely used @xcite . however , the ann based approaches has very slow convergence during the training phase @xcite . on the other hand , statistical regressive models are computationally very efficient and widely used for short term forecasting @xcite . in the statistical approaches , the forecasting accuracy is highly dependent on the estimated model of the wind power output behaviour . therefore , it is important to identify the estimated model parameters accurately . different methods are widely used to estimate the ar model parameters , such as , ordinary least squares ( ls ) approach , forward backward ( fb ) approach , geometric lattice ( gl ) approach and yule - walker ( yw ) approach , etc @xcite . as the wind power output has variable characteristics , the error function obtained from the estimated model may have many local minima . for short - term load forecasting , it has been shown that the particle swarm optimization ( pso ) , one of the major paradigms of the computational swarm intelligence , converges to the global optimal solution of a complex error surface and finds better solution compared with gradient search based stochastic time - series techniques @xcite . previously , pso has been widely used in different applications of power system @xcite . in this work , a modified variant of pso based on constriction factor ( cf ) is employed to identify the ar parameters more accurately . the proposed cf - pso based identified ar parameters have better error minimization profiles compared to the well - established ls , fb , gl and yw based approaches . the organization of this paper is as follows- the formulation of basic pso and cf - pso is discussed in section [ psos ] . autoregressive model order selection and parameter estimation methodology is described in section [ armodel ] . the proposed ar parameter estimation method based on cf - pso is illustrated in section [ psomodel ] . in section [ rnd ] , results obtained from this experiment are given and compared with four standard techniques . finally , the paper concludes with some brief remarks in section [ secend ] . pso is a multi - objective optimization technique which finds the global optimum solution by searching iteratively in a large space of candidate solutions . the description of basic pso and cf - pso formulation is discussed in the following subsections : this meta - heuristic is initialized by generating random population which is referred as a swarm . the dimension of the swarm depends on the problem size . in a swarm , each individual possible solution is represented as a ` particle ' . at each iteration , positions and velocities of particles are updated depending on their individual and collective behavior . generally , objective functions are formulated for solving minimization problems ; however , the duality principle can be used to search the maximum value of the objective function @xcite . at the first step of the optimization process , an _ n_-dimensional initial population ( swarm ) and control parameters are initialized . each particle of a swarm is associated with the position vector and the velocity vector , which can be written as + velocity vector , @xmath0 $ ] and position vector , @xmath1 $ ] + where n represents the search space dimension . before going to the basic pso loop , the position and velocity of each particle is initialized . generally , the initial position of the @xmath2 particle @xmath3 can be obtained from uniformly distributed random vector u ( @xmath4 ) , where @xmath5 and @xmath6 represents the lower and upper limits of the solution space respectively . during the optimization procedure , position of each particle is updated using ( [ peq ] ) @xmath7 where @xmath8 and @xmath9 . + at each iteration , new velocity for each particle is updated which drives the optimization process . the new velocity of any particle is calculated based on its previous velocity , the particle s best known position and the swarm s best known position . particle s best known position is it s location at which the best fitness value so far has been achieved by itself and swarm s best known position is the location at which the best fitness value so far has been achieved by any particle of the entire swarm @xcite . the velocity equation drives the optimization process which is updated using ( [ veq ] ) @xmath10 in this equation , _ w _ is the inertia weight . @xmath11 represents the ` self influence ' of each particle which quantifies the performance of each particle with it s previous performances . the component @xmath12 represents the ` social cognition ' among different particles within a swarm and quantify the performance relative to other neighboring particles . the learning co - efficients @xmath13 and @xmath14 represent the trade - off between the self influence part and the social cognition part of the particles @xcite . the values of @xmath13 and @xmath14 are adopted from previous research and is typically set to 2 @xcite . in eqn ( [ veq ] ) , @xmath15 is particle s best known position and @xmath16 is swarm s best known position . in the solution loop of pso , the algorithm continues to run iteratively , until one of the following stopping conditions is satisfied @xcite . 1 . number of iterations reach the maximum limit , e.g. , 100 iterations . no improvement is observed over a number of iterations , e.g. , error less than @xmath17 = 0.001 to achieve better stability and convergent behavior of pso , a constriction factor has been introduced by clerc and kennedy in @xcite . the superiority of cf - pso over inertia - weight pso is discussed in @xcite . basically , the search procedure of cf - pso is improved using the eigenvalue analysis and the system behavior can be controlled which ensures a convergent and efficient search procedure @xcite . to formulate cf - pso , ( [ veq ] ) is replaced by ( [ veqcons])-([kcons2])@xcite . @xmath18 \label{veqcons}\ ] ] where @xmath19 and @xmath20 here the value of @xmath21 must be greater than 4 to ensure a stable and convergent behavior @xcite . usually , the value of @xmath21 is set to 4.1 ( @xmath22 ) ; therefore , the value of _ k _ becomes 0.7298 @xcite . boundary condition @xcite , which helps to keep the particles within allowable solution space , is also applied in this research as shown below : @xmath23 + ar is a univariate time - series analysis model that is widely used for model estimation and forecasting . in an ar model , the output variable has linear association with its own previous observations . for a sample period of @xmath24^t$ ] , a @xmath25-order ar model can be written following the expression below @xcite : @xmath26 where , @xmath27 are the lag parameters of the model , @xmath28 is the constant term , and @xmath29 is the white gaussian noise with zero mean . to select the minimal appropriate lag order @xmath25 , akaike information criteria is used following ( [ aiceq ] ) , where @xmath30 is the number of parameters in the ar model , @xmath31 is the effective number of observations , @xmath32 is the maximum likelihood of the estimate of the error covariance @xcite . the best fitted model has the minimum aic value . @xmath33 + firstly , the structure of the ar model is selected based on ( [ areq ] ) for a predetermined order @xmath25 . after that the model parameters are estimated using cf - pso . during the optimization loop , the algorithm determines the optimal parameters by minimizing the residual sum of squares ( rss ) of the estimated model as shown in ( [ rsseq ] ) , where @xmath34 is the actual data that need to be predicted and @xmath35 is the estimated data . @xmath36 the steps of cf - pso based ar parameter estimation are discussed below . 1 . initialize particle s position @xmath37 and velocity @xmath38 in an _ n_-dimensional search space . here , the dimension _ n _ represents the order @xmath25 of the ar model and each vector of the particle position indicates a potential solution . 2 . calculate the rss for each ` initial ' particle positions and determine the particle s best known position @xmath39 and swarm s best known position @xmath40 . 3 . update particle s position @xmath37 and velocity @xmath38 following ( [ peq ] ) and ( [ veq ] ) if necessary . 4 . calculate the rss with the updated velocity @xmath38 and position @xmath37 . repeat ( 3 ) and ( 4 ) until the stopping criteria is satisfied , i.e. , there is not significant change in rss over a good number of iterations or the algorithm reaches it s maximum iteration limit . here maximum iteration is considered 100 . the proposed cf - pso based ar model parameter estimation algorithm is implemented on the practical wind power output data of the _ capital wind farm _ , obtained from the australian energy market operator @xcite . the algorithm is implemented using matlab and standard ls , fb , gl and yw based approaches are evaluated using ` system identification toolbox ' @xcite . to compare the performance of the proposed method and four aforementioned well - established methods , following evaluation indices are used : 1 . mean square error ( mse ) : @xmath41 where @xmath42 is the actual data and @xmath43 is the data from the estimated model . 2 . akaike s final prediction error ( fpe ) @xcite : @xmath44 where , @xmath45 is the number of parameters in the estimated model , @xmath46 is the number of values of the model and @xmath47 is the loss function . 3 . normalized mean square error ( nmse ) @xcite : @xmath48 ^ 2\\ { \delta^2}=\frac{1}{n-1 } \sum_{i=1}^n[{y_i } - mean({y_i})]^2\\ \end{split}\ ] ] where @xmath42 is the actual data and @xmath43 is the data from the estimated model . the value of nmse varies between ` -inf ' to ` 1 ' . while ` -inf ' indicates a bad fit , 1 represents the perfect fit of the data . firstly , the appropriate lag order is determined . in order to do that the value of @xmath49 is varied from @xmath50 to @xmath51 and for each value of @xmath49 , the aic is calculated following the information criterion in ( [ aiceq ] ) . once all aic value is known , @xmath49 is selected for that fitted ar(@xmath49 ) model which leads to the minimum aic value . considering @xmath52 , in this analysis aic value is observed when @xmath53 for all four standard methods as shown in fig . [ aiclbl ] . now , the wind power output data ( first week of march 2012 with 5 minute interval ) of the ` capital wind farm ' is used to evaluate the performance of the five approaches including the cf - pso based ar model . for the proposed method , the results are documented considering the mean value of 30 individual runs . the obtained results are summarized in the table . [ case1results ] and the best results among all approaches are highlighted . results presented in table . [ case1results ] shows that the proposed method outperforms the other approaches for this test case . considering the error value ( mse and fpe ) of the ls approach as a base scenario , the error minimization performance ( emp ) of other approaches are evaluated following ( [ myindex ] ) , where @xmath54 is the mse or fpe of the ls method and @xmath55 is the corresponding mse or fpe of other approaches . the positive value of emp indicates an improvement of error minimization performance over ls approach while negative value represents that the performance is worse than the ls approach . @xmath56 in order to justify the performance , the proposed method is employed considering another data set , second week 5s interval data of the month march 2012 for capital wind farm . results from this method is shown in table [ case1results2 ] . in this test case , the proposed cf - pso based ar model reduces the error indices most . compared with ls , almost 40% reduction of error is achieved for this test data . moreover , the nmse for the standard ls , fb , gl and yw based approaches are around 96.7% while proposed method experiences 98.9% , as shown in table . [ case1results2 ] . as the value of nmse close to unity indicates the best performance , proposed method also shows it superiority for this test case . .performance of ar parameter estimation considering the first week data of march 2012 [ cols="^,^,^,^,^,^ " , ] fig . [ actest ] shows the actual wind output data and the estimated model data using the proposed method . the convergence characteristics of the cf - pso based proposed algorithm is shown in fig . [ cfpsolbl ] . from the figure , the algorithm convergence within 40 iterations . according to the results shown in table . [ case1results ] and table . [ case1results2 ] , the best improvement is observed considering the cf - pso based ar modelling among these five approaches for both of the test data sets . the cf - pso based ar parameter estimation method finds a better solution compared to the gradient based methods due to its global search capabilities . it is important to mention that these well - established gradient based methods may trap in local minima as referred by huang et al @xcite . on the other hand , cf - pso finds a global optimal solution . in our analysis , we found that the performance of the cf - pso varied based on the wind data characteristics . if wind data has a global minima that is very close to the local minima , the performance of the cf - pso is slightly improved compared with other algorithms ( as observed in table . i ) . on the other hand , if the local minimum is far from the global minima value , significant improvement is observed using the cf - pso algorithm ( as observed in table . in this paper , constriction factor based pso is employed to enhance the performance of the time - series autoregressive model . the proposed algorithm is implemented to estimate the wind power output considering practical wind data . using the global search capable cf - pso based proposed ar model , the results obtained in this experiment show that the algorithm finds the solution very accurately and efficiently ( within 40 iterations ) . to justify the results obtained from the proposed method , four algorithms including the widely used least - square method and yule - walker method are employed for comparison . experimental results conducted in this experiments show that the proposed method enhances the ar estimation model with better accuracy compared to other four well - established method . in this experiment , the exogenous input variables are not considered during the model estimation , which will be included in the future work . since the proposed model enhances the performance of the autoregressive model by minimizing model estimation errors more effectively , in the future work the forecasting performance will also be explored in detail . a. anwar and h. pota , `` optimum allocation and sizing of dg unit for efficiency enhancement of distribution system , '' in _ ieee international power engineering and optimization conference ( peoco ) _ , jun . p. eriksen , t. ackermann , h. abildgaard , p. smith , w. winter , and j. rodriguez garcia , `` system operation with high wind penetration , '' _ ieee power and energy magazine _ , vol . 3 , no . 6 , 6574 , 2005 . a. anwar and h. pota , `` loss reduction of power distribution network using optimum size and location of distributed generation , '' in _ 21st australasian universities power engineering conference ( aupec ) _ , sept . 2011 , pp . 16 . c. lei and l. ran , `` short - term wind speed forecasting model for wind farm based on wavelet decomposition , '' in _ third international conference on electric utility deregulation and restructuring and power technologies _ , 2008 . m. lei , l. shiyan , j. chuanwen , l. hongling , and z. yan , `` a review on the forecasting of wind speed and generated power , '' _ renewable and sustainable energy reviews _ , vol . 13 , no . 4 , pp . 915 920 , 2009 . d. hill , d. mcmillan , k. r. w. bell , and d. infield , `` application of auto - regressive models to u.k . wind speed data for power system impact studies , '' _ ieee transactions on sustainable energy _ , vol . 3 , no . 1 , pp . 134141 , 2012 . p. poggi , m. muselli , g. notton , c. cristofari , and a. louche , `` forecasting and simulating wind speed in corsica by using an autoregressive model , '' _ energy conversion and management _ , vol . 44 , no . 20 , pp . 3177 3196 , 2003 . g. venayagamoorthy , k. rohrig , and i. erlich , `` one step ahead : short - term wind power forecasting and intelligent predictive control based on data analytics , '' _ ieee power and energy magazine _ , vol . 10 , no . 5 , pp . 7078 , 2012 . k. methaprayoon , c. yingvivatanapong , w .- j . lee , and j. liao , `` an integration of ann wind power estimation into unit commitment considering the forecasting uncertainty , '' _ ieee transactions on industry applications _ , vol . 43 , no . 6 , pp . 14411448 , 2007 . huang , c .- j . huang , and m .- l . wang , `` a particle swarm optimization to identifying the armax model for short - term load forecasting , '' _ ieee transactions on power systems _ , vol . 20 , no . 2 , pp . 11261133 , 2005 . m. karen , v. ramos , and t. razafindralambo , `` using efficiently autoregressive estimation in wireless sensor networks , '' in _ international conference on computer , information , and telecommunication systems ( cits ) _ , 2013 . y. del valle , g. venayagamoorthy , s. mohagheghi , j .- c . hernandez , and r. harley , `` particle swarm optimization : basic concepts , variants and applications in power systems , '' _ ieee transactions on evolutionary computation _ 12 , no . 2 , pp . 171195 , 2008 . r. eberhart and y. shi , `` comparing inertia weights and constriction factors in particle swarm optimization , '' in _ proceedings of the 2000 congress on evolutionary computation _ , vol . 1 , 2000 , pp . 8488 . h. zeineldin , e. el - saadany , m. salama , a. alaboudy , and w. woon , `` optimal sizing of thyristor - controlled impedance for smart grids with multiple configurations , '' _ ieee transactions on smart grid _ , vol . 2 , no . 3 , pp . 528537 , 2011 . m. clerc and j. kennedy , `` the particle swarm - explosion , stability , and convergence in a multidimensional complex space , '' _ ieee transactions on evolutionary computation _ , vol . 6 , no . 1 , 5873 , feb 2002 . j. vlachogiannis and k. lee , `` a comparative study on particle swarm optimization for optimal steady - state performance of power systems , '' _ ieee transactions on power systems _ , vol . 21 , no . 4 , 17181728 , nov . r. a. krohling and l. dos santos coelho , `` coevolutionary particle swarm optimization using gaussian distribution for solving constrained optimization problems , '' _ ieee transactions on systems , man , and cybernetics , part b : cybernetics _ , vol . 36 , no . 6 , pp . 14071416 , dec . 2006 . j. heo , k. lee , and r. garduno - ramirez , `` multiobjective control of power plants using particle swarm optimization techniques , '' _ ieee transactions on energy conversion _ 21 , no . 2 , pp . 552561 , june 2006 .	accurate forecasting is important for cost - effective and efficient monitoring and control of the renewable energy based power generation . wind based power is one of the most difficult energy to predict accurately , due to the widely varying and unpredictable nature of wind energy . although autoregressive ( ar ) techniques have been widely used to create wind power models , they have shown limited accuracy in forecasting , as well as difficulty in determining the correct parameters for an optimized ar model . in this paper , constriction factor particle swarm optimization ( cf - pso ) is employed to optimally determine the parameters of an autoregressive ( ar ) model for accurate prediction of the wind power output behaviour . appropriate lag order of the proposed model is selected based on akaike information criterion . the performance of the proposed pso based ar model is compared with four well - established approaches ; forward - backward approach , geometric lattice approach , least - squares approach and yule - walker approach , that are widely used for error minimization of the ar model . to validate the proposed approach , real - life wind power data of _ capital wind farm _ was obtained from australian energy market operator . experimental evaluation based on a number of different datasets demonstrate that the performance of the ar model is significantly improved compared with benchmark methods . constriction factor particle swarm optimization ( cf - pso ) , ar model , wind power prediction	6,274	348
in this paper we present a novel algorithm to parallelize the direct summation method for astrophysical @xmath1-body problems , either with and without the individual timestep algorithm . the proposed algorithm works also with the ahmad - cohen neighbor scheme @xcite , or with grape special - purpose computers for @xmath1-body problems @xcite . our algorithm is designed to offer better scaling of the communication - computation ratio on distributed - memory multicomputers such as beowulf pc clusters @xcite compared to traditional algorithms . this paper will be organized as follows . in section 2 we describe the traditional algorithms to parallelize direct summation method on distributed - memory parallel computers , and the scaling of communication time and computational time as functions of the number of particles @xmath1 and number of processor @xmath2 . it will be shown that for previously known algorithms the calculation time scales as @xmath10 , while communication time is @xmath11 . thus , even with infinite number of processors the total time per timestep is still @xmath12 , and we can not use more than @xmath12 processors without losing efficiency . @xmath12 sounds large , but the coefficient is rather small . thus , it was not practical to use more than 10 processors for systems with a few thousand particles , on typical beowulf clusters . in section 3 we describe the basic idea of our new algorithm . it will be shown that in this algorithm the communication time is @xmath13 . thus , we can use @xmath0 processors without losing efficiency . this implies a large gain in speed for relatively small number of particles such as a few thousands . we also briefly discuss the relation between our new algorithm and the hyper - systolic algorithm @xcite . in short , though the ideas behind the two algorithms are very different , the actual communication patterns are quite similar , and therefore the performance is also similar for the two algorithms . our algorithm shows a better scaling and also is much easier to extend to individual timestep and ahmad - cohen schemes . in section 4 we discuss the combination of our proposed algorithm and individual timestep algorithm and the ahmad - cohen scheme . in section 5 , we present examples of estimated performance . in section 6 we discuss the combination of our algorithm with grape hardwares . in section 7 we sum up . the parallelization of the direct method has been regarded simple and straightforward [ see , for example , @xcite ] . however , it is only so if @xmath14 and if we use simple shared - timestep method . in this section , we first discuss the communication - calculation ratio of previously known algorithms for the shared timestep method , and then those for individual timestep algorithm with and without the ahmad - cohen scheme . most of the textbooks and papers discuss the ring algorithm . suppose we calculate the force on @xmath1 particles using @xmath2 processors . we connect the processors in a one dimensional ring , and distribute @xmath1 particles so that each processor has @xmath15 particles(figure [ fig : ring ] ) . here and hereafter , we assume that @xmath1 is integer multiple of @xmath2 , to simplify the discussion . the ring algorithm calculates the forces on @xmath1 particles in the following steps . 1 . each processor calculates the interactions between @xmath15 particles within it . calculation cost of this step is @xmath16 , where @xmath17 is the time to calculate interaction between one pair of particles . each processor sends all of its particles to the same direction . here we call that direction `` right '' . thus all processors sends its particles to their right neighbors . the communication cost is @xmath18 , where @xmath19 is the time to send one particle to the neighboring processor and @xmath20 is the startup time for communication . each processor accumulates the force from particles they received to its own particles . calculation cost is @xmath21 . if force from all particles is accumulated , go to step 5 . each processor then sends the particles it received in the previous step to its right neighbor , and goes back to previous step . force calculation completed . = 8 cm the time for actual calculation is given by @xmath22 and the communication time @xmath23 the total time per one timestep of this algorithm is @xmath24 here , we neglect small correction factors of order @xmath25 . for fixed number of particles , the calculation cost ( first term in equation [ eq : tring ] ) scales as @xmath26 while communication cost _ increases_. therefore , for large @xmath2 we see the decrease in the efficiency . here we define efficiency as @xmath27 which reduces to @xmath28 thus , to achieve the efficiency better than 50% , the number of processor @xmath2 must be smaller than @xmath29 equation ( [ eq : phalfring ] ) can be simplified in the two limiting cases @xmath30 in most of distributed - memory multicomputers , @xmath31 . for example , with a 1 gflops processor , we have @xmath32 . if this processor is connected to other processor with the communication link of the effective speed of 10mb / s , @xmath33 . the value of @xmath20 varies depending on both networking hardware and software . table 1 gives the order - of - magnitude values for these coefficients for several platforms . .time coefficients in seconds [ cols="<,^,^,^,^,^",options="header " , ] even so , the number of processors we can use with this 2d algorithm is significantly larger than that for 1d ring , for any value of @xmath1 . if @xmath34 , we can use @xmath0 processors . even if @xmath35 , we can still use @xmath36 processors . in this 2d ring algorithm , the @xmath37 term in the communication cost limits the total performance . we can reduce this term by using the extension of the copy algorithm to 2d . = 8 cm instead of using the ring algorithm in the first stage , processors @xmath38 broadcast their data to all other processors in the same row . after this broadcast processor @xmath39 has both group @xmath40 and group @xmath41 . then each processor calculates the force on particles they received ( group @xmath40 ) from particles they originally have ( group @xmath41 ) . in this scheme , the communication cost is reduced to @xmath42 if the network switch supports the broadcast . if the network does not support the broadcast , the cost varies between @xmath43 for the case of a ring network and @xmath44 for a full crossbar . in the second stage , summation is now taken over the processors in the same row . here , result for row @xmath40 must be obtained on processor @xmath38 , which then broadcasts the forces to all processors in the same column . after this broadcast , all processors have the forces on all particles in them . they can then use this forces to integrate the orbits of particles . in this algorithm , the time integration calculation is duplicated over @xmath45 processors in the same column , but in most cases this does not matter . one alternative is that processor @xmath38 performs the time integration and broadcasts the updated data of particles to other processor in the same column . yet another possibility is to let each of @xmath45 processor to integrate @xmath46 particles , which each of them then broadcasts within the column . which approach is the best depends on the ratio between calculation speed , communication speed and communication startup overhead . the total time per one timestep of this algorithm is @xmath47 for the number of processors @xmath48 for which the efficiency is 50% , we have a cubic equation . for the two limiting cases , the solution is given as @xmath49 & ( $ n > n_{\rm c,2dbcast}$ ) , } \label{eq : p2dbcast}\ ] ] where @xmath50 is defined as @xmath51 the critical value of @xmath1 , @xmath50 , is larger than that for the 2d ring version of the algorithm . this is because we reduced the @xmath52 term in the communication cost to @xmath53 . more importantly , even for @xmath54 , @xmath55 is only logarithmically smaller than @xmath0 . thus , with this broadcast version of the algorithm we can really use @xmath0 processors and still achieve high efficiency . now the relation between our algorithm and the hyper - systolic algorithm @xcite must be obvious . the `` regular bases '' version of the hyper - systolic algorithm applied to @xmath56 processors works in the essentially the same way as the ring version of our algorithm works , though in order to derive our algorithm we do not need to use any complex concepts like @xmath57-range problem or additive number theory . to put things in a slightly different way , the hyper - systolic algorithm is a complex way to reconstruct combination of rowwize ring and columwize summation on a 2d network by a sequence of shift operations in 1-d ring network . thus , as far as the @xmath58 term is small , our algorithm and the hyper - systolic algorithm show the same scaling . however , since @xmath58 term would almost always limit the scaling , the broadcast version of our algorithm is almost always better than the hyper - systolic algorithm . in addition , our algorithm is by far easier to understand and implement . this simplicity of our algorithm makes it possible to extend our algorithm to the individual timestep scheme and even to the ahmad - cohen scheme , which will be discussed in the following sections . if we use the broadcast version as the base , the extension to the individual timestep method is trivial . instead of broadcasting all particles in the first stage , we broadcast only the particles in the current block . in the following steps , we always send only data related with the particles in the current block . everything else is the same as in the case of the shared timestep algorithm . using the same assumption of @xmath59 , we have @xmath60 and @xmath61 where @xmath62 is now given by the following implicit equation @xmath63 for the example values in table 1 , the value of @xmath62 is fairly small . so we can use only @xmath64 processors . however , this is still much larger than the number of processors that can be used with 1d implementation of the individual timestep algorithm . the same load - balance problem as we have discussed in the case of the copy algorithm occurs with this method . we need some load - balancing strategy to actually use this method . the difference from the individual timestep scheme is that the neighbor list is created / used to calculate the forces . the neighbor list for forces from particles in group @xmath41 to particles in group @xmath40 is created , stored and used only by processor @xmath39 . therefore , there is no increase in the communication cost , except for the summation of the number of neighbors . the total calculation time and the 50% efficiency processor count are given by : @xmath65 and @xmath66 where @xmath67 is now given by the following implicit equation @xmath68 note that , in this case , whether or not @xmath69 makes very small difference for the number of processors , since the difference is only of the order of @xmath70 . thus , practically we can say that we can use @xmath71 processors with the 2d version of the ahmad - cohen scheme . in this section , we present the theoretical comparison of the proposed algorithm and the traditional one - dimensional algorithm . first we show the result for the case of myrinet - like fast network . figures [ fig : n3 ] to [ fig : n5 ] show the efficiencies for three different values of @xmath1 as the function of the number of processors @xmath2 . it is clear that 2d algorithms allow us to use much larger number of processors compared to their 1d counterparts . the gain is larger for larger @xmath1 , but becomes smaller as we use more advanced algorithms . the gain for individual timestep is smaller than that for shared timestep , and that for the ahmad - cohen scheme is even smaller . even so , the gain in the processor count is more than a factor of 5 , for the case of the ahmad - cohen scheme and @xmath72 . we believe this is quite a large gain in the parallel efficiency . = 8 cm = 8 cm = 8 cm figure [ fig : n5fe ] shows the efficiencies for the case of the fast ethernet . with the 2d algorithm we can use more than 500 processors even with ahmad cohen scheme , for @xmath73 . pc clusters with inexpensive networks seem to be very attractive platforms to implement parallel version of the ahmad - cohen scheme . the only thing grape does is to greatly reduce the value of @xmath17 . thus , the same 2d network of processors each with one grape processor works fine , if the cost of the frontend is less than that of a grape processor . grape-6 achieves essentially the same effect as this 2d processor grid , but using only @xmath45 hosts and @xmath56 grape processors connected with a rather elaborate multistage networks . in hindsight , such an elaborate network is unnecessary , if the fast network is available for a low cost . from the point of view of the scaling relations , what a grape hardware changes is simply @xmath17 . if we attach a 1tflops grape hardware to a 1 gflops host , we reduce @xmath17 by a factor of @xmath74 . this means that the limiting factor for the number of processors is almost always @xmath19 and not @xmath20 . thus , for a parallel grape system , high - throughput , high - latency network such as gigabit ethernet is a practical choice . figures [ fig : n5g6 ] and [ fig : n6g6 ] shows the efficiencies for grape-6 system . since @xmath17 is much smaller , the number of processors we can use becomes much smaller . of course , in the case of 1d algorithms , the actual speed we can achieve does not depend on @xmath17 , since the total speed is @xmath75 and @xmath55 is proportional to @xmath17 . figure [ fig : n6g6 ] indicates that we can achieve the speed of multiple petaflops with currently available technology , by configuring several thousand grape-6 boards into a 2d network . = 8 cm = 8 cm we described a new two - dimensional algorithm to implement the direct summation method on distributed - memory parallel computers . the basic idea of the new algorithm is to organize processors to @xmath76 2d network , and let the data be shared both rowwize and columnwize . in this way , we can reduce the communication cost from @xmath12 of the previously know algorithms to @xmath77 . for the case of the shared timestep algorithm , the new algorithm behaves in essentially the same as the `` regular bases '' version of the hyper - systolic algorithm does . however , with the broadcast version of our algorithm the communication overhead is reduced , which resulted in the better scaling . also , our algorithm is much simpler , which helped us to extend our algorithm to individual timestep and the ahmad - cohen scheme , as well as combination with grape hardwares . for all cases , compared to the previously known algorithm , the number of processors we can use without losing efficiency is almost _ squared_. this is a quite large improvement in the efficiency of a parallel algorithm . usually , a paper which proposes a new parallel algorithm should offer the verification of the concept , by means of the timing measurement of actual implementation . in this paper we omit this verification , because we believe it s important to let those who working on hyper - systolic algorithms be aware of simpler alternatives . in this paper , we assumed a network with full connectivity . this assumption is okay with small pc clusters , but not true on large mpps . in this case , parallel efficiency of 1d algorithm is significantly reduced , and relative gain of 2d algorithm would become much larger . i thank rainer spurzem and yoko funato for valuable discussions . this work is supported in part by the research for the future program of japan society for the promotion of science ( jsps - rftp97p01102 ) .	we present a novel , highly efficient algorithm to parallelize @xmath0direct summation method for @xmath1-body problems with individual timesteps on distributed - memory parallel machines such as beowulf clusters . previously known algorithms , in which all processors have complete copies of the @xmath1-body system , has the serious problem that the communication - computation ratio increases as we increase the number of processors , since the communication cost is independent of the number of processors . in the new algorithm , @xmath2 processors are organized as a @xmath3 two - dimensional array . each processor has @xmath4 particles , but the data are distributed in such a way that complete system is presented if we look at any row or column consisting of @xmath5 processors . in this algorithm , the communication cost scales as @xmath6 , while the calculation cost scales as @xmath7 . thus , we can use a much larger number of processors without losing efficiency compared to what was practical with previously known algorithms . _ pacs : 02.60.cb;95.10.ce ; 98.10.+z _ celestial mechanics , stellar dynamics;methods : numerical # 1@xmath8 # 1@xmath9 # 1[#1 piet ] # 1[#1 jun ] = cmbx10 scaled 2	4,138	342
the purpose of this paper is to outline the fundamental role the mittag - leffler function in renewal processes that are relevant in the theories of anomalous diffusion . as a matter of fact the interest in this function in statistical physics and probability theory has recently increased as is shown by the large number of papers published since 1990 of which a brief ( incomplete ) bibliography includes . in this paper we develop a theory for long - time behaviour of a renewal process with a generic power law waiting distribution of order @xmath1 , @xmath2 ( thereby for easy readability dispensing with decoration by a slowly varying function ) . to bring the distant future into near sight we change the unit of time from @xmath3 to @xmath4 , @xmath5 . for the random waiting times @xmath6 this means replacing @xmath6 by @xmath7 . then , having very many events in a moderate span of time we compensate this compression by respeeding the whole process , actually slowing it down so that again we have a moderate number of events in a moderate span of time . we will relate the rescaling factor @xmath8 and the respeeding factor @xmath9 in such a way that in the limit @xmath10 we have a reasonable process , namely one whose waiting time distribution is the mittag - leffler waiting time distribution whose density is @xmath11 with the mittag - leffler function @xmath12 we will call the renewal process with waiting time density @xmath13 the _ mittag - leffler ( renewal ) process_. this process can be seen as a fractional generalization of the poisson process , see @xcite . our method is , in some sense , analogous to the one applied in the sixties of the past century by gnedenko and kovalenko @xcite in their analysis of _ thinning _ ( or _ rarefaction _ ) of a renewal process . they found , under certain power law assumptions , in the infinite thinning limit , for the waiting time density the laplace transform @xmath14 but did not identify it as a mittag - leffler type function . in section 2 , we provide , in our notation , an outline of the thinning theory for renewal processes essentially following gnedenko and kovalenko . their method has inspired us for the reatment of our problems . as we consider our renewal process formally as a _ continuous time random walk _ ( ctrw ) with constant non - random jumps 1 in space ( for the counting function @xmath15 , in section 3 we embed ab initio our theory into that of the ctrw , thus being in the position to treat the theory of a time fractional ctrw as limiting case of a ctrw with power law waiting time distribution . in this context the pioneering paper by balakrishnan @xcite of 1985 deserves to be mentioned . balakrishnan already found the importance of the laplace transform @xmath16 in the time fractional ctrw and time fractional diffusion , but also did not identify it as the laplace transform of @xmath13 . then , in 1995 hilfer and anton @xcite , see also @xcite , showed that this waiting time density is characteristic for the time fractional ctrw and can be expressed in terms of the mittag - leffler function in two parameters , that is @xmath17 with the generalized mittag - leffler function @xmath18 the form ( 1.3 ) is equivalent to the form ( 1.1 ) that we prefer as it exhibits visibly also the cumulative probability function , the _ survival function _ , @xmath19 . we explain in section 4 two manipulations , _ rescaling _ and _ respeeding _ and use these in section 5 to deduce the asymptotic universality of the mittag - leffler waiting time density under a power law assumption for the original waiting time . then , in section 6 , assuming a suitable power law also for the spatial jumps we show that by a rescaling of the jump widths by a positive factor @xmath20 ( that means a change of the unit of space from 1 to @xmath21 to bring into near sight the far - away space ) another respeeding is effected , now an acceleration , that in the limit @xmath22 ( under a proper relation between @xmath20 and @xmath8 ) leads to space - time fractional diffusion . in section 7 , we pass to a properly scaled limit for the counting function @xmath15 of a renewal process ( again under power law assumption ) and obtain the time fractional drift process ( viewing @xmath15 as a spatial variable ) . we will extensively work with the transforms of laplace and fourier , so easing calculations and proofs of _ convergence in distribution _ ( also called weak convergence " ) for our passages to the limit . essentially , we treat in this paper three topics . first , in section 2 , the thinning of a pure renewal process . second , in sections 3 - 6 , under power law assumption for the waiting time , the asymptotic relevance of the mittag - leffler law , and then the general ctrw with special attention to space and time transition limits to fractional diffusion . as a third topic , in section 7 , we investigate the long time behaviour of the mittag - leffler renewal process . essential properties of the derivative of fractional order in time and in space are given in appendix a and appendix b , respectively . finally , in appendix c we give details on the two special functions of the mittag - leffler type that play a fundamental role in this paper , the mittag - leffler survival probability and the mittag - leffler waiting time density . the _ thinning _ theory for a renewal process has been considered in detail by gnedenko and kovalenko @xcite . we must note that other authors , like szntai @xcite speak of _ rarefaction _ in place of thinning . let us sketch here the essentials of this theory : in the interest of transparency and easy readability we avoid the possible decoration of the relevant power law by multiplying it with a _ slowly varying function_. as usual we call a ( measurable ) positive function @xmath23 _ slowly varying at zero _ if @xmath24 with @xmath25 for every @xmath26 , _ slowly varying at infinity _ if @xmath24 with @xmath27 for every @xmath26 . a standard example of a slowly varying function at zero and at infinity is @xmath28 , with @xmath29 . denoting by @xmath30 , @xmath31 the time instants of events of a renewal process , assuming @xmath32 , with @xmath33 waiting times @xmath34 for @xmath35 , ( generically denoted by t ) , _ thinning _ ( or _ rarefaction _ ) means that for each positive index @xmath36 a decision is made : the event happening in the instant @xmath37 is deleted with probability @xmath38 or it is maintained with probability @xmath39 , @xmath40 . this procedure produces a _ thinned _ or _ rarefied _ renewal process with fewer events ( very few events if @xmath41 is near zero , the case of particular interest ) in a moderate span of time . to compensate for this loss we change the unit of time so that we still have not very few but still a moderate number of events in a moderate span of time such change of the unit of time is equivalent to rescaling the waiting time , multiplying it with a positive factor @xmath8 so that we have waiting times @xmath42 , and instants @xmath43 , in the rescaled process . our intention is , vaguely speaking , to dispose on @xmath8 in relation to the rarefaction parameter @xmath41 in such a way that for @xmath41 near zero in some sense the average " number of events per unit of time remains unchanged . in an asymptotic sense we will make these considerations precise . denoting by @xmath44 the probability distribution function of the ( original ) waiting time @xmath6 , by @xmath45 its density ( @xmath45 is a generalized function generating a probability measure ) so that @xmath46 , and analogously by @xmath47 and @xmath48(t ) the distribution and density , respectively , of the sum of @xmath36 waiting times , we have recursively @xmath49 observing that after a maintained event the next one of the original process is kept with probability @xmath41 but dropped in favour of the second - next with probability @xmath50 and , generally , @xmath51 events are dropped in favour of the @xmath52-th - next with probability @xmath53 , we get for the waiting time density of the thinned process the formula @xmath54 with the modified waiting time @xmath55 we have @xmath56 hence the density @xmath57 , and analogously for the density of the sum of @xmath52 waiting times @xmath58 . the density of the waiting time of the rescaled ( and thinned ) process now turns out as @xmath59 in the laplace domain we have @xmath60 hence ( using @xmath61 ) @xmath62 from which by laplace inversion we can , in principle , construct the waiting time density of the thinned process . by rescaling we get @xmath63 being interested in stronger and stronger thinning ( _ infinite thinning _ ) let us now consider a scale of processes with the parameters @xmath64 ( of _ rescaling _ ) and @xmath41 ( of _ thinning _ ) , with @xmath41 tending to zero _ under a scaling relation @xmath65 yet to be specified_. we have essentially two cases for the waiting time distribution : its expectation value is finite or infinite . in the first case we put @xmath66 in the second case we assume a queue of power law type ( dispensing with a possible decoration by a function slowly varying at infinity ) @xmath67 then , by the karamata theory ( see @xcite ) the above conditions mean in the laplace domain @xmath68 with a positive coefficient @xmath69 and @xmath2 . the case @xmath70 obviously corresponds to the situation with finite first moment ( 2.6a ) , whereas the case @xmath71 is related to a power law queue with @xmath72 now , passing to the limit of @xmath73 of infinite thinning under the scaling relation @xmath74 between the positive parameters @xmath41 and @xmath8 , the laplace transform of the rescaled density @xmath75 in ( 2.5 ) of the thinned process tends for fixed @xmath76 to @xmath77 which corresponds to the mittag - leffler density @xmath78 let us remark that gnedenko and kovalenko obtained ( 2.9 ) as the laplace transform of the limiting density but did not identify it as the laplace transform of a mittag - leffler type function . observe that in the special case @xmath79 we have @xmath80 , hence as the limiting process the poisson process , as formerly shown in 1956 by rnyi @xcite . the name _ continuous time random walk _ ( ctrw ) became popular in physics after montroll , weiss and scher ( just to cite the pioneers ) in the 1960 s and 1970 s published a celebrated series of papers on random walks for modelling diffusion processes on lattices , see @xcite , and the book by weiss @xcite with references therein . ctrws are rather good and general phenomenological models for diffusion , including processes of anomalous transport , that can be understood in the framework of the classical renewal theory , as stated in the booklet by cox @xcite . in fact a ctrw can be considered as a compound renewal process ( a simple renewal process with reward ) or a random walk _ subordinated _ to a simple renewal process . a spatially one - dimensional ctrw is generated by a sequence of independent identically distributed ( @xmath81 ) positive random waiting times @xmath82 each having the same probability density function @xmath83 @xmath84 and a sequence of @xmath81 random jumps @xmath85 in @xmath86 each having the same probability density @xmath87 @xmath88 let us remark that , for ease of language , we use the word density also for generalized functions in the sense of gelfand & shilov @xcite , that can be interpreted as probability measures . usually the _ probability density functions _ are abbreviated by @xmath89 . we recall that @xmath90 with @xmath91 and @xmath92 with @xmath93 . setting @xmath94 @xmath95 for @xmath96 the wandering particle makes a jump of length @xmath97 in instant @xmath30 , so that its position is @xmath98 for @xmath99 and @xmath100 for @xmath101 we require the distribution of the waiting times and that of the jumps to be independent of each other . so , we have a compound renewal process ( a renewal process with reward ) , compare @xcite . by natural probabilistic arguments we arrive at the _ integral equation _ for the probability density @xmath102 ( a density with respect to the variable @xmath103 ) of the particle being in point @xmath103 at instant @xmath104 see @xcite , @xmath105\,dt'\ , , \eqno(3.1)\ ] ] in which the _ survival function _ @xmath106 denotes the probability that at instant @xmath107 the particle is still sitting in its starting position @xmath108 clearly , ( 3.1 ) satisfies the initial condition @xmath109 . note that the _ special choice _ @xmath110 gives the _ pure renewal process _ , with position @xmath111 , denoting the _ counting function _ , and with jumps all of length 1 in positive direction happening at the renewal instants . for many purposes the integral equation ( 3.1 ) of ctrw can be easily treated by using the laplace and fourier transforms . writing these as @xmath112 @xmath113 then in the laplace - fourier domain eq . ( 3.1 ) reads @xmath114 introducing formally in the laplace domain the auxiliary function @xmath115 and assuming that its laplace inverse @xmath116 exists , we get , following @xcite , in the laplace - fourier domain the equation @xmath117 = \left [ \widehat w(\kappa ) -1\right]\ , \widehat{\widetilde p}(\kappa , s ) \ , , \eqno(3.6)\ ] ] and in the space - time domain the generalized kolmogorov - feller equation @xmath118 with @xmath119 . if the laplace inverse @xmath116 of the formally introduced function @xmath120 does not exist , we can formally set @xmath121 and multiply ( 3.6 ) with @xmath122 . then , if @xmath123 exists , we get in place of ( 3.7 ) the alternative form of the generalized kolmogorov - feller equation @xmath124\ , dt'\ , , \eqno(3.7')\ ] ] with @xmath119 . special choices of the memory function @xmath116 are @xmath125 and @xmath126 , see eqs ( 3.8 ) and ( 3.12 ) : @xmath127 giving the _ exponential waiting time _ with @xmath128 in this case we obtain in the fourier- laplace domain @xmath129\ , \widehat{\widetilde p}(\kappa , s ) \ , , \eqno(3.10)\ ] ] and in the space - time domain the _ classical kolmogorov - feller equation _ @xmath130 @xmath131 giving the _ mittag - leffler waiting time _ with @xmath132 in this case we obtain in the fourier - laplace domain @xmath133 = \lt [ \widehat w(\kappa ) -1\right]\ , \widehat{\widetilde p}(\kappa , s ) \ , , \eqno(3.14)\ ] ] and in the space - time domain the _ time fractional kolmogorov - feller equation _ @xmath134 where @xmath135 denotes the fractional derivative of of order @xmath136 in the caputo sense , see appendix a. the time fractional kolmogorov - feller equation can be also expressed via the riemann - liouville fractional derivative @xmath137 , see again appendix a , that is @xmath138 , \eqno(3.16)\ ] ] with @xmath109 . the equivalence of the two forms ( 3.15 ) and ( 3.16 ) is easily proved in the fourier - laplace domain by multiplying both sides of eq . ( 3.14 ) with the factor @xmath139 . we note that the choice @xmath140 may be considered as a limit of the choice @xmath126 as @xmath70 . in fact , in this limit we find @xmath141 so @xmath142 ( according to a formal representation of the dirac generalized function @xcite ) , so that eqs . ( 3.6)-(3.7 ) reduce to ( 3.10)-(3.11 ) , respectively . in this case the order of the caputo derivative reduces to 1 and that of the r - l derivative to 0 , whereas the mittag - leffler waiting time law reduces to the exponential . in the sequel we will formally unite the choices ( * i * ) and ( * ii * ) by defining what we call the mittag - leffler memory function @xmath143 whose laplace transform is @xmath144 thus we will consider the whole range @xmath2 by extending the mittag - leffler waiting time law in ( 3.13 ) to include the exponential law ( 3.9 ) . equation ( 3.7 ) clearly may be supplemented by an arbitrary initial probability density @xmath145 . the corresponding replacement of @xmath146 by @xmath147 in ( 3.1 ) then requires in ( 3.4 ) multiplication of the term @xmath148 by @xmath149 and in ( 3.6 ) replacement of the lhs by @xmath150 $ ] . with @xmath151 we obtain in @xmath102 the fundamental solution of ( 3.7 ) we now consider two types of manipulations on the ctrw by acting on its governing equation ( 3.7 ) in its laplace - fourier representation ( 3.6 ) . + * ( a ) : rescaling the waiting time * , hence the whole time axis ; + * ( b ) : respeeding the process. means change of the unit of time ( measurement ) . we replace the random waiting time @xmath6 by a waiting time @xmath7 , with the positive _ rescaling factor _ @xmath152 . our idea is to take @xmath153 in order to bring into near sight the distant future . in a moderate span of time we will so have a large number of jump events . for @xmath154 we get the rescaled waiting time density @xmath155 by decorating also the density @xmath38 with an index @xmath8 we obtain the rescaled integral equation of the ctrw in the laplace - fourier domain as @xmath156 = \lt [ \widehat w(\kappa ) -1\rt]\ , \widehat{\widetilde { p}}_\tau ( \kappa , s)\ , , \eqno(4.2)\ ] ] where , in analogy to ( 3.5 ) , @xmath157 means multiplying the quantity representing @xmath158 by a factor @xmath159 , where @xmath160 is the _ respeeding factor _ : @xmath161 means _ acceleration _ , @xmath162 means _ deceleration_. in the laplace - fourier representation this means multiplying the rhs of eq . ( 3.6 ) by the factor @xmath9 since the expression @xmath163 $ ] corresponds to @xmath164 . we now chose to consider the procedures of rescaling and respeeding in their combination so that the equation in the transformed domain of the rescaled and respeeded process has the form @xmath165 = a\ , \lt [ \widehat w(\kappa ) -1\rt]\ , \widehat{\widetilde { p}}_{\tau , a } ( \kappa , s)\ , , \eqno(4.4)\ ] ] clearly , the two manipulations can be discussed separately : the choice @xmath166 means _ pure rescaling _ , the choice @xmath167 means _ pure respeeding _ of the original process . in the special case @xmath168 we only respeed the original system ; if @xmath169 we can counteract the compression effected by rescaling to again obtain a moderate number of events in a moderate span of time by respeeding ( decelerating ) with @xmath170 . these vague notions will become clear as soon as we consider power law waiting times . defining now @xmath171 we finally get , in analogy to ( 3.6 ) , the equation @xmath172 = \lt [ \widehat w(\kappa ) -1\rt]\ , \widehat{\widetilde { p}}_{\tau , a}(\kappa , s)\ , . \eqno(4.6)\ ] ] what is the combined effect of rescaling and respeeding on the waiting time density ? in analogy to ( 3.5 ) and taking account of ( 4.5 ) we find @xmath173 and so , for the deformation of the waiting time density , the _ essential formula _ @xmath174 : the formula ( 4.8 ) has the same structure as the thinning formula ( 2.5 ) by identification of @xmath9 with @xmath41 . in both problems we have a rescaled process defined by a time scale @xmath152 , and we send the relevant factors @xmath152 , @xmath9 and @xmath41 to zero under a proper relationship . however in the thinning theory the relevant independent parameter going to 0 is that of thinning ( actually respeeding ) whereas in the present problem it is the rescaling parameter @xmath152 . we have essentially two different situations for the waiting time distribution according to its first moment ( the expectation value ) being finite or infinite . in other words we assume for the waiting time @xmath89 @xmath175 either @xmath176 or @xmath177 for convenience we have dispensed in ( 5.2 ) with decorating by a slowly varying function at infinity the asymptotic power law . then , by the standard tauberian theory ( see @xcite ) the above conditions ( 5.1)-(5.2 ) mean in the laplace domain the ( comprehensive ) asymptotic form @xmath178 where we have @xmath179 then , _ fixing @xmath76 _ as required by the continuity theorem of probability theory for laplace transforms , taking @xmath180 and _ sending @xmath8 to zero _ , we obtain in the limit the mittag - leffler waiting time law . in fact , eqs . ( 4.8 ) and ( 5.3 ) imply as @xmath10 with @xmath2 , @xmath181 } { 1 - ( 1-\lambda \tau^\beta)\ , \left[1-\lambda \tau^\beta s^\beta + o(\tau^\beta s^\beta)\right ] } \to \frac{1}{1 + s^\beta } \ , , \eqno(5.6)\ ] ] the laplace transform of @xmath182 , see ( 1.1 ) and appendix c. this formula expresses the asymptotic universality of the mittag - leffler waiting time law * that includes the exponential law for @xmath80 . it can easily be generalized to the case of power laws decorated with slowly varying functions , thereby using the tauberian theory by karamata ( see again @xcite ) . the formula ( 5.6 ) says that our general power law waiting time density is gradually deformed into the mittag - leffler waiting time density as @xmath8 tends to zero . let us stress here the distinguished character of the mittag - leffler waiting time density @xmath183 defined in ( 1.1 ) . considering its laplace transform @xmath184 we can easily prove the identity @xmath185 ^ml_,a ( s ) = ^ml ( s / a^1/ ) > 0 , a>0 . ( 5.8)@xmath185 note that eq . ( 5.8 ) states the _ self - similarity _ of the combined operation _ rescaling - respeeding _ for the mittag - leffler waiting time density . in fact , ( 5.8 ) implies @xmath186 with @xmath187 which means replacing the random waiting time @xmath188 by @xmath189 . as a consequences , choosing @xmath190 we have @xmath185 ^ml_,^ ( s ) = ^ml ( s ) > 0 . ( 5.9)@xmath185 hence _ the mittag - leffler waiting time density is invariant against combined rescaling with @xmath152 and respeeding with @xmath191_. observing ( 5.6 ) we can say that @xmath192 is a @xmath10 attractor for any power law waiting time ( 5.2 ) under simultaneous rescaling with @xmath152 and respeeding with @xmath193 . in other words , this attraction property of the mittag - leffler probability distribution with respect to power law waiting times ( with @xmath194 ) is a kind of analogy to the attraction of sums of power law jump distributions by stable distributions . we have again two different situations for the jump - width distribution but according to its second moment being finite or infinite . in other words we assume for the jump - width probability density @xmath195 ( assumed for simplicity to be symmetric : @xmath196 ) either @xmath197 or @xmath198 then we have the asymptotic relation , compare with @xcite , @xmath199 where @xmath200 the above asymptotic relations are known in the framework of the attraction properties of the stable densities . we note that the classical book by gnedenko and kolmogorov @xcite has unfortunately the wrong constant @xmath201 for @xmath202 . as before we dispense with the possible decoration of the relevant power law by a slowly varying function . by another respeeding , in fact an acceleration , we can pass over to space - time fractional diffusion processes . for this we have * three choices * : + * ( a ) : diffusion limit in space only , for general waiting time * , + * ( b ) : diffusion limit in space only , for ml waiting time * , + * ( c ) : joint limit in time and space ( with power laws in both ) with scaling relation. note hat ( b ) is just a special case of ( a ) but of particular relevance ( as we shall see ) . in all three cases we rescale the jump density by a factor @xmath203 , replacing the random jumps @xmath204 by @xmath205 . this means changing the unit of measurement in space from @xmath3 to @xmath21 , with @xmath206 , so bringing into near sight the far - away space . we get the rescaled jump density as @xmath207 , corresponding to @xmath208 . . + starting from the eq . ( 3.6 ) , the laplace - fourier representation of the ctrw equation , without special assumption on the waiting time density , we fix the fourier variable @xmath209 and accelerate the spatially rescaled process by the respeeding factor @xmath210 , arriving at the equation ( using @xmath211 as new dependent variable ) @xmath212 = \frac { \widehat w(h \kappa ) -1}{\mu h^\alpha } \ , \widehat{\widetilde q}_h(\kappa , s)\ , . \eqno(6.5)\ ] ] then , _ fixing @xmath213 _ as required by the continuity theorem of probability theory for fourier transforms , and _ sending @xmath20 to zero _ we get , noting that @xmath214/(\mu h^\alpha ) \to -\|\kappa\|^\alpha$ ] , and writing @xmath215 in place of @xmath216 , @xmath217 = - \|\kappa \|^\alpha \,\widehat{\widetilde u}(\kappa , s)\ , , \eqno(6.6)\ ] ] where we still have , consistently with ( 3.5 ) , @xmath218 being @xmath219 the original waiting time density . in physical space - time we have the integro - pseudo - differential equation @xmath220 with @xmath221 as the symbol of the riesz pseudo - differential operator @xmath222 usually referred to as the riesz fractional derivative of order @xmath223 , see appendix b. : by this rescaling and acceleration the jumps become smaller and smaller , their number in a given span of time larger and larger , the waiting times between jumps smaller and smaller . in the limit there are no waiting times anymore , the original waiting time density @xmath219 is now only spiritual , but still determines via @xmath116 the memory of the process . ( 6.7 ) offers a great variety of diffusion processes with memory depending on the choice of the function @xmath116 . . + we now choose in eq . ( 6.7 ) the mittag - leffler memory function ( 3.17 ) , namely @xmath224 corresponding to the mittag - leffler waiting time law @xmath225 consistently with the time fractional kolmogorov - feller equation ( 3.15 ) , that includes for @xmath80 the classical kolmogorov - feller equation ( 3.11 ) . as a consequence of our spatial diffusion limit , compare with @xcite , we so arrive immediately at the _ space - time fractional diffusion equation _ @xmath226 . + assuming the behaviour for the waiting time density as in eqs . ( 5.1)-(5.2 ) , and for the jump - width density as in eqs . ( 6.1)-(6.2 ) , rescaling as described the waiting times and the jumps by factors @xmath8 and @xmath20 , starting from ( 4.4 ) , decelerating by a factor @xmath227 in time , then accelerating for space by a factor @xmath228 , we obtain ( compare to section 4 , case ( b ) ) , fixing @xmath76 and @xmath209 and setting , for convenience @xmath229 @xmath230 = a(\tau , h)\ , \left [ \widehat w_h(\kappa ) - 1\right]\ , \widehat{\widetilde { p}}_{\tau , a(\tau , h ) } ( \kappa , s)\,,\eqno(6.10)\ ] ] with @xmath231 and @xmath232 fixing @xmath233 to the constant value 1 , which means introducing the relationship of _ well - scaledness _ @xmath234 between the rescaling of time and space , we get @xmath235 because of @xmath236 we finally get the limiting equation @xmath237 = - \|\kappa \|^\alpha \ , \widehat{\widetilde u}(\kappa , s)\,,\eqno(6.15)\ ] ] corresponding to eq . ( 6.8 ) , the space - time fractional diffusion equation . ( @xmath223 ) * the mittag - leffler waiting time ( choice ( b ) ) , obeying the power law asymptotics ( 5.2 ) with @xmath238 leads from ( 6.7 ) directly to the space - time fractional diffusion equation ( 6.8 ) , without requirement of rescaling and deceleration in time , and with these procedures we arrive likewise at ( 6.8 ) . this strange fact is caused by the invariance of the mittag - leffler density to the combined effects of rescaling by @xmath8 and deceleration by @xmath239 , expressed in eq . ( 5.9 ) . * ( @xmath136 ) * going again through our preceding deductions , we observe that the combined ( well - scaled ) passage of @xmath152 and @xmath20 , under the relation ( 6.12 ) , towards zero can be split in two distinct ways into two separate passages . : keep @xmath20 fixed letting @xmath152 tend to zero , then in the resulting model send also @xmath20 to zero . : interchange the order played by @xmath20 and @xmath152 in the first way . under our power law assumptions we can transform ( 3.7 ) , the basic integral equation of ctrw , into eq . ( 3.15 ) ( time fractional ctrw ) by rescaling - respeeding manipulation only in the time variable , and then by rescaling in space followed by an acceleration into ( 6.8 ) , the space - time fractional diffusion equation . or we can transform ( 3.7 ) by rescaling in space followed by an acceleration into eq . ( 6.7 ) ( general space fractional diffusion with memory ) , and then by by rescaling - respeeding in the time variable arrive at ( 6.8 ) . * ( @xmath240 ) * where have the waiting times gone in the space - time fractional diffusion equation ( 6.8 ) ? we can answer this question by interpreting eq . ( 6.10 ) under the scaling relation ( 6.12 ) as the laplace - fourier representation @xmath241 = \left [ \widehat w_h(\kappa ) - 1\right]\ , \widehat{\widetilde { p}}_{\tau , 1 } ( \kappa , s)\,,\eqno(6.16)\ ] ] of our original ctrw ( 3.1 ) , whose laplace - fourier representation ( 3.6 ) coincides with ( 6.16 ) if there we delete all decorations with indices . ( 6.16 ) represents the same physical process as ( 3.1 ) but expressed in terms of new units @xmath242 and @xmath21 of time and space , respectively . however , the respeeding factor @xmath233 being fixed to 1 , there is no change of physical speed . when these new units are made smaller and smaller , moderate spans of time and space become numerically smaller and smaller , shrinking towards zero as @xmath152and @xmath20 tend to zero , and likewise the waiting times and the jump widths shrink to zero . the distant future and the far - away space come numerically into near sight . as long as @xmath152 and @xmath20 are positive , we always have the same physical process , only measured in other units . the finally resulting space - time fractional diffusion process ( 6.8 ) remembers the power laws for waiting times and jumps in form of the orders @xmath136 and @xmath223 of fractional differentiation . * ( @xmath243 ) * an objection could be raised against the somewhat mystical actions of respeeding . namely , if the respeeding factor @xmath9 in eq . ( 4.4 ) differs from 1 , the underlying renewal process and consequently the whole ctrw are distorted . however , for the ctrw we carry out the actions of deceleration and acceleration in either order in succession or simultaneously in combination , and by our special choice of these factors they cancel each other in effect , so that there remains no physical distortion . this is particularly obvious in our choice ( c ) , see the above comment ( @xmath240 ) . * ( @xmath244 ) * let us finally point out an advantage of splitting the passages @xmath10 and @xmath22 . whereas by the combined passage as in choice ( c ) , if done in the well - scaled way ( 6.12 ) , the mystical concept of respeeding can be avoided , there arises the question of correct use of the continuity theorems of probability . there is one continuity theorem for the laplace transform , one for the fourier transform , see @xcite . possible doubts whether their simultaneous use is legitimate vanish by applying them in succession , as in our two splitting methods . in our investigations we have met four types of spatially one - dimensional stochastic processes for the sojourn probability density @xmath102 or @xmath245 . for the reader s convenience let us give a list of these processes in physical coordinates , referring to the preceding text for details , and remind briefly how they can be connected by appropriate scaling and passages to the limit . let us note that in all these processes the initial condition @xmath246 for @xmath247 or @xmath248 can be replaced by a more general probability density function @xmath147 . the integral equation for the ctrw is , see ( 3.1 ) with ( 3.2 ) , @xmath249\,dt'\ ] ] is equivalent , by the introduction of the memory function @xmath116 , see ( 3.5 ) , to the generalized kolmogorov - feller equation , see ( 3.7 ) , @xmath250 the time fractional kolmogorov - feller equation , see ( 3.15 ) , @xmath251 the integro - pseudo - differential equation of space fractional diffusion with general memory , see ( 6.7 ) , @xmath252 the space - time fractional diffusion equation , see ( 6.8 ) , @xmath253 we now sketch shortly how these four evolution equations are connected in our theory . ( i ) goes over in eq . ( ii ) , likewise eq . ( iii ) in eq . ( iv ) by the special choice @xmath254 for the memory function , see ( 3.17 ) . under our power law assumption for the waiting time , see ( 5.1 ) and ( 5.2 ) , these transitions can be achieved asymptotically by manipulation via rescaling and respeeding of the underlying renewal process . under our power law assumption for the jumps , see ( 6.1)and ( 6.2 ) , the transition from eq . ( i ) to eq . ( iii ) and from ( ii ) to ( iv ) can be achieved asymptotically by passage to the diffusion limit only in space . under our power law assumption for time space there is a direct way from eq . ( i ) to eq . ( iv ) , namely _ the well - scaled passage to the diffusion limit _ , for which the condition ( 6.12 ) is relevant . it is instructive to study the spatial transition to the diffusion limit for the _ mittag - leffler renewal process_. as said in section 3 this renewal process , viewed as a ctrw by treating its counting number @xmath255 as a spatial variable @xmath103 , is obtained by choosing @xmath256 as the jump width density , see eq . its waiting time density is , see ( 1.1 ) , ( 3.13 ) , @xmath257 we have @xmath258 , @xmath259 , hence @xmath260 = \lt ( \e^{i\kappa}-1\rt)\ , \widehat{\widetilde { p } } ( \kappa , s ) \ , . \eqno(7.1)\ ] ] rescaling in space by a factor @xmath20 and accelerating ( because of @xmath261 for @xmath262 ) this pure renewal process by the factor @xmath21 we get a process @xmath263 = \rec{h } \lt ( \e^{i h \kappa}-1\rt)\ , \widehat{\widetilde { q}_h } ( \kappa , s ) \,,\ ] ] which as @xmath264 and @xmath209 fixed gives @xmath265 = i\kappa \ , \widehat{\widetilde { u } } ( \kappa , s ) \ , , \eqno(7.2)\ ] ] which implies @xmath266 we note that eq . ( 7.2 ) corresponds to the time fractional drift equation @xmath267 by using the known scaling rules for the fourier and laplace transforms , @xmath268 we infer directly from ( 7 . 3 ) ( thus without inverting the two transforms ) the following _ scaling property _ of the ( fundamental ) solution @xmath269 consequently , introducing the _ similarity variable _ @xmath270 we can write @xmath271 where @xmath272 to determine the solution in the space - time domain we can follow two alternative strategies related to the different order in carrying out the inversion of the fourier - laplace transforms in ( 7 . 3 ) . indeed we can + ( s1 ) : invert the fourier transform getting @xmath273 and then invert this laplace transform , + ( s2 ) : invert the laplace transform getting @xmath274 and then invert this fourier transform . recalling the fourier transform pair , @xmath275 where @xmath276 denotes the unit step heaviside function , we get @xmath277 in view of the fact that @xmath278 is the laplace transform of the extremal unilateral stable density of order @xmath136 , @xmath279 ( see for notation appendix b ) , we recognize that the solution in the space - time domain can be expressed in terms of a fractional integral ( see appendix a ) of such density , namely @xmath280\ , . \eqno(7.7)\ ] ] working in the laplace domain we can note that the fundamental solution of our fractional drift equation ( 7.4 ) is simply related to that of the time fractional diffusion - wave equation @xmath281 equipped with the initial conditions @xmath282 if @xmath283 and @xmath284 if @xmath285 . in fact , the solution of ( 7.8 ) turns out the half of the solution ( 7.7 ) of our time fractional drift equation ( 7.4 ) , extended in a symmetric way to all of @xmath286 , as can be seen by factorizing eq . ( 7.8 ) as @xmath287 indeed eq . ( 7.8 ) was solved by using the laplace transform strategy by mainardi in the 1990 s , see e.g. @xcite where the reader can find mathematical details of the proof and instructive plots of the fundamental solution . then , based on mainardi s analysis , we can state that the required solution of eq . ( 7.4 ) reads @xmath288 where @xmath289 denotes the function of wright type defined in the complex plane @xmath290 } \!=\ ! \rec{\pi}\ , \sum_{n=1}^{\infty}\,{(-z)^{n-1 } \over ( n-1)!}\ , \gamma(\beta n ) \,\sin ( \pi \beta n ) . \eqno(7.10)\ ] ] the @xmath289 function is a special case of the wright function defined by the series representation , valid in the whole complex plane , @xmath291 indeed , we recognize @xmath292 originally , wright introduced and investigated this function with the restriction @xmath293 in a series of notes starting from 1933 in the framework of the asymptotic theory of partitions . only later , in 1940 , he considered the case @xmath294 . we note that in the handbook of the bateman project @xcite ( see vol . 18 ) , presumably for a misprint , @xmath69 is restricted to be non negative . for further mathematical details on the @xmath289-wright function we recommend @xcite . for our time fractional drift equation ( 7.4 ) we note the particular case @xmath295 for which we obtain @xmath296\ , , \q x\ge 0 \,,\q t\ge 0\ , . \eqno(7.13)\ ] ] in the limiting case @xmath80 we recover the rightward pure drift , @xmath297 in view of the fact that that @xmath289-wright function of order @xmath136 is related to the extremal unilateral stable density of order @xmath136 , see @xcite , we conclude by displaying the alternative form of the solution of the time fractional drift equation : @xmath298 which , compared with ( 7.7 ) , shows the effect of the fractional integral on the stable density function @xmath299 . recalling the laplace transform pair , see e.g. @xcite , @xmath300 we get @xmath301 from which @xmath302 where @xmath303 denotes the cauchy principal value . because , see @xcite , vol . 3 , chapter xviii on miscellaneous functions , section 18.1 eq . ( 7 ) , @xmath304 we see that @xmath305 does not tend to zero fast enough for the integral ( 7.17 ) to exist as a regular improper riemann integral . but there should be no problem for existence as a cauchy principal value integral . it can be shown that the present strategy based on fourier integral ( 7.17 ) provides the result ( 7.9 ) . : not wanting to overload our paper we have deliberately avoided the concept of _ subordination _ in fractional diffusion . but , referring to @xcite , let us say that if in ( 7.15 ) we replace @xmath103 by @xmath306 we get the _ subordinator _ , i.e. the probability law for generating the operational time @xmath306 from the physical time @xmath107 , see eq . ( 5.20 ) in @xcite , and , in other notation , @xcite . because of its relation ( 7.16 ) via fourier transform to the mittag - leffler function with imaginary argument , the probability law governing the process ( 7.15 ) sometimes is called the _ mittag - leffler distribution _ , see e.g. @xcite . although so named it must not be confused with our _ mittag - leffler waiting time distribution _ whose density is given by ( 1.1 ) . the basic role of the mittag - leffler waiting time probability density in time fractional continuous time random walk ( ctrw ) has become well known by the fundamental paper of 1995 by hilfer and anton @xcite . earlier in the theory of thinning ( rarefaction ) of a renewal process under power law assumptions , see the 1968 book by gnedenko and kovalenko @xcite , this density had been found as limit density by a combination of thinning followed by rescaling of time and imposing a proper relation between the rescaling factor and the thinning parameter . likewise one arrives at this law when wanting to construct a certain special class of anomalous random walks , see the 1985 paper by balakrishnan @xcite , the anomaly defined by growth of the second moment of the sojourn probability density like a power of time with exponent between 0 and 1 . balakrishnan s paper , having appeared a few years before the fundamental paper of 1989 by schneider and wyss @xcite , is difficult to read as it is written in a style different from the present one , so we will here not go into details . but let it be said that by well - scaled passage to the limit from ctrw ( again under suitable power law assumptions in space and time ) he obtained the space - time fractional diffusion equation in form of an equivalent integro - differential equation . unfortunately , balakrishnan s paper did not find the attention it would have deserved . however , due to the sad fact that the mittag - leffler function too long played a rather neglected role in treatises on special functions balakrishnan as well as gnedenko and kovalenko contented themselves with presenting their results only in the laplace transform domain ; they did not identify their limit density as a mittag - leffler type function . having worked ourselves for some time on questions of well - scaled passage to the diffusion limit from continuous time random walks to fractional diffusion , see @xcite , we got from the theory of thinning the idea that it should be possible to carry out the passages to the limit separately in space and in time . in time this can be done by a combination of re - scaling time and respeeding the underlying renewal process ( formally treating it as a ctrw with unit steps in space ) . in fact , _ thinning _ in the sense of gnedenko and kovalenko transforms the original renewal process into one that is running more slowly and this effect can be balanced by proper choice of the rescaling factor . the result of our combination of rescaling and respeeding for a ctrw governed by a given renewal process with a generic power law waiting time law is a time fractional ctrw . by another rescaling in space ( now under power law assumption for the jumps ) which can be interpreted as a second respeeding we arrive at the already classical space - time fractional diffusion equation . in this way we shed new light on the long time and wide space behaviour of continuous time random walks . in a series of comments at the end of section 6 , we have explained how , by what we call _ well - scaled _ passage to the diffusion limit , the transition from the ctrw to the space - time fractional diffusion process actually can be obtained by merely rescaling time and space without any respeeding at all . however , the separate passages to the limit are more satisfying with respect to mathematical rigour . finally , in section 7 , we have treated the time fractional drift process as a properly scaled limit of the counting function of a pure renewal process governed by a waiting time law of mittag - leffler type . our trick in finding the limiting waiting time law of this renewal process consists in treating it as a ctrw with positive jumps of size @xmath3 so that its counting number acts as a spatial variable . then , by suitably rescaling this spatial variable , we obtain as an interesting side result the long time behaviour of the mittag - leffler renewal process . for a sufficiently well - behaved function @xmath45 ( @xmath307 ) we define the _ caputo time fractional derivative _ of order @xmath308 with @xmath71 through @xmath309 so that @xmath310 such operator has been referred to as the _ caputo _ fractional derivative since it was introduced by caputo in the late 1960 s for modelling the energy dissipation in the rheology of the earth , see @xcite . soon later this derivative was adopted by caputo and mainardi in the framework of the linear theory of viscoelasticity , see @xcite . the reader should observe that the _ caputo _ fractional derivative differs from the usual _ riemann - liouville _ ( r - l ) fractional derivative @xmath311 \ , , \q 0<\beta<1 \ , . \eqno(a.3)\ ] ] both derivatives are related to the riemann liouville ( r - l ) fractional integral that is defined for any order @xmath312 as @xmath313 so that @xmath314 . incidentally @xmath315 for @xmath316 . then , in virtue of eqs ( a.2)-(a.4 ) , the two fractional derivatives read : @xmath317 @xmath318 in particular , the r - l derivative of order @xmath136 is the left inverse of the corresponding r - l fractional integral in that @xmath319 . we note the relationships between the two fractional derivatives ( when both of them exist ) , for @xmath320 , @xmath321 \ , = \ , _ td^\beta \ , f(t ) - \frac{t^{-\beta } } { \gamma(1-\beta)}\,f(0^+ ) \ , . % % \q 0<\beta<1\ , . \eqno(a.7)\ ] ] as a consequence we can interpret the caputo derivative as a sort of regularization of the r - l derivative as soon as @xmath322 is finite ; in this sense such fractional derivative was independently introduced in 1968 by dzherbashyan and nersesian @xcite , as pointed out by kochubei , see @xcite . in this respect the regularized fractional derivative is sometimes referred to as the _ caputo - dzherbashyan derivative_. we observe the different behaviour of the two fractional derivatives ( a.2 ) , ( a.3 ) at the end points of the parameter interval @xmath323 , as it can be noted from their definitions in operational terms ( a.5 ) , ( a.6 ) . in fact , whereas for @xmath324 both derivatives reduce to @xmath325 , due to the fact that the operator @xmath326 commutes with @xmath325 , for @xmath327 we have @xmath328 the above behaviours have induced us to keep for the riemann - liouville derivative the same symbolic notation as for the standard derivative of integer order , while for the caputo derivative to decorate the corresponding symbol with subscript @xmath329 . for the r - l derivative the laplace transform reads for @xmath320 @xmath330 thus the rule ( a.9 ) is more cumbersome to be used than ( a.1 ) since it requires the initial value of an extra function @xmath331 related to the given @xmath45 through a fractional integral . however , when @xmath332 is finite we recognize @xmath333 . in the limit @xmath324 both derivatives reduce to the derivative of the first order so we recover the corresponding standard formula for the laplace transform : @xmath334 we conclude this appendix noting that in a proper way both derivatives can be generalized for any order @xmath335 , see @xcite . let us first recall that a generic linear pseudo - differential operator @xmath336 , acting with respect to the variable @xmath337 is defined through its fourier representation , namely @xmath338 where @xmath339 is referred to as the symbol of @xmath336 , formally given as @xmath340 the fractional _ riesz _ derivative @xmath341 is defined as the pseudo - differential operator with symbol @xmath342 this means that for a sufficiently well - behaved ( generalized ) function @xmath147 ( @xmath343 ) we have @xmath344 the symbol of the riesz fractional derivative is nothing but the logarithm of the characteristic function of the generic symmetric _ ( in the lvy sense ) probability density , see @xcite . noting @xmath345 we recognize that @xmath346 in other words , the riesz derivative is a symmetric fractional generalization of the second derivative to orders less than 2 . in an explicit way the riesz derivative reads , for @xmath202 , @xmath347 where in the l.h.s we have also adopted the alternative and illuminating notation introduced by zaslavsky , see @xcite . this operator is referred to as the _ riesz fractional derivative _ since it is obtained from the inversion of the fractional integral originally introduced by marcel riesz in the late 1940 s , known as the _ riesz potential _ , see @xcite . it is based on a suitable regularization of a hyper - singular integral , according to a method formerly introduced by marchaud in 1927 . : straightforward generalization to the _ riesz - feller derivative _ of order @xmath223 and skewness @xmath348 is possible . such pseudo - differential operator is denoted by us as @xmath349 in this case we have @xmath350 in an explicit way the riesz - feller derivative reads , for @xmath202 , @xmath351 \ , \int_0^\infty { f(x+\xi)- f(x ) \over { \xi}^{1+\alpha}}\ , d \xi \right.}\\ & + \,{\ds \left . \sin \,[(\alpha-\theta ) \pi/2 ] \ , \int_0^\infty { f(x-\xi)- f(x ) \over { \xi}^{1+\alpha}}\ , d \xi \right\}}\ , . \end{array } \eqno ( b.8)\ ] ] notice in ( b.7 ) that @xmath352 $ ] . thus the symbol of the riesz - feller fractional derivative is the logarithm of the characteristic function of the more general ( strictly ) _ stable _ probability density , closely following the feller parameterization , see @xcite revisited by the present authors in @xcite . according to our notation , the strictly stable density of order @xmath223 and skewness @xmath348 is denoted by @xmath353 . we note that the allowed region for the parameters @xmath354 and @xmath348 turns out to be a diamond in the plane @xmath355 with vertices in the points @xmath356 , @xmath357 , @xmath358 , @xmath359 , that we call the _ feller - takayasu diamond _ , see fig . 1 . for more details we refer the reader to @xcite , where series representations and numerical plots of the stable densities @xmath353 are found . in particular , we recall that the extremal stable densities obtained for @xmath360 with @xmath361 are unilateral , with support in @xmath362 , respectively . -0.1truecm the mittag - leffler function with parameter @xmath136 is defined as @xmath363 it is an entire function of order @xmath1 and reduces for @xmath80 to @xmath364 for detailed information on the functions of mittag - leffler type the reader may consult and references therein . hereafter , we find it convenient to summarize the most relevant features of the functions @xmath365 @xmath366 that turn out to be the most relevant functions of mittag - leffler type for our purposes . both of them reduce to the exponential function @xmath367 in the limit as @xmath368 . we begin to quote their expansions in power series of @xmath369 ( convergent for @xmath307 ) and their asymptotic representations for @xmath370 , @xmath185 ( t ) = _ n=0^ ( -1)^n ~ , ( c.4 ) @xmath371 ( t ) = _ n=0^ ( -1)^n ~ . ( c.5 ) @xmath185 the laplace transforms of @xmath372 and @xmath175 can easily be obtained by transforming the series ( c.4 ) , ( c.5 ) term by term , respectively : they read @xmath373 for @xmath71 both functions @xmath372 , @xmath175 keep the complete monotonicity of the limiting exponential function of @xmath374 complete monotonicity of a function @xmath45 means , for @xmath375 , and @xmath307 , @xmath376 , or equivalently , its representability as ( real ) laplace transform of a non - negative function or measure , see @xcite . recalling the theory of the mittag - leffler functions of order less than 1 , we obtain for @xmath71 the following representations , see @xcite , @xmath377 @xmath378 in figs 2 and 3 we exhibit plots of the functions @xmath372 and @xmath175 , respectively in logarithmic and linear scales . e. barkai and r.j . silbey , fractional kramers equation , _ j. phys b _ * 104 * ( 2000 ) , 3866 - 3874 . e. barkai and i.m . sokolov on hilfer s objection to the fractional time diffusion equation , _ physica a _ * 373 * ( 2007 ) , 231 - 236 . , _ renewal theory _ , 2-nd edn . , methuen , london ( 1967 ) . , _ integral transforms and representations of functions in the complex plane_. moscow , nauka ( 1966 ) . in russian . 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a gamma - ray burst ( grb ) event comprises two phases , prompt emission and afterglow . the prompt @xmath1-ray emission is usually highly variable , with many pulses overlapping within a short duration ( fishman & meegan 1995 ) . the power density spectra ( pdss ) of the light curves are typically a power law with a possible turnover at high frequencies ( beloborodov et al . 2000 ) . the light curves may be decomposed as the superposition of an underlying slow component and a more rapid fast component ( gao et al . the fast component tends to be more significant in high energies , and becomes less significant at lower frequencies ( vetere et al . 2006 ) . it has been shown that the external shock model has difficulty producing grb variability while maintaining a high radiative efficiency ( sari & piran 1997 ; cf . dermer & mitman 1999 ) . the detection of the steep decay phase following grb prompt emission ( tagliaferri et al . 2005 ) suggests that the prompt emission region is detached from the afterglow emission region ( zhang et al . this nails down the internal origin of grb prompt emission for the majority of grbs . for an internal origin of grb prompt emission , the variability is usually attributed to the erratic activity of the central engine ( e.g. , rees & mszros 1994 ; kobayashi et al . 1997 ) . it is envisaged that the ejecta launched from the central engine is composed of multiple shells with variable bulk lorentz factors . faster late shells catch up and collide with slower early shells . part of the kinetic energy of the ejecta is converted to energy of non - thermal particles in these internal shocks , a fraction of which is released as the observed non - thermal radiation . in this model , different variability timescales are related to the angular spreading time of colliding shells at different internal shock radii . in order to account for superposed slow and fast variability components , one has to assume that the central engine itself carries these two variability components in the time history of jet launching ( hascot et al . 2012 ) , whose physical origin is unclear . the internal shock model also suffers a list of criticisms ( e.g. , zhang & yan 2011 for a review ) , including low radiation efficiency ( e.g. , kumar 1999 ; panaitescu et al . 1999 ) , fast cooling ( ghisellini et al . 2000 ; kumar & mcmahon 2008 ) , spectrum . however , a requirement is that the emission region has to be large where the magnetic field is weak . this corresponds to an unconventional internal shock radius , but is consistent with the icmart model . ] , particle number excess ( daigne & mochkovitch 1998 ; shen & zhang 2009 ) , inconsistency with some empirical relations ( amati et al . 2002 ; zhang & mszros 2002 ; liang et al . 2010 ) , and overpredicting the brightness of the photosphere emission component ( daigne & mochkovitch 2002 ; zhang & peer 2009 ) . alternatively , the grb variability can be interpreted as locally doppler - boosted emission in a relativistic bulk flow , such as relativistic mini - jets ( lyutikov & blandford 2003 ; yamazaki et al . 2004 ) or relativistic turbulence ( narayan & kumar 2009 ; kumar & narayan 2009 ; lazar et al . 2009 ) in a bulk relativistic ejecta . some criticisms have been raised to these models . for example , relativistic turbulence damps quickly so that the emission from the turbulence can not be sustained ( zrake & macfadyen 2012 ) . the simulated light curves are composed of well - separated sharp pulses without an underlying slow component ( narayan & kumar 2009 ; lazar et al . 2009 ) . also the pulse was calculated to have a symmetric shape for the turbulence model ( lazar et al . 2009 ) , which is in contradiction with the data . recently , zhang & yan ( 2011 , hereafter zy11 ) proposed an internal - collision - induced magnetic reconnection and turbulence ( icmart ) model to explain prompt emission of grbs . like the traditional internal shock scheme , the icmart model envisages internal interactions of shells within the ejecta wind . the main difference is that the ejecta is poynting flux dominated , with the magnetization parameter @xmath2 in the collision region , where @xmath3 and @xmath4 are poynting flux and matter flux , respectively . this was motivated by the non - detection of a bright photosphere thermal component in grb 080916c ( zhang & peer 2009 ) and most other large area telescope grbs ( zhang et al . 2011 ) . for a helical magnetic field structure , the initial collisions only serve to distort the magnetic field configurations . as multiple collisions proceed , the field configurations would be distorted to a critical point when a cascade of reconnection and turbulence occurs . charged particles can be accelerated in these reconnection regions , leading to intense gamma - ray radiation . within this model , a grb light curve is supposed to have two variability components : a broad ( slow ) component that tracks central engine activity , and an erratic ( fast ) component with multiple sharp pulses superposed on the slow component , which is related to numerous reconnection sites during the icmart event . in this paper , we simulate grb light curves and their corresponding pdss within the framework of the icmart model . in section 2 we describe the basic model and the simulation method . the simulation results are presented in section 3 . section 4 summarizes the findings with some discussion . we first summarize the basic ideas of the icmart model ( zy11 ) . magnetized shells with initial @xmath5 are envisaged to collide , leading to distortion of magnetic field lines until a threshold is reached and a runaway magnetic dissipation is triggered . during such an `` avalanche''-like reconnection / turbulence cascade , it is envisaged that fast reconnection seeds in the moderately high @xmath0 regime would inject moderately relativistic outflows in the emission regions ( zy11 ; lyubarsky 2005 ) , which would excite relativistic turbulence . the turbulence would facilitate more reconnection events , which trigger further turbulence . the magnetic energy is converted to particle energy and efficient radiation . during the growth of the reconnection / turbulence cascade , the number of reconnection sites as observed at any instant increases rapidly with time , so that multiple mini - emitters contribute simultaneously to the observed gamma - ray emission . rapid evolution of individual reconnection sites leads to rapid variability of the observed grb light curves . the cascade stops as @xmath0 drops around or below unity when most magnetic energy is converted into radiation or kinetic energy . during the growth of an icmart event , turbulence is not quickly damped due to the continuous injection of particle energy from the reconnection events , which continuously drives turbulence . with these preparations , we can model the light curve of a grb within the framework of the icmart model . lacking full numerical simulations of magnetic turbulence and reconnection , in this paper we perform a monte carlo simulation based on some simplest assumptions . we define each reconnection event as a fundamental mini - emitter , which carries a local lorentz boost with respect to the bulk of the emission outflow . each reconnection event can be modeled as a pulse , which can be bright and spiky if the mini - emitter beams toward the observer , but dim and broad if the mini - emitter beams away from the observer s direction . the observed light curve is the superposition of the emission from all these mini - emitters . for simplicity , we assume that the characteristic brightness ( peak luminosity ) of each reconnection event in the rest frame of the reconnection outflow is the same . we also take the shape of each pulse as a gaussian form for simplicity ( e.g. , narayan & kumar 2009 ; lazar et al . our goal is to try to simulate the superposed slow and fast components , and the precise shape of each pulse does not matter too much . in any case , we note that the shape of a spike within the icmart model is mainly defined by the time history of each reconnecting mini - jet rather than the time history of an ideal eddy , so the pulse profile may not necessarily be symmetric with peak time . this is different from the previous models ( narayan & kumar 2009 ; lazar et al . 2009 ) that invoke relativistic turbulence . more importantly , the shape of a broad pulse in the model is asymmetric : the rising portion is defined by the timescale of the reconnection - turbulence cascade process , while the decay portion is controlled by high - latitude emission after the icmart cascade ceases . there are three rest frames in this model : the first is the rest frame of the mini - jet , i.e. the outflow of the individual reconnection event . these mini - jets are moving with a relative lorentz factor @xmath1 with respect to the jet bulk . we denote parameters in this frame as ( @xmath6 ) . the second frame is the rest frame of the jet bulk , which moves with a lorentz factor @xmath7 with respect to the central engine . we denote parameters in this frame as ( @xmath8 ) . the third one is the rest frame of the observer ( with the cosmological expansion effect ignored ) . the quantities within these three frames are connected through two doppler factors , i.e. , @xmath9^{-1}\ ] ] and @xmath10^{-1},\ ] ] where @xmath11 and @xmath12 are the corresponding dimensionless velocities with respect to @xmath7 and @xmath1 , respectively , @xmath13 is the latitude of the mini - jet with respect to the line of sight ( i.e. the angle between the line of sight and the radial direction of the bulk ejecta at the location of the mini - jet ) , and @xmath14 is the angle between the mini - jet direction and radial direction of the ejecta bulk within the comoving frame of the ejecta bulk . each reconnection event is supposed to give rise to a single pulse in the grb light curve . since several reconnection events may occur simultaneously , some pulses can superpose with each other . for a naive sweet - parker reconnection , one has ( e.g. , see zweibel & yamada 2009 and references therein ) @xmath15 where @xmath16 is the inflow velocity of the reconnection layers , @xmath17 is the outflow velocity , and @xmath18 and @xmath19 are the width and length of the reconnection layer , respectively . reconnection physics demands @xmath20 , so that @xmath21 . on the other hand , what defines the duration of the reconnection event is the thickness of the bunch of magnetic field lines that continuously approach each other , and we assume that it is also of the order of @xmath19 . as a result , in the bulk comoving frame ( the @xmath8 frame ) , the duration of each pulse can be approximated as @xmath22 . in the observer frame , this is translated to @xmath23 , which corresponds to the duration of a certain pulse in the observer frame . for simplicity , we assume that the radiation intensity arising from each reconnection event has the same spectral form , i.e. , the band function ( see band et al . 1993 ) , in the comoving frame of the mini - jet ( the @xmath6 frame ) , @xmath24 the observed flux can be calculated as @xmath25 where @xmath26 is the distance of the grb to the observer . in a high-@xmath0 flow , @xmath17 can eventually reach a relativistic speed ( with lorentz factor @xmath1 ) , and @xmath27 can reach a maximum value of @xmath28 ( e.g. lyubarsky 2005 and references herein ) . therefore , @xmath29 . the lorentz factor of the mini - jet is related to @xmath0 and would drop to unity when @xmath0 drops below unity . the detailed dependence is related to the complicated physics of relativistic reconnection . in this paper , we adopt @xmath30 ( i.e. , @xmath1 is proportional to the relativistic alfvn lorentz factor ) . we also investigated other dependences between @xmath1 and @xmath31 . the general conclusions regarding how the simulated light - curve properties depend on various parameters are essentially similar . in the rest of the paper , we only focus on the @xmath30 assumption . in the simulations , we fix the band function parameters as the following : @xmath32 , @xmath33 , and the peak frequency @xmath34 is chosen such that @xmath35 kev is satisfied , where 300 kev is the typical observed value of grb spectral peak , and @xmath36 is the average value of the product of the two doppler factors . based on these assumptions , we calculate the received flux in the detector band of _ swift _ burst alert telescope ( bat ; i.e. , 15 - 150 kev ) . in our monte carlo simulation , four random parameters have been introduced . they are : ( 1 ) comoving length of the reconnection region @xmath37 , which is assumed to either have a typical value or have a power - law distribution with index @xmath38 below a typical value ; ( 2 ) the mini - jet direction ( angle @xmath14 with respect to the bulk motion direction ) in the bulk comoving frame , which is taken as isotropic or a gaussian distribution with respect to @xmath39 ( see more discussion below ) ; ( 3 ) the latitude of a mini - jet @xmath13 with respect to the viewing direction , which is random within the cone of the jet opening angle ; and ( 4 ) the epoch when a mini - jet occurs , which is taken to satisfy a distribution of exponential growth with time , i.e. @xmath40 . the total number @xmath41 of the mini - jets is a free parameter , which is defined by the requirement that they dissipate most magnetic energy in the local emission regions , so that the local @xmath0 is brought to below unity after each icmart event . of the ejecta can be still above unity , if the filling factor @xmath42 , since the majority of magnetic energy is still not dissipated . a small @xmath43 seems to be required by the central engine study of lei et al . ( 2013 ) , who obtained @xmath0 values greater than the measured typical lorentz factors of grbs ( liang et al . assuming that the magnetic energy density is roughly uniform within the emission region , this number can be simply written as the ratio between the total dissipated volume ( i.e. , total volume multiplied by the filling factor @xmath43 ) and the volume of the region affected by each reconnection event that powers a mini - jet . within the @xmath44 cone , this number is @xmath45 where @xmath46 is the radius of the emission region from the central engine . other input parameters include the radius of the emission region @xmath46 , the jet opening angle @xmath47 , the initial values of @xmath7 , and @xmath0 ( which defines the initial @xmath1 ) . for each reconnection event , we assume that half of the dissipated magnetic energy is released in the form of photons , while the other half is deposited to the jet bulk and used to boost the kinetic energy of the bulk . therefore , @xmath7 , @xmath0 , and @xmath1 are all functions of time during each icmart event . the exponential growth of magnetic dissipation eventually ends when the local @xmath0 drops around or below unity . without numerical simulations , it is unclear how abrupt the ending process is . in this paper we just assume an abrupt cessation of the cascade process , so that the number of new mini - jets drops to 0 after a particular time . the observed `` tail '' emission after this epoch is therefore contributed by the high - latitude emission from other mini - jets not along the line of sight due to the `` curvature effect '' delay . this delay timescale is calculated as @xmath48 with respect to the last emission along the line of sight , where @xmath49 . we calculate the contribution of all the mini - jets within @xmath50 . although most of the received emission comes from the mini - jets within the @xmath44 cone , those mini - jets outside the @xmath44 cone make some contribution to the high - latitude emission . we calculate the delay timescale of each mini - jet , apply its doppler factor to calculate the amplitude and shape of the pulse , and superpose these mini - jets to get the curvature tail of each icmart event . we run a series of monte carlo simulations to generate sample light curves . we first focus on the light curves for only one icmart episode . the light curve of one grb could be then modeled by superposing multiple icmart events . we first take the following nominal parameters : @xmath51 cm , @xmath52 cm , @xmath53 , @xmath54 , and @xmath55 . considering an exponential growth , i.e. , that each reconnection seed would eject a bipolar outflow and would stir up the ambient medium to trigger two reconnection events , one may estimate the generation number of successive reconnection events , @xmath56 , through the requirement @xmath57 . the timescale for each generation in the bulk comoving frame may be estimated as @xmath58 s , which corresponds to an observer frame timescale @xmath59 s. this is the typical `` @xmath60-folding '' timescale . the total duration ( rising timescale ) of an icmart event is therefore @xmath61 times larger , i.e.m @xmath62 s , which we adopt in the simulations . we also assume that the observer s line of sight is along the jet axis , and we take a redshift @xmath63 for simplicity . for a power - law distribution of @xmath19 , in principle , @xmath19 can extend to much smaller values . in our simulations , reconnection regions with @xmath64 cm are not considered , since the observed durations of these events already meet the detector s variability limit . in the following we test various factors that may affect the shape of the light curves . we first test how the simulated light curve depends on the unknown distribution of @xmath14 in the bulk comoving frame . we first assume an isotropic distribution and calculate the light curve . the result is shown in figure [ fig : direction](a ) . one can immediately see that the light curve has a broad component , with some spiky small pulses superposed on top . the broad component is due to the contributions of all the mini - jets beaming toward random directions in the bulk motion rest frame . the rising of the broad pulse corresponds to the exponential growth of the number of mini - jets , while the decay is controlled by the high - latitude effect . + + since an icmart event corresponds to an event of destroying the initial ordered magnetic field , the magnetic configurations in the icmart region , even near the end of the cascade , should not be completely random . the initial magnetic field configuration should be parallel to the ejecta plane ( e.g. , spruit et al . 2001 ; zhang & kobayashi 2005 ) . this is because the toroidal component falls with radius much slower than the poloidal component . such a configuration should still leave an imprint on the @xmath14 distribution . we consider a distribution of @xmath14 that has a gaussian distribution with respect to the original field line direction , i.e. , @xmath39 . in figures [ fig : direction](b ) and [ fig : direction](c ) we show the gaussian angle to be @xmath65 and @xmath66 , respectively . one can see that the simulated light curves have progressively less flux as the distribution angle becomes smaller . this is because with a smaller distribution angle , only rare mini - jets could beam toward the observer , which have a relatively lower flux ( than the larger gaussian angle distribution ) with respect to the majority of mini - jets that beam away from the observer and only contribute to the background . the overall shape of the light curves does not differ significantly . we next compare the effect of lorentz factor contrast in the icmart region . we keep the initial value of the bulk lorentz factor @xmath7 constant , i.e. , @xmath53 , and vary @xmath67 . this corresponds to different values of the initial magnetization @xmath68 . in figure [ fig : gamma - ratio ] , we compare three sets of simulations , with ( a ) @xmath69 ; ( b ) @xmath70 and ( c ) @xmath71 . other parameters are the same as those adopted to calculate figure [ fig : direction ] , and the gaussian @xmath14-distribution model with typical angle @xmath65 has been adopted . we show that the light curves become progressively more erratic and spikier when the @xmath67 becomes larger . this is because a larger @xmath67 would give rise to larger @xmath72 , and thus a larger value of the total doppler factor @xmath73 . a larger @xmath67 also tends to give a more significant evolution of the parameters ( figure [ fig : evolution ] ) . initially , a constant @xmath74 corresponds to a constant @xmath75 cone , so that observed numbers of mini - jets are the same in all these cases . however a larger @xmath67 can give rise to a larger @xmath7 near the end of evolution , thus a smaller @xmath44 cone . the slow component is not as significant , so that the light curves become spikier . + + in order to show the evolution of the physical parameters during the icmart cascade event , in figure [ fig : evolution ] we display the evolution of the bulk lorentz factor @xmath7 , the mini - jet lorentz factor @xmath1 , and the emission region magnetization @xmath0 as a function of time . it can be seen that evolution is more significant for a larger @xmath67 ( and equivalently a larger @xmath68 ) . + + next , we test how the total number of mini - jets @xmath41 within the @xmath44 cone affects the light curves . according to equation [ eq : n ] , varying @xmath41 is effectively varying the filling factor @xmath43 . by varying @xmath41 , the total number of e - folding steps @xmath61 is slightly modified , as is the rising time @xmath76 . in figure 4 , we compare the simulated light curves for different @xmath41 values , i.e. , @xmath77 , and @xmath78 , respectively . it can be seen that in general the light curves appear smoother with increasing @xmath41 . this can be readily understood : the larger the @xmath41 , the more reconnection events happen simultaneously , so that more mini - jets beaming to different directions tend to enhance the slow component . the short - timescale structures are smeared out , and the light curves become smoother . + next , we explore the effect of the emission region radius @xmath46 . figures [ fig : r](a ) , [ fig : r](b ) , and [ fig : r](c ) show the results for @xmath79 cm , @xmath80 cm , and @xmath81 , respectively . one can see that the larger the @xmath46 , the longer and stronger the high - latitude emission tail . this is because the length of the high - latitude tail is defined by @xmath82 . we notice again that the rising time is the growth time of the cascade , which is the @xmath60-folding time of consuming most of the magnetic energy in the emission region , which is defined by the total number @xmath41 of the mini - jets and the characteristic scale @xmath19 of each mini - jet . since the rising and falling times are related to different parameters , the pulse is usually asymmetric ( e.g. , figure [ fig : r ] ) . the simulated light curve is more consistent with data if the emission radius @xmath46 is large . zy11 suggested that icmart events should happen at larger radii , say , @xmath83 cm , in order to reach the critical condition of triggering a reconnection / turbulence cascade . it is intriguing to see that such large - radius icmart events make light curves more resemble the observed ones . + + we also discuss the effect of different sizes of reconnection regions . we make two sets of simulations . in the first set , we vary @xmath19 while keeping @xmath46 constant . we also keep @xmath84 , so effectively , we are varying the filling factor @xmath43 . since @xmath85 , the rising time @xmath86 is modified correspondingly . the results are presented in figure [ fig : size1 ] , which shows the simulated light curves for @xmath87 , @xmath88 , and @xmath89 cm , respectively . it can be seen that the smaller the @xmath19 , the spikier the light curve . this is because a smaller @xmath19 corresponds to a shorter duration of each reconnection event . for the @xmath90 cm case , short - timescale structures are missing , and the light curve is very smooth . + + next , we keep both @xmath41 and @xmath43 constant . by varying @xmath19 , we are effectively varying @xmath46 as well , so that the ratio @xmath91 is a constant . the results are shown in figure [ fig : size2 ] , in which light curves for @xmath87 , @xmath88 , and @xmath89 cm are simulated . the general trend as discussed above is still there , but since @xmath46 is changed accordingly , the contrasts are less significant , namely , the smaller @xmath19 cases are less spiky and larger @xmath19 cases are less smooth with respect to the case where @xmath46 is fixed ( figure [ fig : size1 ] . since the decay phase is defined by @xmath46 ( section [ sec : r ] above ) , varying @xmath46 with @xmath19 also affects the length of the decaying phase . + + we next test the effect of size distribution of the reconnection regions . we try two possibilities : the power - law distribution with an index @xmath38 ( the kolmogorov type ) ( figure [ fig : size - distribution](a ) ) and a uniform distribution ( figure [ fig : size - distribution](b ) ) . one can see that the uniform distribution has a smoother shape . in this case , the observed small pulse width distribution is solely determined by the distribution of the doppler factors . for the power - law distribution case , an extra factor ( the intrinsic distribution ) plays a role to make small pulses , so that the light curves are spikier . + finally we calculate the light curves for different energy bands . we consider three cases here , below the peak of the band spectrum ( figure [ fig : bands](a ) ) , i.e. , 15 - 150 kev ( also the observation band for _ swift _ bat ) , above the peak ( 500 - 650 kev , figure [ fig : bands](b ) ) , and across the entire energy band ( 15 - 650 kev , figure [ fig : bands](c ) ) . the high - energy light curve is slightly narrower and spikier , as observed in real grbs . in general , the overall shape of the light curves does not differ significantly . + + as suggested by zy11 , a real grb light curve may consist of multiple icmart events . in figure [ fig : grb ] , we simulate three emission episodes and superpose them together to make a mock grb light curve . we have varied @xmath74 and @xmath67 around the values @xmath92 and @xmath93 , respectively , with small fluctuations in different episodes . other parameters are the same as those adopted in figure [ fig : direction ] with a @xmath65 gaussian @xmath14-distribution . the simulated light curve shows reasonable features as observed in some grbs . we note that in reality the parameters of different icmart events could be more different , so that a variety of light curves could be made , which may account for the diverse prompt emission light curves as observed . in order to test whether our simulated light curves mimic the observed ones , we also perform a pds analysis of our results . in order to get robust pds slopes , for each set of parameters , we perform 10 different monte carlo simulations to get 10 different light curves , derive the pds slope of each light curve , and calculate the average slope to stand for this particular set of parameters . some examples of pdss are presented in figure [ fig : pds ] . generally , the pdss can be fit with a power law , with indices generally steeper than @xmath94 . the averaged pds indices for all the cases corresponding to figures 1 , 2 , and 4 - 9 are collected in table 1 . observationally the pds slopes are steeper in softer bands ( e.g. _ swift _ ; guidorzi et al . 2012 ) than harder bands ( e.g. batse ; beloborodov et al . our simulations recover this trend . the presented pds values are taken from the _ swift _ band . it is encouraging to see that the simulated values are generally consistent with the _ swift _ data ( guidorzi et al . our simulations also show a turnover of pdss in the high - frequency regime with a steeper index . such a feature is seen in some grbs . + .pds slopes of simulated light curves [ cols="^,^",options="header " , ] from table 1 , one can see that various parameters can affect the slope of a pds . generally speaking , spikier light curves have more power in high frequencies and therefore have a shallower pds slope . most pds indices listed in table 1 can be understood this way . for figure [ fig : direction ] , it is seen that more isotropic distributions give steeper slopes . this is because the more isotropic cases give more mini - jets contributing to the broad component , and thus enhance the low - frequency power . similarly , as shown in figure [ fig : n ] , a smaller number @xmath41 gives richer spiky features , and therefore gives a shallower pds slope . the @xmath46-dependence ( figure [ fig : r ] ) can be understood as the following : a larger @xmath46 corresponds to a longer curvature decay tail , on top of which rapid variability can be observed , so that the pds slope is shallower . for the size effect ( figure [ fig : size1 ] ) , a smaller @xmath19 can give rise to pulses with shorter duration and hence , a more dominant high - frequency power and shallower pds ( figure [ fig : size1 ] ) . when both @xmath46 and @xmath19 co - vary , this effect is still relevant , but somewhat compensated by the @xmath46 effect ( figure [ fig : size2 ] ) . next , without a size distribution , the pds is steep ( figure [ fig : size - distribution](b ) ) . by introducing a size distribution , one has more contributions to short - time variability from smaller sizes , so the pds becomes shallower . finally , the light curves in a higher energy band are somewhat spikier ( figure [ fig : bands ] ) and hence have a shallower pds . this is consistent with the finding of guidorzi et al . ( 2012 ) and beloborodov et al . ( 2000 ) : using the _ swift _ bat data , guidorzi et al . ( 2012 ) obtained a steeper pds slope than beloborodov et al . ( 2000 ) , who used the batse data ( higher energy band ) to perform the analysis . it is interesting to investigate the change of pds slope due to the change of the initial lorentz factor contrast . as shown in figure [ fig : evolution ] , in principle one can have strong parameter evolution during one icmart event , which causes complicated evolution of the pds behavior . to avoid such strong evolution , we first fix @xmath53 , and vary @xmath67 so that the ratio @xmath95 evolves in the range of @xmath96 . in figure [ fig : pds1 ] , we present the pds slope as a function of @xmath95 . the triangles ( and dotted line ) are calculated by turning off parameter evolution ( i.e. , keeping @xmath1 and @xmath7 unchanged throughout ) , and the squares ( and solid line ) are calculated by turning on the parameter evolution ( figure [ fig : evolution ] ) . one can see that the pds slope becomes progressively shallower as @xmath97 increases . this is understandable , since a larger @xmath67 corresponds to a stronger fast emission component , and therefore the light curves are spikier ( see figure [ fig : gamma - ratio ] ) . one can tentatively draw the conclusion that a more magnetized outflow tends to make spikier light curves . since the final lorentz factor of the ejecta at the deceleration time is proportional to @xmath98 , and since observationally the lorentz factor at the onset of afterglow does not have a wide distribution ( e.g. , liang et al . 2010 ) , it is interesting to investigate how the pds slope depends on the lorentz factor contrast when @xmath98 is set to constant . in figure [ fig : pds2 ] , we present the case of @xmath99 for cases both without and with parameter evolution . the range of the contrast is set to @xmath100 ( i.e. @xmath101 , @xmath102 ) to @xmath103 ( i.e. , @xmath104 ) . the convention is the same as figure 10 . the dependence shows more complicated patterns . for the case without evolution ( triangles and dotted line ) , in general one can see decrease of pds slope when @xmath95 increases ( except the slight tilt at very large @xmath95 ) . this can be understood in the following way . as @xmath95 increases , one has two competing effects . the increase of @xmath67 tends to enhance the small timescale variability . on the other hand , the decrease of @xmath7 tends to enlarge the @xmath44 cone , so that many more mini - jets not beaming toward the observer could contribute to the slow component . the net result after competition is that the latter effect wins , so that the long - time variability is more enhanced , and hence , a steeper pds is obtained . this trend is overturned when @xmath67 exceeds @xmath74 near the end of the curve . when evolution is taken into account ( squares and solid line ) , the situation is even more complicated . when @xmath67 is small enough , the above - mentioned trend is retained . however , when @xmath67 becomes large enough , evolution of @xmath1 and @xmath7 becomes significant ( figure [ fig : evolution ] ) , so that quickly one can reach a regime with small @xmath1 and large @xmath7 . the average pds would be dominated by this late phase , so that the general trend is reversed from the no - evolution case . in reality , since a real grb light curve would usually be the superposition of multiple icmart events , the clean evolution expected in a single icmart event would be smeared out . in this paper we have simulated a sample of grb prompt emission light curves and pdss within the framework of the icmart model ( zy11 ) . this model was developed to model grbs whose jet composition is still somewhat poynting flux dominated in the emission region . this was motivated by the non - detection of the photosphere component in some grbs ( zhang & peer 2009 ; zhang et al . since the emission region has a moderately high @xmath0 , in order to generate a reconnection / turbulence cascade envisaged by zy11 , the energy dissipation region must have many locally lorentz - boosted emission regions , or mini - jets . the detected emission would be the superposition of emissions from all these mini - jets , which beam to random directions in the bulk comoving frame . other global magnetic dissipation models for grb prompt emission have been proposed in the literature ( e.g. , lyutikov & blandford 2003 ; giannios & spruit 2006 ) . if these models invoke runaway generation of mini - jets at a relatively large emission radius , then the simulations in this paper also apply to those scenarios . lacking detailed numerical simulations for a reconnection / turbulence cascade , we carried out a monte carlo simulation by inputting many mini - jets with certain directional and temporal distributions within the icmart scenario . we investigated the roles of the directional distribution , lorentz factor contrast , number of reconnection regions , emission radius , size of the mini - jet , mini - jet size distribution , energy dependence , etc . , in defining the light curves and their pdss . we adopt our simulation parameters according to observations ( e.g. , typical length of reconnection region @xmath52 cm corresponding to observed variability timescale @xmath59 s , 15 - 150 kev band for simulated light curves corresponding to _ swift _ bat band , and so on ) , as well as the requirements of the icmart model itseft ( e.g. , emission region radius @xmath51 cm in order to make sure that runaway reconnection can happen , and exponential growth of the number of reconnection events with time ) . within the icmart framework , most of our parameters are physically related to each other self - consistently . even though some simplified assumptions are introduced so that the light curves may not fully represent the complex physics in an icmart event , our simulated light curves nonetheless show some encouraging features that are consistent with the grb prompt emission data . the most noticeable feature is the superposition of an underlying slow component and more erratic fast component , which seems to be consistent with the data ( gao et al . 2012 ; vetere et al . the slow component is caused by the superposition of emission from all the mini - jets in the emission region , while the fast component is related to those mini - jets that happen to beam toward the observer . we follow the physics of an icmart event , including the exponential growth of the reconnection region , dissipation of the magnetic field energy ( so that @xmath0 drops with time ) , and acceleration of the bulk ejecta during the energy dissipation process and find that the erratic grb light curves as observed can be generally reproduced within the model . among all the model parameters , the lorentz factor contrast and the number of mini - jets play an important role in defining the `` spikiness '' of the light curve . we also derived the pds slopes of the simulated light curves , and found that they are generally consistent with the data . generally speaking , the larger the contrast @xmath95 ( keeping @xmath74 constant ) , the shallower the pds slope . besides grbs , the `` jet - in - the - jet '' scenario has been discussed in other astrophysical contexts . giannios et al . ( 2010 ) interpreted the fast tev variability of active galactic nuleus jets using the mini - jet scenario . yuan et al . ( 2011 ) applied the scenario to account for the gamma - ray flares of the crab nebula . compared with earlier work of narayan & kumar ( 2009 ) , kumar & narayan ( 2009 ) , and lazar et al . ( 2009 ) , the new ingredient introduced in our paper is the exponential growth of the number of mini - jets as a function of time , as envisaged in the icmart model ( zy11 ; see also stern & svensson 1996 ) . as a result , our model allows many mini - jets emitting simultaneously at any instant . this is the key ingredient to define the broad component of each icmart event . a grb light curve is composed of multiple icmart events ( figure 8) , which are controlled by the erratic central engine activity . in order to set up the monte carlo simulations , we had to introduce a number of assumptions . these include power - law distribution of the size of reconnection regions , gaussian shape of each pulse , same intrinsic radiation spectrum for all emitters , exponential growth of numbers of pulses with time , isotropic or gaussian distribution of the mini - jet directions , and so on . some factors are still missing . for example , in the comoving frame of the jet bulk but outside the mini - jets , there would also be particles that give rise to radiation . the effects of this inter - mini - jet emission should be investigated ( e.g. , lin et al . 2013 ) . the physical conditions of real grbs must be more complex than what is modeled here , so that one may not reproduce the full observational features of grbs with the simulations presented in this paper . nonetheless , our simulations show the encouraging results that the simulated light curves based on these simplified assumptions can indeed reproduce some key features of the observations , e.g. the slow and fast variability components and a variety of degree of spikiness of the light curves . by changing parameters ( e.g. , @xmath14-distribution , lorentz factor contrast , jet opening angle ) , diverse light curves can be generated , ranging from relatively smooth to relatively spiky ones . the pdss of the simulated light curves are also generally consistent with the data . all these suggest that the icmart model may be a good candidate to interpret grb prompt emission . within the icmart theoretical framework , the following constraints can be made to the model parameters . ( 1 ) to reproduce the general fast - rising slower decay shape of broad pulses , the emission radius should be relatively large ( @xmath105 cm and beyond ) . ( 2 ) since many grbs show high - amplitude rapid variability , the grb initial magnetization parameter @xmath68 in the emission region could be high ( e.g. , from several to hundreds ) . ( 3 ) the observed minimum variability timescale constrains that @xmath19 can not be too large and has to be @xmath106 cm . ( 4 ) in order not to smear these peaks by overgenerating mini - jets , one also requires a filling factor @xmath42 , suggesting that in these cases the global @xmath0 of the outflow after the icmart event may not drop to unity . ( 5 ) erratic light curves with multiple episodes suggest that the grb central engine acts multiple times to eject highly magnetized shells so that multiple icmart events can be generated within one burst . ( 6 ) the existence of smooth - pulse grbs suggests that in some cases the @xmath68 is not much larger than unity ( so that @xmath67 is not much larger than unity ) , or there are so many mini - jets operating simultaneously . other information ( e.g. polarization properties and prompt emission efficiency ) is needed to break the degeneracy . we thank kohta murase , pawan kumar , zi - gao dai , he gao , da - bin lin , and chun li for useful discussion , and an anonymous referee for very helpful suggestions . this work is partially supported by nsf through grant ast-0908362 . bo zhang acknowledges a scholarship from china scholarship council for support . amati , l. , frontera , f. , tavani , m. , et al . 2002 , , 390 , 81 band , d. , matteson , j. , ford , l. , et al . 1993 , , 413 , 281 beloborodov , a. m. , stern , b. e. , & svensson , r. 2000 , , 535 , 158 cho , j. , lazarian , a. , & vishniac , e. t. 2003 , turbulence and magnetic fields in astrophysics , 614 , 56 daigne , f. , & mochkovitch , r. 1998 , , 296 , 275 daigne , f. , & mochkovitch , r. 2002 , , 336 , 1271 dermer , c. d. , & mitman , k. e. 1999 , , 513 , l5 drenkhahn , g. & spruit , h. c. 2002 , , 391 , 1141 fishman , g. j. , & meegan , c. a. 1995 , , 33 , 415 gao , h. , zhang , b .- b . , & zhang , b. 2012 , , 748 , 134 giannios , d. , & spruit , h. c. 2006 , , 450 , 887 giannios , d. , uzdensky , d. a. , & begelman , m. c. 2010 , , 402 , 1649 ghisellini , g. , celotti , a. , & lazzati , d. 2000 , , 313 , l1 goldreich , p. , & sridhar , s. 1995 , , 438 , 763 guidorzi , c. , margutti , r. , amati , l. , et al . 2012 , , 422 , 1785 hascot , r. , daigne , f. , & mochkovitch , r. 2012 , , 542 , l29 kobayashi , s. , piran , t. , & sari , r. 1997 , , 490 , 92 kumar , p. 1999 , , 523 , l113 kumar , p. , & mcmahon , e. 2008 , , 384 , 33 kumar , p. , & narayan , r. 2009 , , 395 , 472 lazar , a. , nakar , e. , & piran , t. 2009 , , 695 , l10 lazarian , a. , & vishniac , e. t. 1999 , , 517 , 700 lei , w .- h . , zhang , b. , & liang , e .- w . 2013 , , 765 , 125 liang , e .- w . , yi , s .- x . , zhang , j. , et al . 2010 , , 725 , 2209 lin , d .- b . , gu , w .- m . , hou , s .- j . , liu , t. sun , m .- y . , lu , j .- f . 2013 , , 776 , 41 lyubarsky , y. e. 2005 , , 358 , 113 lyutikov , m. , & blandford , r. 2003 , arxiv : astro - ph/0312347 narayan , r. , & kumar , p. 2009 , , 394 , l117 panaitescu , a. , spada , m. , & mszros , p. 1999 , , 522 , l105 rees , m. j. , & meszaros , p. 1994 , , 430 , l93 sari , r. , & piran , t. 1997 , , 485 , 270 shen , r .- f . , & zhang , b. 2009 , , 398 , 1936 spruit , h. c. , daigne , f. , & drenkhahn , g. 2001 , , 369 , 694 stern , b. e. , & svensson , r. 1996 , , 469 , l109 tagliaferri , g. , goad , m. , chincarini , g. , et al . 2005 , , 436 , 985 uhm , z. l. , & zhang , b. 2013 , arxiv:1303.2704 vetere , l. , massaro , e. , costa , e. , soffitta , p. , & ventura , g. 2006 , , 447 , 499 yamazaki , r. , ioka , k. , & nakamura , t. 2004 , , 607 , l103 yuan , q. , yin , p .- f . , wu , x .- f . , et al . 2011 , , 730 , l15 zhang , b. , & mszros , p. 2002 , , 581 , 1236 zhang , b. , & kobayashi , s. 2005 , , 628 , 315 zhang , b. , fan , y. z. , dyks , j. , et al . 2006 , , 642 , 354 zhang , b. , & peer , a. 2009 , , 700 , l65 zhang , b. , & yan , h. 2011 , , 726 , 90 ( zy11 ) zhang , b .- b . , zhang , b. , liang , e .- w . , et al . 2011 , , 730 , 141 zrake , j. , & macfadyen , a. i. 2012 , , 744 , 32 zweibel , e. g. , & yamada , m. 2009 , , 47 , 291	in this paper , we simulate the prompt emission light curves of gamma - ray bursts ( grbs ) within the framework of the internal - collision - induced magnetic reconnection and turbulence ( icmart ) model . this model applies to grbs with a moderately high magnetization parameter @xmath0 in the emission region . we show that this model can produce highly variable light curves with both fast and slow components . the rapid variability is caused by many locally doppler - boosted mini - emitters due to turbulent magnetic reconnection in a moderately high @xmath0 flow . the runaway growth and subsequent depletion of these mini - emitters as a function of time define a broad slow component for each icmart event . a grb light curve is usually composed of multiple icmart events that are fundamentally driven by the erratic grb central engine activity . allowing variations of the model parameters , one is able to reproduce a variety of light curves and the power density spectra as observed .	13,808	256
markov s principle has a special status in constructive mathematics . one way to formulate this principle is that if it is impossible that a given algorithm does not terminate , then it does terminate . it is equivalent to the fact that if a set of natural number and its complement are both computably enumerable , then this set is decidable . this form is often used in recursivity theory . this principle was first formulated by markov , who called it `` leningrad s principle '' , and founded a branch of constructive mathematics around this principle @xcite . this principle is also equivalent to the fact that if a given real number is _ not _ equal to @xmath0 then this number is _ apart _ from @xmath0 ( that is this number is @xmath1 or @xmath2 for some rational number @xmath3 ) . on this form , it was explicitly _ refuted _ by brouwer in intuitionistic mathematics , who gave an example of a real number ( well defined intuitionistically ) which is not equal to @xmath0 , but also not apart from @xmath0 . ( the motivation of brouwer for this example was to show the necessity of using _ negation _ in intuitionistic mathematics @xcite . ) the idea of brouwer can be represented formally using topological models @xcite . in a neutral approach to mathematics , such as bishop s @xcite , markov s principle is simply left undecided . we also expect to be able to prove that markov s principle is _ not _ provable in formal system in which we can express bishop s mathematics . for instance , kreisel @xcite introduced _ modified realizability _ to show that markov s principle is not derivable in the formal system @xmath4 . similarly , one would expect that markov s principle is _ not _ derivable in martin - lf type theory @xcite , but , as far as we know , such a result has not been established yet . we say that a statement @xmath5 is _ independent _ of some formal system if @xmath5 can not be derived in that system . a statement in the formal system of martin - lf type theory ( @xmath6 ) is represented by a closed type . a statement / type @xmath5 is derivable if it is inhabited by some term @xmath7 ( written @xmath8 ) . this is the so - called propositions - as - types principle . correspondingly we say that a statement @xmath5 ( represented as a type ) is independent of @xmath6 if there is no term @xmath7 such that @xmath8 . the main result of this paper is to show that markov s principle is independent of martin - lf type theory . the main idea for proving this independence is to follow brouwer s argument . we want to extend type theory with a `` generic '' infinite sequence of @xmath0 and @xmath9 and establish that it is both absurd that this generic sequence is never @xmath0 , but also that we can not show that it _ has to _ take the value @xmath0 . to add such a generic sequence is exactly like adding a _ cohen real _ @xcite in forcing extension of set theory . a natural attempt for doing this will be to consider a _ topological model _ of type theory ( sheaf model over cantor space ) , extending the work @xcite to type theory . however , while it is well understood how to represent universes in _ presheaf _ model @xcite , it has turned out to be surprisingly difficult to represent universes in _ sheaf _ models , see @xcite and @xcite . our approach is here instead a purely _ syntactical _ description of a forcing extension of type theory ( refining previous work of @xcite ) , which contains a formal symbol for the generic sequence and a proof that it is absurd that this generic sequence is never @xmath0 , together with a _ normalization _ theorem , from which we can deduce that we _ can not _ prove that this generic sequence has to take the value @xmath0 . since this formal system is an extension of type theory , the independence of markov s principle follows . as stated in @xcite , which describes an elegant generalization of this principle in type theory , markov s principle is an important technical tool for proving termination of computations , and thus can play a crucial role if type theory is extended with general recursion as in @xcite . this paper is organized as follows . we first describe the rules of the version of type theory we are considering . this version can be seen as a simplified version of type theory as represented in the system agda @xcite , and in particular , contrary to the work @xcite , we allow @xmath10-conversion , and we express conversion as _ judgment_. markov s principle can be formulated in a natural way in this formal system . we describe then the forcing extension of type theory , where we add a cohen real . for proving normalization , we follow tait s computability method @xcite , but we have to consider an extension of this with a computability _ relation _ in order to interpret the conversion judgment . this can be seen as a forcing extension of the technique used in @xcite . using this computability argument , it is then possible to show that we can not show that the generic sequence has to take the value @xmath0 . we end by a refinement of this method , giving a consistent extension of type theory where the _ negation _ of markov s principle is provable the syntax of our type theory is given by the grammar : @xmath11 we use the notation @xmath12 as a short hand for the term @xmath13 , where @xmath14 is the successor constructor . we describe a type theory with one universe la russell , natural numbers , functional extensionality and surjective pairing , hereafter referred to as @xmath6.-conversion and surjective pairing . ] * natural numbers : * * booleans : * * unit type : * * empty type : * * dependent functions : * * dependent pairs : * * universe : * * congruence : * the following four rules are admissible in the this type system @xcite , we consider them as rules of our type system : markov s principle can be represented in type theory by the type @xmath15\ ] ] where @xmath16 is defined by @xmath17 . note that @xmath18 is inhabited when @xmath19 and empty when @xmath20 . thus @xmath21 is inhabited if there is @xmath22 such that @xmath19 . the main result of this paper is the following : thmmainresult [ thm : mainresult ] there is no term @xmath7 such that @xmath23 . an _ extension _ of @xmath6 is given by introducing new objects , judgment forms and derivation rules . this means in particular that any judgment valid in @xmath6 is valid in the extension . consistent _ extension is one in which the type @xmath24 is uninhabited . to show theorem [ thm : mainresult ] we will form a consistent extension of @xmath6 with a new constant @xmath25 where @xmath26 . we will then show that @xmath27 is derivable while @xmath28 is not derivable . thus showing that @xmath29 is not derivable in this extension and consequently not derivable in @xmath6 . while this is sufficient to establish independence in the sense of non - derivability of @xmath30 . to establish the independence of @xmath30 in the stronger sense one also needs to show that @xmath31 is not derivable in @xmath6 . this can achieved by reference to the work of aczel @xcite where it is shown that @xmath6 extended with @xmath32 is consistent . since @xmath33 we have @xmath34 . if we let @xmath35 we get that @xmath36 . by @xmath37 abstraction we have @xmath38 . we can then conclude that there is no term @xmath7 such that @xmath39 . finally , we will refine the result of theorem [ thm : mainresult ] by building a consistent extension of @xmath6 where @xmath31 is derivable . a _ condition _ @xmath40 is a graph of a partial finite function from @xmath41 to @xmath42 . we denote by @xmath43 the empty condition . we write @xmath44 when @xmath45 . we say @xmath46 _ extends _ @xmath40 ( written @xmath47 ) if @xmath40 is a subset of @xmath46 . a condition can be thought of as a basic compact open in cantor space @xmath48 . two conditions @xmath40 and @xmath46 are _ compatible _ if @xmath49 is a condition and we write @xmath50 for @xmath49 . if @xmath51 we write @xmath52 for @xmath53 and @xmath54 for @xmath55 . we define the notion of _ partition _ corresponding to the notion of finite covering of a compact open in cantor space . we write @xmath56 to say that @xmath57 is a partition of @xmath40 and we define it as follows : 1 . 2 . if @xmath51 and @xmath59 and @xmath60 then @xmath61 . note that if @xmath56 then any @xmath62 and @xmath63 are incompatible unless @xmath64 . if moreover @xmath65 then @xmath66 . we extend the given type theory by annotating the judgments with conditions , i.e. replacing each judgment @xmath67 in the given type system with a judgment @xmath68 . in addition we add the locality rule : we add a term @xmath25 for the generic point along with the introduction and conversion rules : we add a term @xmath69 and the rule : since @xmath69 inhabits the type @xmath70 , our goal is then to show that no term inhabits the type @xmath71 . it follows directly from the description of the forcing extension that : [ lem : extension ] @xmath72 in standard type theory then @xmath73 . note that if @xmath74 and @xmath68 then @xmath75 ( monotonicity ) . a statement @xmath5 ( represented as a closed type ) is derivable in this extension if @xmath76 for some @xmath7 , which implies @xmath77 for all @xmath40 . similarly to @xcite we can state a conservativity result for this extension . let @xmath78 and @xmath79 be two terms of standard type theory . we say that @xmath80 is compatible with a condition @xmath40 if @xmath80 is such that @xmath81 whenever @xmath45 and @xmath82 otherwise . we say that @xmath83 is compatible with a condition @xmath40 if @xmath80 is compatible with @xmath40 and @xmath83 is given by @xmath84 where @xmath85 is the smallest natural number such that @xmath86 . to see that @xmath83 is well typed , note that by design @xmath87 thus @xmath88 and @xmath89 . we have then @xmath90 , thus @xmath91 . let @xmath92 and @xmath79 be compatible with some condition @xmath40 . if @xmath93 then @xmath94 \vdash j[g/{\mathsf{f}},v/{\mathsf{w}}]$ ] , i.e. replacing @xmath25 with @xmath80 then @xmath69 with @xmath83 we obtain a valid judgment in standard type theory . in particular , if @xmath95 where neither @xmath25 nor @xmath69 occur in @xmath96 or @xmath97 then @xmath72 is a valid judgment in standard type theory . the proof is by induction on the type system and it is straightforward for all the standard rules . for ( @xmath25-eval ) we have @xmath98 \coloneqq g\ , { \sbox{\myboxa}{$\m@thn$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}$ ] and since @xmath80 is compatible with @xmath40 we have @xmath99\vdash g\ , { \sbox{\myboxa}{$\m@thn$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}= p(n){\!:\!}{{\mathsf{n}_2}}$ ] whenever @xmath100 . for ( @xmath69-term ) we have @xmath101 & \coloneqq ( { \mathsf{w}}{\!:\!}\neg\neg ( \sigma(x{\!:\!}{\mathsf{n}})\ , { \mathsf{iszero}}\,(g\,x)))[v/{\mathsf{w}}]\\ & \coloneqq v{\!:\!}\neg\neg ( \sigma(x{\!:\!}{\mathsf{n}})\,{\mathsf{iszero}}\,(g\,x ) ) . \end{aligned}\ ] ] for ( loc ) the statement follows from the observation that when @xmath80 is compatible with @xmath40 and @xmath56 then @xmath80 is compatible with exactly one @xmath102 in this section we outline a semantics for the forcing extension given in the previous section . we will interpret the judgments of type theory by computability predicates and relations defined by reducibility to computable weak head normal forms . we extend the @xmath103 conversion with @xmath104 whenever @xmath45 . to ease the presentation of the proofs and definitions we introduce _ evaluation contexts _ following @xcite . @xmath105 \mid \mathbb{e}\ , u \mid \mathbb{e}{.1}\mid \mathbb{e}{.2}\mid { \mathsf{s}}\,\mathbb{e } \mid { \mathsf{f}}\ , \mathbb{e}\\ & { { \mathsf{rec}_{{\mathsf{n}_0}}}}(\lambda x. c)\,\mathbb{e } \mid { { \mathsf{rec}_{{\mathsf{n}_1}}}}(\lambda x. c)\,a\,\mathbb{e } \mid { { \mathsf{rec}_{{\mathsf{n}_2}}}}(\lambda x. c)\ , a_0\,a_1\,\mathbb{e}\mid { { \mathsf{rec}_{\mathsf{n}}}}(\lambda x. c)\ , c_z\,g\,\mathbb{e}\end{aligned}\ ] ] an expression @xmath106 $ ] is then the expression resulting from replacing the hole @xmath107 $ ] by @xmath108 . we have the following reduction rules : note that we reduce under @xmath14 . also note that the relation @xmath109 is monotone , that is if @xmath47 and @xmath110 then @xmath111 . in the following we will show that @xmath109 is also local , i.e. if @xmath56 and @xmath112 for all @xmath62 then @xmath113 . [ untypedredislocal ] if @xmath114 and @xmath115 and @xmath116 then @xmath113 . by induction on the derivation of @xmath115 . if @xmath115 is derived by ( @xmath25-red ) then @xmath117 and @xmath118 for some @xmath119 . but since we also have a reduction @xmath120 , we have @xmath121 which could only be the case if @xmath122 . thus we have a reduction @xmath123 . if on the other hand we have a derivation @xmath124 , then we have @xmath113 directly . if the derivation @xmath115 has the form @xmath106{\rightarrowtriangle}_{p(m\mapsto { \mathsf{0 } } ) } \mathbb{e}[e']$ ] then we have @xmath125 and the statement follows by induction . [ reductionislevelbehaved ] let @xmath74 . if @xmath112 then @xmath113 or @xmath7 has the form @xmath126 $ ] for some @xmath127 . by induction on the derivation of @xmath112 . if the reduction @xmath112 has the form @xmath128 then either @xmath129 and the statement follows or @xmath122 and we have @xmath113 . if on the other hand we have @xmath124 then @xmath113 immediately . if @xmath112 has the form @xmath106 { \rightarrowtriangle}_q \mathbb{e}[e']$ ] then @xmath130 and the statement follows by induction . [ eitherreduceorstuck ] let @xmath115 and @xmath131 for some @xmath132 . if @xmath133 then @xmath113 ; otherwise , @xmath7 has the form @xmath126 $ ] . define @xmath134 to mean @xmath113 and @xmath135 and write @xmath136 for @xmath137 . note that it holds that if @xmath138 and @xmath139 then @xmath140 $ ] and if @xmath141 then @xmath142 and @xmath143 $ ] . we define a closure for this relation as follows : we will write @xmath144 as a short hand for @xmath145 . a term @xmath7 is _ in _ @xmath40-whnf if whenever @xmath146 then @xmath147 . a whnf is _ canonical _ if it has one of the forms : @xmath148 a @xmath40-whnf is _ proper _ if it is canonical or it is of the form @xmath149 $ ] for @xmath129 . a canonical @xmath40-whnf has no further reduction at any @xmath74 . a non - canonical proper @xmath40-whnf , i.e. of the form @xmath149 $ ] for @xmath129 , have further reduction at some @xmath74 , namely when @xmath150 . we have the following corollaries to lemma [ untypedredislocal ] and corollary [ eitherreduceorstuck ] . [ typedreduceorstuck ] let @xmath114 . let @xmath151 and @xmath152 . if @xmath133 then @xmath134 ; otherwise @xmath7 has the form @xmath126 $ ] . [ onestepredmonolocal ] let @xmath56 and @xmath153 for all @xmath62 . we have @xmath134 . by induction on @xmath57 . if @xmath154 the the statement follows . assume the statement holds for @xmath155 and @xmath156 and let @xmath157 . by ih , @xmath158 and @xmath159 . from lemma [ untypedredislocal ] , @xmath113 . since @xmath160 and @xmath161 , then @xmath162 . thus @xmath163 . note that if @xmath74 and @xmath164 then @xmath165 . however if @xmath166 and @xmath167 it is not necessarily the case that @xmath168 . e.g. we have that @xmath169 and @xmath170 but it is _ not _ true that @xmath171 for a closed term @xmath77 , we say that @xmath7 _ has _ a @xmath40-whnf if @xmath164 and @xmath172 is in @xmath40-whnf . if @xmath172 is canonical , respectively proper , we say that @xmath7 has a canonical , respectively proper , @xmath40-whnf . since the reduction relation is deterministic we have : [ lem : uniquenessofwhnf ] a term @xmath77 has at most one @xmath40-whnf . [ properwhnflocal ] let @xmath77 and @xmath114 . if @xmath7 has proper @xmath173-whnf and a proper @xmath174-whnf then @xmath7 has a proper @xmath40-whnf . let @xmath166 and @xmath175 with @xmath172 in proper @xmath173-whnf and @xmath83 in proper @xmath174-whnf . if @xmath147 or @xmath176 then @xmath7 is already in proper @xmath40-whnf . alternatively we have reductions @xmath177 and @xmath178 . by corollary [ typedreduceorstuck ] either @xmath7 is in proper @xmath40-whnf or @xmath179 and @xmath180 . it then follows by induction that @xmath181 , and thus @xmath7 , has a proper @xmath40-whnf . we define inductively a forcing relation @xmath182 to express that a type @xmath5 is computable at @xmath40 . mutually by recursion we define relations @xmath183 ( @xmath184 computable of type @xmath5 at @xmath40 ) , @xmath185 ( @xmath5 and @xmath186 are computably equal at @xmath40 ) , and @xmath187 ( @xmath184 is computably equal to @xmath188 of type @xmath5 at @xmath40 ) . we write @xmath189 for @xmath190 and @xmath191 . the definition fits the generalized mutual induction - recursion schema @xcite . the following rules have an implicit ( hidden ) premise @xmath192 1 . assuming @xmath182 by @xmath193 1 . assuming @xmath194 then @xmath185 if 1 . 2 . @xmath196 $ ] , @xmath197 and @xmath198 . @xmath199 for all @xmath7 . @xmath200 for all @xmath7 and @xmath172 . 2 . assuming @xmath182 by @xmath201 1 . assuming @xmath194 then @xmath185 if 1 . 2 . @xmath196 $ ] , @xmath197 and @xmath198 . 2 . @xmath203 if 1 . @xmath204 2 . @xmath205{\!:\!}a$ ] , @xmath129 and @xmath206 . 3 . assuming @xmath207 and @xmath208 then @xmath209 if 1 . @xmath204 and @xmath210 . 2 . @xmath204 and @xmath211 { \!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath212 . @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath212 . 3 . assuming @xmath182 by @xmath213 1 . assuming @xmath194 then @xmath185 if 1 . @xmath214 2 . @xmath196 $ ] , @xmath197 and @xmath198 . 2 . @xmath203 if 1 . @xmath215 , @xmath216 . 2 . @xmath205 $ ] , @xmath129 and @xmath206 . 3 . assuming @xmath207 and @xmath208 then @xmath209 if 1 . @xmath215 and @xmath217 , @xmath216 . @xmath215 , @xmath216 and @xmath211{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath212 . @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath212 . 4 . assuming @xmath182 by @xmath218 1 . assuming @xmath194 then @xmath185 if 1 . 2 . @xmath196 $ ] , @xmath197 and @xmath198 . 2 . @xmath203 if 1 . @xmath220 . 2 . @xmath205 $ ] , @xmath129 and @xmath206 . 3 . assuming @xmath207 and @xmath208 then @xmath209 if 1 . @xmath220 and @xmath221 . 2 . @xmath220 and @xmath211{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath212 . @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath212 . 5 . assuming @xmath182 by @xmath222 ( let @xmath223 ) . 1 . assuming @xmath194 and @xmath224 then @xmath185 if 1 . @xmath225 and @xmath226 and @xmath227=e[a])$ ] . 2 . @xmath196 $ ] , @xmath197 and @xmath198 . 2 . assuming @xmath77 then @xmath203 if 1 . @xmath228 $ ] and @xmath229)$ ] . 3 . assuming @xmath207 and @xmath208 and @xmath162 then @xmath209 if 1 . @xmath230)$ ] 6 . assuming @xmath182 by @xmath231 ( let @xmath232 ) . 1 . assuming @xmath194 and @xmath224 then @xmath185 if 1 . @xmath233 and @xmath226 and @xmath227=e[a])$ ] . 2 . @xmath196 $ ] , @xmath197 and @xmath198 . 2 . assuming @xmath77 then @xmath203 if 1 . @xmath234 and @xmath235 $ ] 3 . assuming @xmath207 and @xmath208 and @xmath162 then @xmath209 if 1 . @xmath236 and @xmath237 $ ] 7 . assuming @xmath182 by @xmath238 ( i.e. @xmath239 ) . 1 . assuming @xmath194 then @xmath185 if @xmath240 2 . assuming @xmath241 then @xmath242 if 1 . @xmath243 with @xmath244 2 . @xmath245 and @xmath246 and + @xmath247 { \!:\!}a)$ ] and @xmath248=g[b ] { \!:\!}a)$ ] . 3 . @xmath249 $ ] , @xmath129 and @xmath250 . 3 . assuming @xmath251 and @xmath252 and @xmath253 then @xmath254 if 1 . @xmath243 and @xmath255 for @xmath256 . 2 . @xmath257 and @xmath258 and + @xmath259 and @xmath247 = e[a ] { \!:\!}a)$ ] 3 . @xmath260 and @xmath261 and + @xmath259 and @xmath247 = e[a ] { \!:\!}a)$ ] 4 . @xmath262 and @xmath263 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath264 . 5 . @xmath265 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath264 assuming @xmath182 by @xmath266 ( i.e. @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath268 ) . 1 . assuming @xmath194 and @xmath269 then @xmath185 if @xmath198 2 . assuming @xmath77 then @xmath203 if @xmath206 . 3 . assuming @xmath207 and @xmath208 and @xmath162 then @xmath209 if @xmath270 . note from the definition that when @xmath185 then @xmath182 and @xmath194 , when @xmath183 then @xmath182 and when @xmath187 then @xmath183 and @xmath271 . it follows also from the definition that @xmath272 whenever @xmath190 . the clause ( @xmath273 ) gives semantics to _ variable types_. for example , if @xmath274 and @xmath275 the type @xmath276 has reductions @xmath277 and @xmath278 . thus @xmath279 and @xmath280 and since @xmath281 we have @xmath282 . [ forcedtypeslocallyreducetocanonical ] if @xmath182 then there is a partition @xmath56 where @xmath5 has a canonical @xmath46-whnf for all @xmath62 . if @xmath185 then there is a partition @xmath56 where @xmath5 and @xmath186 have similar canonical @xmath46-whnf for all @xmath62 , i.e. @xmath283 and @xmath284 where @xmath285 is of the form @xmath286 , @xmath287 , @xmath288 , @xmath289 , @xmath290 , @xmath291 , or @xmath292 . the statement follows from the definition by induction on the derivation of @xmath182 [ existenceofpwhnfislocal ] let @xmath56 . if @xmath293 for all @xmath62 then @xmath5 has a proper @xmath40-whnf . follows from lemma [ forcedtypeslocallyreducetocanonical ] and corollary [ properwhnflocal ] by induction . we will now proceed to prove properties of the forcing relation , monotonicity , locality , reflexivity , symmetry and transitivity . in the premise of any forcing @xmath190 there are a number of typing judgments . since the type system satisfy these properties we will largely ignore these typing judgment in the proofs . [ lem : typeintromonotone ] if @xmath182 and @xmath74 then @xmath293 . let @xmath182 and @xmath74 . by induction on the derivation of @xmath182 1 . ( derivation by @xmath193 , @xmath201 , @xmath213 , @xmath218 or @xmath238 . ) let @xmath294 for @xmath295 . since the reduction is monotone we have @xmath296 , thus @xmath293 . ( derivation by @xmath222 or @xmath231 . ) let @xmath297 . from the premise @xmath298 , by ih , it follows that @xmath299 . from @xmath300)$ ] and @xmath301=g[b])$ ] it follows directly that @xmath302)$ ] and @xmath303=g[b])$ ] . hence @xmath293 . ( derivation by @xmath266 . ) let @xmath304 , m\notin { \textup{dom(}p\textup{)}}$ ] . if @xmath305 then @xmath306 . since @xmath307 with a derivation strictly smaller than the derivation of @xmath182 then by ih @xmath293 . alternatively , @xmath308 $ ] but then @xmath309 . by ih we have @xmath310 and thus @xmath293 . [ lem : typeeqmonotone ] if @xmath185 and @xmath74 then @xmath311 . let @xmath185 and @xmath74 . we have then that @xmath182 and @xmath194 . by lemma [ lem : typeintromonotone ] we have that @xmath293 and @xmath312 . by induction on the derivation of @xmath182 1 . [ floctypeq ] let @xmath182 by @xmath266 , i.e. @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath313 . by induction on the derivation of @xmath185 . if @xmath314 then @xmath315 . since the derivation of @xmath316 is strictly smaller than that of @xmath317 , by ih @xmath311 otherwise , @xmath318 and @xmath319 $ ] and since @xmath320 , by ih , @xmath321 . by the definition @xmath311 . 2 . let @xmath182 by @xmath218 ( i.e. @xmath322 ) . by induction on the derivation of @xmath185 1 . let @xmath219 . we have directly that @xmath311 by monotonicity of the reduction . 2 . let @xmath196 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath313 . the statement then follows similarly to ( [ floctypeq ] ) . + the statement follows similarly when @xmath182 holds by @xmath323 . 3 . let @xmath223 . by induction on the derivation of @xmath185 1 . let @xmath324 and @xmath226 and @xmath325=e[a])$ ] . by ih @xmath326 . directly we have @xmath327=e[a])$ ] . thus @xmath311 . 2 . let @xmath196 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath313 . the statement then follows similarly to ( [ floctypeq ] ) . + the statement follows similarly when @xmath182 holds by @xmath231 . ( derivation by @xmath238 ) we have then that @xmath240 and thus @xmath311 . [ lem : termintromonotone ] if @xmath207 and @xmath74 then @xmath328 . let @xmath207 and @xmath74 . from the definition we have that @xmath182 . from lemma [ lem : typeintromonotone ] we have that @xmath293 . by induction on the derivation of @xmath182 . 1 . let @xmath182 by @xmath218 . by induction on the derivation of @xmath207 let @xmath329 . then @xmath330 and @xmath328 . 2 . let @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath331 . if @xmath150 then @xmath332 , since the derivation of @xmath333 is strictly smaller than the derivation @xmath207 , by ih , @xmath328 . otherwise , @xmath334 and @xmath335 { \!:\!}a$ ] . but @xmath336 and by ih @xmath337 . by the definition @xmath328 . + the statement follows similarly when @xmath182 is derived by @xmath201 or @xmath213 . 2 . let @xmath182 by @xmath238 . the statement follows similarly to lemma [ lem : typeintromonotone ] . 3 . let @xmath182 by @xmath222 and let @xmath223 . from @xmath338)$ ] and @xmath339)$ ] we have directly that @xmath340)$ ] and @xmath341)$ ] . thus @xmath328 . 4 . let @xmath182 by @xmath231 and let @xmath232 . by induction on the derivation of @xmath207 . from @xmath234 and @xmath235 $ ] , by ih , @xmath342 and @xmath343 $ ] , thus @xmath328 . let @xmath182 by @xmath266 and let @xmath267 , k\notin { \textup{dom(}p\textup{)}}$ ] and @xmath268 . by induction on the derivation of @xmath207 . if @xmath314 then @xmath315 from the premise @xmath331 , by ih , @xmath328 . otherwise , @xmath334 and @xmath319 $ ] ( i.e. @xmath293 by @xmath266 ) . since @xmath344 , by the ih , @xmath337 . by the definition @xmath328 . [ lem : termeqmonotone ] if @xmath345 and @xmath74 then @xmath346 . let @xmath209 and @xmath74 . we have then that @xmath182 , @xmath207 , and @xmath208 . by lemma [ lem : typeintromonotone ] @xmath293 . by lemma [ lem : termintromonotone ] @xmath347 and @xmath348 . by induction on the derivation @xmath182 . 1 . let @xmath182 by @xmath218 . by induction on the derivation of @xmath345 let @xmath329 and @xmath349 . by monotonicity of reduction @xmath346 . [ localtermeqforcingismonotone ] let @xmath329 and @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath350 . if @xmath150 then @xmath332 , by ih , @xmath346 . otherwise , @xmath335{\!:\!}a$ ] . but @xmath336 and by ih @xmath351 . by the definition @xmath346 . 3 . let @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] . the statement follows similarly to ( [ localtermeqforcingismonotone ] ) . + the statement follows similarly for when @xmath182 holds by @xmath201 or @xmath213 . 2 . let @xmath182 by @xmath238 . the statement follows by a proof similar to that of lemma [ lem : typeeqmonotone ] . 3 . let @xmath182 by @xmath222 and let @xmath223 . from @xmath352)$ ] we have directly that @xmath353)$ ] . hence @xmath346 . 4 . let @xmath182 by @xmath231 and let @xmath232 . by induction on the derivation of @xmath345 . from @xmath354 and @xmath355 $ ] , by ih we have @xmath356 and @xmath357 $ ] , thus @xmath346 . 5 . let @xmath182 by @xmath266 and let @xmath267 , k \notin{\textup{dom(}p\textup{)}}$ ] . by induction on the derivation of @xmath209 . if @xmath150 then the statement follows by ih . if @xmath334 then @xmath319 $ ] ( i.e. @xmath293 by @xmath266 ) and since @xmath358 , by ih , @xmath359 . hence @xmath346 . we collect the results of lemmas [ lem : typeintromonotone ] , [ lem : termintromonotone ] , [ lem : termeqmonotone ] , and [ lem : typeeqmonotone ] in the following corollary . [ cor : monotonicity ] if @xmath190 and @xmath74 then @xmath191 . we write @xmath360 when @xmath361 . by monotonicity @xmath360 iff @xmath190 for all @xmath40 . [ lem : localitysublemma1 ] let @xmath182 and @xmath194 . if @xmath362 and @xmath363 then @xmath185 . by induction on the derivation of @xmath182 . 1 . let @xmath182 by @xmath218 . by induction on the derivation of @xmath194 1 . if @xmath194 by @xmath218 then @xmath185 immediately . [ typeqrightloc ] if @xmath194 by @xmath266 . the statement follows similarly to ( [ loctypeqloc ] ) below . + the statement follows similarly when @xmath182 is derived by @xmath364 and @xmath213 . 2 . let @xmath182 by @xmath222 and let @xmath223 . by induction on the derivation of @xmath194 1 . let @xmath194 by @xmath222 and let @xmath225 . since @xmath182 and @xmath194 we have @xmath298 and @xmath365 . from the premise @xmath366 and by ih @xmath226 . + let @xmath74 and @xmath367 . if @xmath368 then @xmath369=e[a]$ ] and @xmath185 . otherwise , since @xmath309 , by monotonicity @xmath370=e[a]$ ] . from @xmath182 we have that @xmath369 $ ] and from @xmath194 we have that @xmath371 $ ] . by ih @xmath369 = e[a]$ ] . we thus have @xmath185 . 2 . let @xmath194 by @xmath266 . the statement then follows similarly to ( [ loctypeqloc ] ) below . + the statement follows similarly when @xmath182 is derived by @xmath231 . 3 . if @xmath182 by @xmath238 then @xmath372 and @xmath185 . [ loctypeqloc ] if @xmath182 by @xmath266 . let @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath268 if @xmath373 then we have @xmath185 by the definition . 2 . if @xmath374 . by monotonicity @xmath268 and @xmath375 and @xmath376 and @xmath377 . the derivation of @xmath378 is strictly smaller than the derivation of @xmath182 . by ih we have @xmath379 . by the definition @xmath185 . [ lem : typeintrolocal ] if @xmath380 for some @xmath114 then @xmath182 . the proof is by by induction on the derivations @xmath381 . note that from @xmath380 we have that @xmath5 has proper @xmath173-whnf and @xmath174-whnf and by corollary [ existenceofpwhnfislocal ] @xmath5 has a proper @xmath40-whnf . 1 . let @xmath381 by @xmath218 1 . if @xmath5 has a canonical @xmath40-whnf then @xmath322 and @xmath182 . 2 . otherwise , @xmath382 , k\notin{\textup{dom(}p\textup{)}}$ ] . since @xmath5 has a canonical @xmath173-whnf we have that @xmath373 and by the definition we have @xmath182 by @xmath266 + the statement follows similarly when @xmath383 holds by @xmath193 , @xmath201 or @xmath213 . 2 . let @xmath381 by @xmath222 1 . if @xmath5 has a canonical @xmath40-whnf then @xmath223 . from @xmath384 we have @xmath385 . by ih @xmath298 . let @xmath74 1 . if @xmath305 then @xmath306 . assume , w.l.o.g , @xmath386 , then @xmath387)$ ] and @xmath388=g[b])$ ] . 2 . if @xmath389 then @xmath390 . let @xmath367 , by monotonicity @xmath391 . since @xmath309 , by the definition @xmath370 $ ] and by ih @xmath369 $ ] . let @xmath392 , by monotonicity @xmath393 . but then @xmath394 $ ] and @xmath395 $ ] and @xmath394=g[b]$ ] . by lemma [ lem : localitysublemma1 ] we have @xmath369=g[b]$ ] . + the statement follows similarly when @xmath383 holds by @xmath231 . 3 . let @xmath381 by @xmath238 then @xmath396 and @xmath182 . 4 . let @xmath381 by @xmath266 . since @xmath5 does nt have a canonical @xmath173-whnf @xmath5 does nt have a canonical @xmath40-whnf . since @xmath5 has a proper @xmath40-whnf we have @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] . 1 . if @xmath373 then by the definition we have @xmath182 by @xmath266 2 . if @xmath374 then @xmath397 , k\notin{\textup{dom(}p\textup{)}}$ ] . hence @xmath398 by @xmath266 . we have then @xmath399 and @xmath400 . by ih @xmath401 and @xmath402 . by the definition @xmath182 . [ lem : typeeqlocal ] if @xmath362 and @xmath363 for @xmath132 then @xmath185 . by the definition @xmath380 and @xmath403 . by lemma [ lem : typeintrolocal ] @xmath182 and @xmath194 . by lemma [ lem : localitysublemma1 ] @xmath185 . [ lem : termintroeqlocal ] 1 . if @xmath404 and @xmath405 for some @xmath114 then @xmath207 . 2 . if @xmath406 and @xmath407 for some @xmath114 then @xmath345 . we prove the two statements mutually by induction . 1 . [ termintromontone ] from @xmath408 we have @xmath380 and by lemma [ lem : typeintrolocal ] @xmath182 . by induction on the derivation of @xmath182 [ natlocaltermforced ] let @xmath182 by @xmath218 . since @xmath7 has proper @xmath173-whnf and @xmath174-whnf . by lemma [ properwhnflocal ] @xmath7 has a proper @xmath40-whnf . by induction on the derivation of @xmath409 . 1 . let @xmath410 1 . if @xmath7 has a canonical @xmath40-whnf then @xmath329 and @xmath207 directly . 2 . otherwise , @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] . but then we have that @xmath373 and by the definition @xmath207 . 2 . let @xmath411,k\notin{\textup{dom(}p(m\mapsto { \mathsf{0}})\textup{)}}$ ] and @xmath412 . by monotonicity @xmath413 . by ih @xmath331 and by the definition @xmath207 . + the statement follows similarly when @xmath182 by @xmath201 and @xmath213 . [ pilocaltermforced ] let @xmath182 by @xmath222 and @xmath223 . let @xmath74 . if @xmath368 then we have directly @xmath414 $ ] and @xmath415 $ ] . otherwise , we have @xmath309 . let @xmath367 . by monotonicity @xmath416 and we have @xmath417 $ ] . by ih we have @xmath418 $ ] . let @xmath392 . by monotonicity @xmath393 and we have @xmath419 $ ] . by ih ( [ termeqlocal ] ) @xmath420 $ ] . thus we have @xmath207 . [ sigmalocaltermforced ] let @xmath182 by @xmath231 and let @xmath232 . we have @xmath421 and @xmath422 $ ] . by ih @xmath234 and @xmath235 $ ] . thus @xmath207 . 4 . let @xmath182 by @xmath238 . the statement then follows similarly to lemma [ lem : typeintrolocal ] . [ locallocaltermforced ] let @xmath182 by @xmath266 and let @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] . if @xmath373 then by the definition @xmath207 . if @xmath374 then by monotonicity @xmath423 and @xmath424 . by ih @xmath331 . by the definition @xmath207 . [ termeqlocal ] from @xmath425 we have @xmath408 and @xmath426 and @xmath380 . by lemma [ lem : typeintrolocal ] @xmath182 . by induction on the derivation of @xmath182 let @xmath182 by @xmath218 . by ( [ natlocaltermforced ] ) we have @xmath207 . by induction on the derivation of @xmath207 . 1 . if @xmath329 by induction on the derivation of @xmath208 1 . if @xmath172 has a canonical @xmath40-whnf then @xmath349 and @xmath345 . [ natermeqlocalright ] otherwise , @xmath427{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] . if @xmath373 then by the definition @xmath209 . if @xmath374 then by monotonicity @xmath428 and @xmath429 . by ih , @xmath350 . by the definition @xmath345 . 2 . if @xmath205{\!:\!}a , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath331 . the statement follows similarly to ( [ natermeqlocalright ] ) . + the statement follows similarly when @xmath182 holds by @xmath201 and @xmath213 . 2 . let @xmath182 by @xmath222 and @xmath223 . by ( [ pilocaltermforced ] ) we have @xmath207 and @xmath208 . let @xmath74 . if @xmath368 then we have @xmath430 $ ] . otherwise , we have @xmath309 . let @xmath367 . by monotonicity @xmath416 and we have @xmath431 $ ] . by ih we have @xmath432 $ ] . thus @xmath345 . [ sigmalocaltermforced ] let @xmath182 by @xmath231 and let @xmath232 . by ( [ sigmalocaltermforced ] ) @xmath207 and @xmath208 . we have @xmath433 and @xmath434 $ ] . by ih @xmath354 . since we have @xmath435 $ ] then by ih([termintromontone ] ) @xmath436 $ ] . by ih we have @xmath237 $ ] . thus we have @xmath345 . 4 . let @xmath182 by @xmath238 . the statement then follows similarly to lemma [ lem : typeeqlocal ] . let @xmath182 by @xmath266 and let @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] . by ( [ locallocaltermforced ] ) @xmath207 and @xmath208 . if @xmath373 then by the definition @xmath345 . if @xmath374 then by monotonicity @xmath437 and @xmath438 . by ih @xmath350 . by the definition @xmath345 . [ lem : localcharacter ] if @xmath439 and @xmath440 for all @xmath62 then @xmath190 . follows from lemma [ lem : typeintrolocal ] , lemma [ lem : termintroeqlocal ] , and lemma [ lem : typeeqlocal ] by induction . [ lem : basetypecanonicity ] let @xmath441 . if @xmath207 then there is a partition @xmath56 where @xmath7 has a canonical @xmath46-whnf for all @xmath62 . if @xmath345 then there is a partition @xmath56 where @xmath7 and @xmath172 have the same canonical @xmath46-whnf for each @xmath62 . follows from the definition . [ lem : equivalenttypeshaveequalpers ] let @xmath185 . 1 . if @xmath442 then @xmath443 and if @xmath444 then @xmath208 . 2 . if @xmath445 then @xmath446 and if @xmath447 then @xmath448 . by induction on the derivations of @xmath182 , @xmath194 and @xmath185 1 . let @xmath322 and @xmath219 1 . let @xmath207 . by lemma [ lem : basetypecanonicity ] there is a partition @xmath56 where for each @xmath62 @xmath330 for some @xmath449 . but then @xmath450 and @xmath451 . by local character @xmath452 . similarly if @xmath444 then @xmath208 . 2 . let @xmath345 then there is a partition @xmath56 where for each @xmath62 @xmath330 and @xmath453 for some @xmath449 . but then @xmath450 and @xmath454 and @xmath455 . by local character @xmath456 . similarly @xmath457 whenever @xmath458 2 . let @xmath223 and @xmath225 let @xmath207 and @xmath74 . let @xmath459 . from @xmath185 we get @xmath226 and by monotonicity @xmath326 . by ih @xmath367 . thus we have @xmath418 $ ] and since @xmath369=e[a]$ ] we get @xmath460 $ ] . similarly if @xmath461 by monotonicity @xmath461 and by ih @xmath392 . thus @xmath462 $ ] and since @xmath369=e[a]$ ] we get @xmath463 $ ] . similarly @xmath208 when @xmath444 . 2 . let @xmath345 and @xmath74 . let @xmath459 . similarly to the above we get @xmath464 $ ] . thus showing @xmath455 . similarly we have @xmath465 when @xmath466 3 . let @xmath232 and @xmath233 let @xmath207 . we have @xmath234 and @xmath235 $ ] . from @xmath185 we get @xmath226 and by ih @xmath467 . from @xmath185 we get @xmath468=e[t{.1}]$ ] and by ih @xmath469 $ ] . thus @xmath452 . similarly we have @xmath208 when @xmath444 . 2 . let @xmath345 . we have @xmath354 and @xmath355 $ ] . from @xmath185 we get @xmath226 and by ih @xmath470 . from @xmath185 we get @xmath468=e[t{.1}]$ ] and by ih @xmath471 $ ] . thus @xmath456 . similarly we have @xmath457 when @xmath458 . if either @xmath5 or @xmath186 does not reduce to a canonical @xmath40-whnf then by lemma [ forcedtypeslocallyreducetocanonical ] we have a partition @xmath56 where for each @xmath102 both @xmath5 and @xmath186 have canonical whnf and we can show the statement for each @xmath62 by the above . by local character the statement follows for @xmath40 . immediately from the definition we have if @xmath472 then @xmath473 . if @xmath185 then @xmath474 . if both @xmath5 and @xmath186 have canonical @xmath40-whnf then the statement follows by induction from the definition and lemma [ lem : equivalenttypeshaveequalpers ] . otherwise , by lemma [ forcedtypeslocallyreducetocanonical ] we have a partition @xmath56 where both @xmath5 and @xmath186 have canonical @xmath46-whnf for all @xmath62 . by monotonicity @xmath311 and it follows by the above that @xmath475 . by local character @xmath474 . [ lem : typeequiv_refl , symm , trans ] if @xmath185 and @xmath476 then @xmath477 . let @xmath185 and @xmath476 . we then have that @xmath182 , @xmath194 and @xmath478 . thus @xmath5 , @xmath186 and @xmath479 have proper @xmath40-whnf . if any of these proper @xmath40-wnf is not canonical then by lemma [ forcedtypeslocallyreducetocanonical ] we can find a partition @xmath56 where all three have canonical @xmath46-whnf for all @xmath62 . by monotonicity @xmath311 and @xmath480 for all @xmath62 . if we can then show that @xmath481 for all @xmath62 then by local character we will have @xmath482 . thus we can assume w.l.o.g that @xmath5 , @xmath186 and @xmath479 have canonical @xmath40-whnf . by induction on the derivations of @xmath182 , 1 . let @xmath182 by @xmath218 . since by assumption @xmath186 has a canonical @xmath40-wnf and @xmath185 then @xmath219 . similarly @xmath483 and we have @xmath482 + the statement follows similarly when @xmath182 holds by @xmath193 , @xmath201 and @xmath213 . 2 . let @xmath182 by @xmath222 and @xmath223 . from @xmath185 and since by assumption @xmath186 has a canonical @xmath40-whnf we have @xmath225 and @xmath226 and @xmath484=e[a])$ ] . since @xmath476 and by assumption @xmath186 has a canonical @xmath40-whnf we have @xmath485 and @xmath486 and @xmath487=r[b])$ ] . + by ih @xmath488 . let @xmath74 and @xmath367 . by monotonicity @xmath326 and by lemma [ lem : equivalenttypeshaveequalpers ] @xmath459 . thus @xmath371=r[a]$ ] . but @xmath369=e[a]$ ] . by ih @xmath369=r[a]$ ] . thus we have @xmath482 . + the statement follows similarly when @xmath182 holds by @xmath231 . 3 . since @xmath185 and @xmath476 , we have @xmath240 and @xmath489 and the statements follows . immediately from the definition we have the following if @xmath207 then @xmath490 . if @xmath345 and then @xmath491 . let @xmath345 . we have @xmath207 , @xmath208 and @xmath182 . by induction on the derivation of @xmath182 . 1 . let @xmath182 by @xmath218 . since @xmath345 we have a partition ( lemma [ lem : basetypecanonicity ] ) @xmath56 where for each @xmath62 we have @xmath330 and @xmath453 for some @xmath449 . hence @xmath492 for all @xmath62 . by local character @xmath345 . + the statement follows similarly when @xmath182 is derived by @xmath201 and @xmath213 . 2 . let @xmath182 by @xmath222 and let @xmath223 . let @xmath74 and @xmath367 we then have @xmath493 $ ] . we have @xmath369 $ ] and by ih @xmath494 $ ] . thus @xmath495 . 3 . let @xmath182 by @xmath231 and let @xmath232 . we have @xmath354 and @xmath237 $ ] . since @xmath299 , by ih @xmath496 . since @xmath182 we have @xmath468=g[u{.1}]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] @xmath497 $ ] . since @xmath498 $ ] , by ih @xmath499 $ ] . thus @xmath495 . 4 . let @xmath182 by @xmath238 . the statement then follows similarly to lemma [ lem : typeequiv_refl , symm , trans ] 5 . let @xmath182 by @xmath266 and let @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath268 . since @xmath345 we have that @xmath500 . by ih @xmath501 . by the definition @xmath495 . if @xmath345 and @xmath502 then @xmath503 . let @xmath345 and @xmath502 . we have @xmath182 , @xmath207 , @xmath208 and @xmath504 . by induction on the derivation of @xmath182 . 1 . let @xmath182 by @xmath218 . by lemma [ lem : basetypecanonicity ] there is a partition @xmath56 where for each @xmath62 we have @xmath330 , @xmath453 , and @xmath505 . thus @xmath506 for all @xmath62 . by local character @xmath506 . + the statement follows similarly when @xmath182 by @xmath201 and @xmath213 2 . let @xmath182 by @xmath222 and let @xmath223 . let @xmath74 and @xmath367 . we have then @xmath493 $ ] and @xmath507 $ ] . by ih @xmath508 $ ] . thus @xmath503 . 3 . let @xmath182 by @xmath231 and let @xmath232 . since @xmath345 we have that @xmath236 and @xmath237 $ ] . similarly we have that @xmath509 and @xmath510 $ ] . since @xmath182 we have that @xmath468=g[u{.1}]$ ] and by lemma [ lem : equivalenttypeshaveequalpers ] @xmath511 $ ] . by ih we have @xmath512 and @xmath513 $ ] . we have then that @xmath514 . 4 . let @xmath182 by @xmath238 . the statement then follows similarly to lemma [ lem : typeequiv_refl , symm , trans ] . let @xmath182 by @xmath266 and let @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath268 . we have that @xmath500 and @xmath515 . by ih @xmath516 . by the definition @xmath503 . in this section we show that the type theory described in section [ sec : forcingextensionoftypetheory ] is sound with respect to the semantics described in section [ sec : semantics ] . i.e. we aim to show that @xmath190 whenever @xmath272 . [ lem : types_reductionimplyequiv ] if @xmath517 and @xmath194 then @xmath182 and @xmath185 . follows from the definition . [ lem : terms_onestepreductionimplyequiv ] let @xmath182 . if @xmath518 and @xmath208 then @xmath207 and @xmath345 . let @xmath518 and @xmath208 . by induction on the derivation of @xmath182 . 1 . let @xmath182 by @xmath238 . the statement follows similarly to lemma [ lem : types_reductionimplyequiv ] . 2 . let @xmath182 by @xmath218 . by induction on the derivation of @xmath208 . if @xmath519 then @xmath520 and the statement follows by the definition . if @xmath211{\!:\!}a , k\notin { \textup{dom(}p\textup{)}}$ ] and @xmath521 then since @xmath522 , by ih @xmath331 and @xmath212 . by the definition @xmath207 and @xmath209 . + the statement follows similarly for @xmath201 and @xmath213 . 3 . let @xmath182 by @xmath222 and let @xmath523 . let @xmath74 and @xmath367 . we have @xmath524 $ ] . by ih @xmath525 $ ] and @xmath493 $ ] . if @xmath526 we similarly get @xmath527 $ ] and @xmath528 $ ] . since @xmath369=g[b]$ ] , by lemma [ lem : equivalenttypeshaveequalpers ] @xmath529 $ ] . but @xmath530 $ ] . by symmetry and transitivity @xmath462 $ ] . thus @xmath207 and @xmath345 . 4 . let @xmath182 by @xmath222 and let @xmath531 . from @xmath163 we have @xmath77 and we have @xmath142 and @xmath532 $ ] . by the induction @xmath234 and @xmath236 . by the induction @xmath533 $ ] and @xmath497 $ ] . but since @xmath182 and we have shown @xmath236 we get @xmath468=g[u{.1}]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] @xmath235 $ ] and @xmath355 $ ] . thus @xmath203 and @xmath534 5 . let @xmath182 by @xmath266 . let @xmath267 , k\notin{\textup{dom(}p\textup{)}}$ ] and @xmath268 . since @xmath208 we have @xmath521 . but we have @xmath522 . by ih @xmath331 and @xmath535 . by the definition @xmath207 and @xmath209 . [ cor : terms_reductionimplyequiv ] let @xmath146 and @xmath182 . if @xmath208 then @xmath207 and @xmath345 . [ cor : genericwelltyped ] @xmath536 . it s direct to see that @xmath537 . for an arbitrary condition @xmath40 let @xmath538 . by lemma [ lem : basetypecanonicity ] we have a partition @xmath56 where for each @xmath62 , @xmath539 for some @xmath540 . we have thus a reduction @xmath541 . if @xmath305 then @xmath542 and by definition @xmath543 . if @xmath544 then @xmath545 and @xmath546 . thus @xmath547 . by the definition @xmath543 . we thus have that @xmath543 for all @xmath62 and by local character @xmath548 . let @xmath549 . by lemma [ lem : basetypecanonicity ] there is a partition @xmath56 where for each @xmath62 , @xmath550 and @xmath551 for some @xmath552 . we then have @xmath553 and @xmath554 . if @xmath305 then @xmath555 and @xmath556 . by corollary [ cor : terms_reductionimplyequiv ] , symmetry and transitivity @xmath557 . if on the other hand @xmath558 then @xmath559 and @xmath560 . by the definition @xmath561 . thus @xmath561 for all @xmath62 . by local character @xmath562 . hence @xmath536 . [ lem : forcingtermnegation ] if @xmath563 and @xmath182 then @xmath564 iff for all @xmath74 there is no term @xmath172 such that @xmath565 . let @xmath182 and @xmath563 . we have directly that @xmath566 . assume @xmath564 . if @xmath567 for some @xmath74 , then @xmath568 which is impossible . conversely , assume it is the case that for all @xmath47 there is no @xmath172 for which @xmath565 . since @xmath569 and @xmath570 never hold for any @xmath571 , the statements `` @xmath572 '' and `` @xmath573 '' hold trivially . [ lem : markovpremisewitnesswelltyped ] @xmath574 . by lemma [ lem : forcingtermnegation ] it is enough to show that for all @xmath46 there is no term @xmath172 for which @xmath575 . assume @xmath575 for some @xmath172 . let @xmath576 we have then @xmath577 thus @xmath578 which is impossible . let @xmath579 and @xmath580 . let @xmath581 $ ] . we write @xmath582 if @xmath583 for all @xmath584 . letting @xmath585 , we write @xmath586 if @xmath587 for all @xmath584 . we write @xmath588 if @xmath589 for all @xmath584 and @xmath590 if @xmath591 for all @xmath584 . [ lem : evalissub ] if @xmath592 then @xmath593 . if @xmath594 then @xmath595 . follows from the definition . 1 . we write @xmath596 if @xmath597 and for all @xmath74 whenever @xmath598 then @xmath599 and whenever @xmath600 then @xmath601 . we write @xmath602 if @xmath603 , @xmath596 and for all @xmath74 whenever @xmath598 then @xmath604 and whenever @xmath605 then @xmath606 . we write @xmath607 if @xmath608 , @xmath596 , @xmath609 and for all @xmath74 whenever @xmath598 then @xmath610 . we write @xmath611 if @xmath612 , @xmath602 , @xmath613 and for all @xmath74 whenever @xmath598 then @xmath614 . in the following we will show that whenever we have a rule in the type system then it holds that . which is sufficient to show soundness . follows from corollary [ lem : localcharacter ] . [ soundpiintro ] let @xmath74 and @xmath598 . let @xmath615 and @xmath616 . we have @xmath617 . since @xmath618 we have @xmath619 and by lemma [ lem : equivalenttypeshaveequalpers ] @xmath620 . thus @xmath621 . we have @xmath622 $ ] and @xmath622=g\sigma[a]$ ] . if moreover @xmath623 then @xmath624 . thus @xmath622=g\rho[b]$ ] . by the definition @xmath625 and @xmath626 . [ soundpieq ] let @xmath74 and @xmath598 . similarly to lemma [ soundpiintro ] , we can show @xmath627 , @xmath628 , @xmath629 , and @xmath630 . from the premise let @xmath615 and @xmath616 . we have then @xmath617 . thus @xmath622=e\rho[a]$ ] . by the definition @xmath632 and @xmath633 . [ soundlambdaintro ] let @xmath74 and @xmath598 . by lemma [ lem : evalissub ] @xmath634 . let @xmath615 and @xmath635 . since @xmath636 we have that @xmath637 . since @xmath638 we have @xmath639{\!:\!}g\rho[d]$ ] . by the reduction rules @xmath640 $ ] . thus @xmath641{\!:\!}g\rho[d]$ ] . but @xmath642 , hence , @xmath643{\!:\!}g\rho[d]$ ] . by lemma [ lem : terms_onestepreductionimplyequiv ] we have that @xmath644 $ ] and @xmath645{\!:\!}g\rho[d]$ ] . let @xmath646 we have similarly that @xmath647{\!:\!}g\rho[e]$ ] . we have also that @xmath648 $ ] , thus @xmath643=t\rho[e]{\!:\!}g\rho[d]$ ] and @xmath649=g\rho[e]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] we have @xmath647{\!:\!}g\rho[d]$ ] . by symmetry and transitivity we have @xmath650 $ ] . thus @xmath651 . let @xmath600 . we get @xmath652 . similarly to the above we can show @xmath653 . let @xmath615 and @xmath616 . by lemma [ lem : equivalenttypeshaveequalpers ] @xmath620 . we then have @xmath621 . thus we have @xmath622=g\sigma[a]$ ] . thus @xmath654 and by lemma [ lem : equivalenttypeshaveequalpers ] @xmath655 . we have @xmath656=t\sigma[a]{\!:\!}g\rho[a]$ ] . but @xmath657{\!:\!}g\rho[a]$ ] and @xmath658 { \!:\!}g\sigma[a]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] @xmath658 { \!:\!}g\rho[a]$ ] . by symmetry and transitivity @xmath659 $ ] . thus @xmath660 . [ soundbeta ] let @xmath74 and @xmath598 . we have @xmath661 . as in lemma [ soundlambdaintro ] @xmath662\rho { \!:\!}g[a]\rho$ ] which by lemma [ lem : terms_onestepreductionimplyequiv ] imply that @xmath663\rho { \!:\!}g\rho[a]$ ] . [ soundapptyp ] let @xmath74 and @xmath598 . we have @xmath664 and @xmath661 . by the definition @xmath665\rho$ ] . let @xmath666 . we have then @xmath667 and @xmath668 . from the definition @xmath669\rho$ ] . from the definition @xmath670\rho$ ] . by transitivity @xmath671\rho$ ] . [ soundappeq ] ( 1 ) ( 2 ) let @xmath74 and @xmath598 . 1 . we have @xmath664 and @xmath672 . from the definition get @xmath673\rho$ ] . we have @xmath674 and @xmath675 . from the definition we get @xmath676\rho$ ] . [ soundfunext ] let @xmath74 and @xmath598 . we have @xmath677 and @xmath678 . let @xmath615 and @xmath616 . we have then that @xmath679 . thus @xmath680 $ ] . by the definition @xmath674 . [ soundpairing ] let @xmath74 and @xmath598 . by the typing rules @xmath681 and @xmath682 $ ] . but @xmath634 . by substitution we have @xmath683 and @xmath684\rho$ ] . but @xmath685 and @xmath686 . thus @xmath687 and @xmath688\rho$ ] . from the premise @xmath661 and @xmath689\rho$ ] . by lemma [ lem : terms_onestepreductionimplyequiv ] @xmath690 and @xmath691\rho$ ] . by lemma [ lem : terms_onestepreductionimplyequiv ] @xmath692 , thus @xmath693 . hence @xmath369\rho = g[(a , b){.1}]\rho$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] @xmath694\rho$ ] . by the definition we have then that @xmath695 . let @xmath605 . similarly we can show @xmath696 . we have that @xmath668 and @xmath697\rho$ ] . we have also @xmath698 we thus have @xmath369\rho = g[a]\sigma$ ] . by lemma [ lem : terms_onestepreductionimplyequiv ] @xmath699\sigma$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] @xmath699\rho$ ] . but we also have by lemma [ lem : terms_onestepreductionimplyequiv ] that @xmath700 . hence , by lemma [ lem : equivalenttypeshaveequalpers ] , we have @xmath701 . by symmetry and transitivity we then have that @xmath702 and @xmath703\rho$ ] . thus we have that @xmath704 . [ soundproj ] \(1 ) ( 2 ) let @xmath74 and @xmath598 . 1 . we have @xmath705 and @xmath706\rho$ ] . by substitution we get @xmath707 . but @xmath708 , thus @xmath709 . we have that @xmath710 . thus by lemma [ lem : terms_onestepreductionimplyequiv ] @xmath711 and @xmath712 . similarly we have @xmath713 $ ] . since @xmath714\rho$ ] , by lemma [ lem : terms_onestepreductionimplyequiv ] , we have that @xmath715\rho$ ] and @xmath716 $ ] . [ soundprojtypeq ] \(1 ) ( 2 ) let @xmath74 and @xmath598 . 1 . we have @xmath717 . by the definition we have @xmath718 and @xmath719\rho$ ] . let @xmath605 . we have that @xmath720 . by the definition @xmath721 and @xmath722\rho$ ] . we have @xmath723 . by the definition @xmath724 and @xmath725\rho$ ] . [ soundpairext ] let @xmath74 and @xmath598 . we have @xmath717 and @xmath726 . we also have @xmath727 and @xmath728\rho$ ] . by the definition @xmath729 . [ soundnat ] \(1 ) ( 2 ) ( 3 ) \(1 ) and ( 2 ) follow directly from the definition while ( 3 ) follows from lemma [ lem : basetypecanonicity ] . [ soundnessnatrec ] let @xmath74 and @xmath598 . we have then that @xmath730 , hence , @xmath731 . let @xmath615 . let @xmath732 , @xmath733 and @xmath734 . by lemma [ lem : basetypecanonicity ] there is a partition @xmath735 such that for each @xmath736 , @xmath737 and @xmath738 . in order to show that @xmath739 we need to show that @xmath740 $ ] , @xmath741 $ ] , and @xmath742 $ ] . by local character it will be sufficient to show that for each @xmath736 we have @xmath743 $ ] , @xmath744 $ ] , and @xmath745 $ ] . we have that @xmath746\\ & s\vdash ( { { \mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\rho\,b { { \rightarrowtriangle}^\ast}({{\mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\rho\ , { \sbox{\myboxa}{$\m@thn$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi } { \!:\!}f\rho[b]\end{aligned}\ ] ] let @xmath747 . by induction on @xmath748 . if @xmath749 then @xmath750\\ & s\vdash ( { { \mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\rho\,b { { \rightarrowtriangle}^\ast}({{\mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\rho\,{\mathsf{0}}{\rightarrowtriangle}a_0\rho { \!:\!}f\rho[b]\end{aligned}\ ] ] by lemma [ lem : equivalenttypeshaveequalpers ] we have then that @xmath751 & \hphantom{space } s{\vdash}({{\mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\rho\,b = a_0\rho { \!:\!}f\rho[b]\end{aligned}\ ] ] since @xmath752 we have @xmath753 and thus @xmath754=f\rho[b]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] , symmetry and transitivity @xmath755 $ ] . assume the statement holds for @xmath756 . let @xmath757 . we have then @xmath758\\ & s\vdash ( { { \mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\rho\,b { { \rightarrowtriangle}^\ast}g\rho \ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\ , ( ( { { \mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\rho\ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi } ) { \!:\!}f\rho[{\mathsf{s}}\ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}]\end{aligned}\ ] ] by ih @xmath759 $ ] . but we have @xmath760\rightarrow f[{\mathsf{s}}\,x])$ ] and thus @xmath761 $ ] . by corollary [ cor : terms_reductionimplyequiv ] , symmetry and transitivity we get that @xmath762 $ ] , @xmath763 $ ] , and @xmath764 $ ] . but @xmath765 , thus , @xmath754=f\rho[{\mathsf{s}}\ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] we get then that @xmath766 $ ] . as indicated above , this is sufficient to show @xmath767 . similarly we can show @xmath768 . to show that @xmath769 we need to show that whenever @xmath616 for some @xmath615 we have @xmath770 $ ] . let @xmath771 for @xmath615 . by lemma [ lem : basetypecanonicity ] we have a partition @xmath735 where for each @xmath736 we have @xmath737 . as above it is sufficient to show @xmath772 $ ] for all @xmath736 . let @xmath773 . by induction on @xmath748 . if @xmath749 then as above @xmath751 & \hphantom{space } s{\vdash}({{\mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\sigma\,a = a_0\sigma { \!:\!}f\sigma[b]\end{aligned}\ ] ] since @xmath774 we have @xmath775 . we have then that @xmath754=f\sigma[a]$ ] . but we also have that @xmath776 $ ] . by lemma [ lem : equivalenttypeshaveequalpers ] , symmetry and transitivity it then follows that @xmath777 $ ] . assume the statement holds for @xmath756 . let @xmath757 . as before we have that @xmath758\\ & s\vdash ( { { \mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\sigma\,a { { \rightarrowtriangle}^\ast}g\sigma \ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}\ , ( ( { { \mathsf{rec}_{\mathsf{n}}}}(\lambda x. f)\,a_0\ , g)\sigma\ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi } ) { \!:\!}f\sigma[{\mathsf{s}}\ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}]\end{aligned}\ ] ] by ih @xmath778 $ ] . but @xmath779\rightarrow f[{\mathsf{s}}\,x]))\rho$ ] , thus @xmath780 $ ] but @xmath781=f\sigma[{\mathsf{s}}\ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}]$ ] and @xmath782=f\rho[{\mathsf{s}}\ , { \sbox{\myboxa}{$\m@th\ell$ } \setbox\myboxb\null \ht\myboxb=\ht\myboxa \dp\myboxb=\dp\myboxa \wd\myboxb=0.75\wd\myboxa \sbox\myboxb{$\m@th\overline{\copy\myboxb}$ } \setlength\mylena{\the\wd\myboxa } \addtolength\mylena{-\the\wd\myboxb } \ifdim\wd\myboxb<\wd\myboxa \rlap{\hskip 0.5\mylena\usebox\myboxb}{\usebox\myboxa } \else \hskip -0.5\mylena\rlap{\usebox\myboxa}{\hskip 0.5\mylena\usebox\myboxb } \fi}]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] , symmetry and transitivity we have then that @xmath783 $ ] which is sufficient to show @xmath784 [ soundnatrec0 ] let @xmath74 and @xmath598 . we have @xmath634 and thus we get that @xmath785 $ ] . but @xmath786 . thus @xmath787 $ ] . but @xmath788 $ ] . by lemma [ lem : terms_onestepreductionimplyequiv ] we have @xmath789 $ ] . [ soundnatrecsuc ] let @xmath74 and @xmath598 . we have @xmath790 . by lemma [ lem : basetypecanonicity ] there is a partition @xmath791 such that for each @xmath736 there is @xmath552 and @xmath792 . thus @xmath793 . we have then that @xmath794\end{aligned}\ ] ] but @xmath795 $ ] . by corollary [ cor : terms_reductionimplyequiv ] , symmetry and transitivity @xmath796 $ ] . since @xmath797 we have that @xmath798 = f\rho[{\mathsf{s}}\ , n\rho]$ ] . by lemma [ lem : equivalenttypeshaveequalpers ] we thus have that @xmath799\rho$ ] . by local character @xmath800\rho$ ] [ soundnatreccong ] the proof follows by an argument similar to that used to prove lemma [ soundnessnatrec ] . for the congruence rules , soundness follows by reflexivity , symmetry and transitivity of the forcing relation . soundness for rules of @xmath24 , @xmath801 and @xmath802 follow similarly to those of @xmath803 . soundness for the rules of @xmath804 follows similarly to soundness of typing rules . we have then the following corollary : [ lem : soundness ] if @xmath93 then @xmath805 [ thm : fundamentalthm ] if @xmath272 then @xmath190 . follows from corollary [ lem : soundness ] . now we have enough machinery to show the independence of @xmath30 from type theory . the idea is that if a judgment @xmath97 is derivable in type theory ( i.e. @xmath806 ) then it is derivable in the forcing extension ( i.e. @xmath807 ) and by theorem [ thm : fundamentalthm ] it holds in the interpretation ( i.e. @xmath360 ) . it thus suffices to show that there no @xmath7 such that @xmath808 to establish the independence of @xmath30 from type theory . first we recall the formulation of @xmath30 . @xmath809\end{aligned}\ ] ] where @xmath16 is given by @xmath810 . [ lem : markovconclusionnowitness ] there is no term @xmath7 such that @xmath811 . assume @xmath811 for some @xmath7 . we then have @xmath812 and @xmath813 . by lemma [ lem : basetypecanonicity ] one has a partition @xmath814 where for each @xmath62 , @xmath815 . hence @xmath816 and by lemma [ lem : types_reductionimplyequiv ] @xmath817 . but , by definition , the partition @xmath57 must contain a condition , say @xmath818 , such that @xmath819 whenever @xmath820 ( this holds vacuously for @xmath821 ) . let @xmath822 . assume @xmath823 , then @xmath824 . by monotonicity , from @xmath813 we get @xmath825 . but @xmath826 thus @xmath827 . hence , by lemma [ lem : equivalenttypeshaveequalpers ] , @xmath828 which is impossible , thus contradicting our assumption . if on the other hand @xmath829 then , since @xmath830 , we can apply the above argument with @xmath831 instead of @xmath818 . [ lem : markovnotforcible ] there is no term @xmath7 such that @xmath808 . assume @xmath808 for some @xmath7 . from the definition , whenever @xmath832 we have @xmath833 . since by corollary [ cor : genericwelltyped ] , @xmath536 we have @xmath834 . since by lemma [ lem : markovpremisewitnesswelltyped ] @xmath835 we have @xmath836 which is impossible by lemma [ lem : markovconclusionnowitness ] . from theorem [ thm : fundamentalthm ] , lemma [ lem : markovnotforcible ] , and lemma [ lem : extension ] we can then conclude : we extend the type system in section [ sec : forcingextensionoftypetheory ] further by adding a generic point @xmath837 for each condition @xmath46 . the introduction and conversion rules for @xmath837 are given by : with the reduction rules : we observe that with these added rules the reduction relation is still monotone . for each @xmath837 we add a term : finally we add a term @xmath838 witnessing the negation of @xmath30 by analogy to corollary [ cor : genericwelltyped ] we have [ lem : qgenericpointforced ] @xmath839 for all @xmath46 . [ lem : qmarkovpermisewitnessforced ] @xmath840 for all @xmath46 . assume @xmath841 for some @xmath40 and @xmath7 . let @xmath842 , we have @xmath843 . thus @xmath844 and @xmath845 which is impossible . [ lem : qmarkovconclusionnowitness ] there is no term @xmath7 for which @xmath846 . assume @xmath846 for some @xmath7 . we then have @xmath847 and @xmath848 . by lemma [ lem : basetypecanonicity ] one has a partition @xmath849 where for each @xmath584 , @xmath850 for some @xmath851 . hence @xmath852 . but any partition of @xmath46 contain a condition , say @xmath853 , where @xmath854 whenever @xmath855 and @xmath856 . assume @xmath857 . if @xmath858 then @xmath859 and if @xmath860 then @xmath861 . thus @xmath862 and by lemma [ lem : types_reductionimplyequiv ] @xmath863 . from @xmath813 by monotonicity and lemma [ lem : equivalenttypeshaveequalpers ] we have @xmath864 which is impossible . if on the other hand @xmath865 then since @xmath866 we can apply the above argument with @xmath867 instead of @xmath853 . @xmath868 assume @xmath869 for some @xmath40 and @xmath7 . thus whenever @xmath74 and @xmath870 then @xmath871 . but we have @xmath872 by lemma [ lem : qgenericpointforced ] . hence @xmath873 . but @xmath874 by lemma [ lem : qmarkovpermisewitnessforced ] . thus @xmath875 which is impossible by lemma [ lem : qmarkovconclusionnowitness ] . we have then that this extension is sound with respect to the interpretation . hence we have shown the following statement . there is a consistent extension of @xmath6 where @xmath31 is derivable . recall that @xmath876 . we have then a term @xmath877 . thus in this extension we have a term @xmath878 . we would like to thank simon huber , thomas streicher , martn escard and chuangjie xu .	in this paper , we show that markov s principle is not derivable in dependent type theory with natural numbers and one universe . one tentative way to prove this would be to remark that markov s principle does not hold in a sheaf model of type theory over cantor space , since markov s principle does not hold for the generic point of this model . it is however not clear how to interpret the universe in a sheaf model @xcite . instead we design an extension of type theory , which intuitively extends type theory by the addition of a generic point of cantor space . we then show the consistency of this extension by a normalization argument . markov s principle does not hold in this extension , and it follows that it can not be proved in type theory .	30,529	184
quintessence ( caldwell et al . 1998 ) or dark energy is a new component of the cosmic medium that has been introduced in order to explain the dimming of distant snia ( riess et al . 1998 ; perlmutter et al . 1999 ) through an accelerated expansion while at the same time saving the inflationary prediction of a flat universe . the recent measures of the cmb at high resolution ( lange et al . 2000 , de bernardis et al . 2000 , balbi et al . 2000 ) have added to the motivations for a conspicuous fraction of unclustered dark energy with negative pressure . in its simplest formulation ( see e.g. silveira & waga 1997 ) , the quintessence component can be modeled as a perfect fluid with equation of state @xmath4 with @xmath5 in the range @xmath6 ( @xmath7 for acceleration ) . when @xmath8 we have pure cosmological constant , while for @xmath9 we reduce to the ordinary pressureless matter . the case @xmath10 mimicks a universe filled with cosmic strings ( see e.g. vilenkin 1984 ) . more realistic models possess an effective equation of state that changes with time , and can be modeled by scalar fields ( ratra & peebles 1988 , wetterich 1995 , frieman et al . 1995 , ferreira and joyce 1998 ) , possibly with coupling to gravity or matter ( baccigalupi , perrotta & matarrese 2000 , amendola 2000 ) . the introduction of the new component modifies the universe expansion and introduces at least a new parameter , @xmath5 , in cosmology . most deep cosmological tests , from large scale structure to cmb , from lensing to deep counting , are affected in some way by the presence of the new field . here we study how a perfect fluid quintessence affects the dyer - roeder ( dr ) distance , a necessary tool for all lensing studies ( dyer & roeder 1972 , 1974 ) . the assumption of constant equation of state is at least partially justified by the relatively narrow range of redshift we are considering , @xmath11 . we rederive the dr equation in quintessence cosmology , we solve it analytically whenever possible , and give a very accurate analytical fit to its numerical solution . finally , we apply the dr solutions to a likelihood determination of @xmath0 through the observations of time - delays in multiple images . the dataset we use is composed of only six time - delays , and does not allow to test directly for quintessence ; however , we will show that inclusion of such cosmologies may have an important impact on the determination of @xmath0 with this method . for instance , we find that @xmath0 is smaller that for a pure cosmological constant . in this work we confine ourselves to flat space and extremal values of the beam parameter @xmath12 ; in a paper in preparation we extend to curved spaces and general @xmath12 . in this section we derive the dr distance in quintessence cosmology , find its analytical solutions , when possible , and its asymptotic solutions . finally , we give a very accurate analytical fit to the general numerical solutions as a function of @xmath1 and @xmath5 . first of all , let us notice that when quintessence is present , the friedmann equation ( the @xmath13 component of the einstein equations in a flat frw metric ) becomes ( in units @xmath14 ) @xmath15 .\ ] ] where @xmath0 is the present value of the hubble constant , @xmath1 the present value of the matter density parameter , and where the scale factor is normalized to unity today . in terms of the redshift @xmath16 we can write @xmath17 where @xmath18 the ricci focalization equation in a conformally flat metric ( such as the frw metric ) with curvature tensor @xmath19 is ( see e.g. schneider , falco & ehlers 1992 ) @xmath20 where @xmath21 is the beam area and @xmath22 is the tangent vector to the surface of propagation of the light ray , and the dot means derivation with respect to the affine parameter @xmath23 . multiplying the einstein s gravitational field equation @xmath24 by @xmath25 and imposing the condition @xmath26 for the null geodesic we obtain @xmath27 ; from ( [ focricci ] ) we obtain @xmath28 considering only ordinary pressureless matter and quintessence the energy - momentum tensor writes @xmath29 multiplying by @xmath25 , putting @xmath30 and inserting in ( [ focricci2 ] ) we have @xmath31 \sqrt{a}=0 . \label{focricci3}\ ] ] now , the angular diameter distance @xmath32 is defined as the ratio between the diameter of an object and its angular diameter . we have then @xmath33 . since @xmath34 , and defining the dimensionless distance @xmath35 , eq . ( [ focricci3 ] ) writes @xmath36 r=0 \label{focricciaffine}\ ] ] where we introduced the affine parameter @xmath37 defined implicitely by the relation @xmath38 where @xmath39 is defined in eq . ( [ h(z ) ] ) . finally we get the dr equation with the redshift as independent variable @xmath40 \frac{d^{2}r}{dz^{2}}+ \\ \\ + \left ( 1+z\right ) \left [ \frac{7}{2}\omega _ { m}\left ( 1+z\right ) ^{3}+\frac{% m+4}{2}\left ( 1-\omega _ { m}\right ) \left ( 1+z\right ) ^{m}\right ] \frac{dr}{dz% } + \\ \\ + \left [ \frac{3}{2}\alpha \omega _ { m}\left ( 1+z\right ) ^{3}+\frac{m}{2}% \left ( 1-\omega _ { m}\right ) \left ( 1+z\right ) ^{m}\right ] r=0 \end{array } \label{drg}\ ] ] the constant @xmath12 in eq . ( [ drg ] ) is the fraction of matter homogeneously distributed inside the beam : when @xmath41 all the matter is clustered ( empty beam ) , while for @xmath42 the matter is spread homogeneously and we recover the usual angular diameter distance ( filled beam ) . notice that in our case the empty beam is actually filled uniformly with quintessence . since the actual value of @xmath12 is unknown ( see however barber et al . 2000 , who argue in favor of @xmath12 near unity ) , we will adopt in the following the two extremal values @xmath41 and @xmath42 . the appropriate boundary conditions are ( see e.g. schneider , falco & ehlers 1992 ) @xmath43 ^{-1/2 } \end{array } \right . \label{condizioni al contorno z}\ ] ] defining @xmath44 these become @xmath45 * analytical solutions. eq . ( [ drg ] ) has an analytical solution only for some values of @xmath12 and @xmath5 . here we list all the known analytical solutions . all the cases for @xmath41 and @xmath46 or @xmath47 are new solutions . for the case @xmath8 see also demiansky et al . ( 2000 ) . case when @xmath8 the quintessence fluid is the cosmological constant ; the dr equation becomes @xmath48 \frac{d^{2}r}{dz^{2}}+ \\ + \left [ \frac{7}{2}\omega _ { m}\left ( 1+z\right ) ^{3}+2\left ( 1-\omega _ { m}\right ) \right ] \frac{dr}{dz}=0 \end{array}\ ] ] whose solution in integral form is @xmath49 that is @xmath50 -\frac{1}{% 1+z}f\left [ -\frac{1}{3},\frac{1}{2};\frac{2}{3},\frac{\omega _ { m}}{\omega _ { m}-1}(1+z)^{3}\right ] \right ) $ % \end{tabular}\ ] ] where @xmath51 $ ] is the gauss hypergeometric function . case the solution is @xmath52 -\sqrt{1+z}f\left [ -\frac{1}{4},1;\frac{5}{4},\frac{\omega _ { m}}{\omega _ { m}-1}\left ( 1+z\right ) ^{2}\right ] } { \left ( 2\omega _ { m}-1\right ) f\left [ -\frac{1}{4},1;\frac{5}{4},\frac{\omega _ { m}}{\omega _ { m}-1}\right ] + \frac{4}{5}\frac{\omega _ { m}}{\omega _ { m}-1}f\left [ \frac{3}{% 4},2;\frac{9}{4},\frac{\omega _ { m}}{\omega _ { m}-1}\right ] } $ % \end{tabular}\ ] ] the solution can be written in terms of the meijer @xmath53 function , but the expression is so complicated that it is not worth reporting it here . case now quintessence coincides with ordinary matter ; however , the choice @xmath54 implies that the ordinary matter is completely clustered , while quintessence remains homogeneous . the solution is @xmath55 \label{soluzione alfa0 m3}\ ] ] where @xmath56 cases for @xmath57 the dr equation is @xmath58 and its solution is @xmath59 $ & $ m\neq 2 $ \\ & \\ $ \frac{1}{1+z}\ln \left ( 1+z\right ) $ & $ m=2$% \end{tabular } \right.\ ] ] while for @xmath60 we get @xmath61 and @xmath62 the case @xmath42 is actually the standard angular diameter distance in a homogeneous universe . we list some particular solution here for completeness ( see also bloomfield - torres & waga 1996 , waga & miceli 1999 ) . case the general solution is @xmath63 which gives for any @xmath5 @xmath64 \right . + $ \\ $ \left . -\frac{1}{\sqrt{\left ( 1+z\right ) ^{m}}}f\left [ \frac{m-2}{2\left ( m-3\right ) } , \frac{1}{2};\frac{3m-8}{2\left ( m-3\right ) } , \frac{\omega _ { m}}{% \omega _ { m}-1}\left ( 1+z\right ) ^{3}\right ] \right ) $ % \end{tabular}\ ] ] case the solution reduces to @xmath65 case for @xmath57 we get @xmath66 ( identical to the case @xmath41 , * @xmath57 ) . for @xmath67 we reduce to the case @xmath42 * , * @xmath9 . * asymptotic limits . * the dr equation can be solved analytically in the limit of small or large @xmath16 . here we give these limits ; they will be used to produce accurate analytical fits to the numerical solutions . for small @xmath16 and @xmath41 we have @xmath68 \frac{dr_{s}}{dz}+\frac{m}{2}\left ( 1-\omega _ { m}\right ) r_{s}=0 \label{drgpiccoli}\ ] ] whose solution is @xmath69 } { a\left ( \omega _ { m},m\right ) } \exp \left\ { -\left [ 3\omega _ { m}+m\left ( 1-\omega _ { m}\right ) + 4\right ] \frac{z}{4}\right\}\ ] ] where @xmath70 ^{2}+24\omega _ { m}}}{4}\ ] ] or , to second order in @xmath16 @xmath71 z^{2}+o\left ( z^{3}\right ) \label{second}\ ] ] for completeness , we quote also the result in the case @xmath72 : @xmath73 } { a^{\alpha } \left ( \omega _ { m},m,\alpha \right ) } \exp \left\ { -\left [ 3\omega _ { m}+m\left ( 1-\omega _ { m}\right ) + 4% \right ] \frac{z}{4}\right\}\ ] ] where @xmath74 ^{2}+24\left ( 1-\alpha \right ) \omega _ { m}}}{4}.\ ] ] the limit to second order is identical to eq . ( [ second ] ) , which shows that for small redshift the degree of emptiness is not relevant . for large @xmath16 ( and @xmath75 ) the dr equation for @xmath76 reduces to @xmath77 if we define @xmath78 we obtain the equation @xmath79 g=0\ ] ] whose solution can be written in terms of bessel functions of first kind : @xmath80 $ & $ m\neq 0 $ \\ & \\ $ c_{2}-\frac{2}{5\left ( 1+z\right ) ^{5/2}}c_{1}$ & $ m=0$% \end{tabular } \right . \label{rlarge}\ ] ] where @xmath81 notice that the large-@xmath16 limit is always a constant : @xmath82 since in the limit of large @xmath16 the @xmath83 term in ( [ rlarge ] ) is negligible , we will consider in the following only the @xmath84 term . again for completeness we quote the analogous result for any @xmath72 : @xmath85\ ] ] * numerical fits . * the rest of this section focuses on the @xmath41 case , for which we do not have a closed solution . although in the application of the next section we employ the exact numerical solutions , it may prove practical to produce analytical fits . we use the asymptotic limits above to build an analytical fit to the full numerical solution , valid for all @xmath1 and all @xmath5 . we search a fit in the form @xmath86 where the function @xmath87 is a step - like curve chosen so as to interpolate from @xmath88 to @xmath89 as @xmath16 goes from @xmath89 to infinity , and @xmath90 interpolates similarly from @xmath89 to @xmath91 we chose @xmath92 $ \\ \\ $ t_{l}\left ( z\right ) = \frac{1}{2}\left [ 1+\tanh \left ( \frac{1}{\delta } \ln \left ( \frac{z}{z_{0}}\right ) \right ) \right ] $ % \end{tabular}\ ] ] so that the transition occurs at @xmath93 and @xmath94 sets its steepness . we have then three parameters @xmath95 to fit as functions of @xmath1 and @xmath5 . the result for @xmath84 in terms of a simple polynomial fit is @xmath96 for @xmath93 and @xmath94 we find convenient to use the following functional form : @xmath97 the values of the fit parameters for @xmath93 and @xmath94 are listed in table i. @xmath98 such fits are accurate to better than 5% over the range @xmath99 and @xmath100 as can be seen from fig . 2 , where we compare our fits to a sample of exact numerical solutions . as a cautionary remark , let us notice that the assumption of a constant @xmath101 over a very large range in redshift is certainly problematic . the results of the next section , however , are obtained in a relatively narrow range of redshifts , so that the approximation should be acceptable . now that we are in possess of the general angular diameter distance in quintessence cosmology we can apply it to real data . as a first application , in this section we use six observed time - delays to measure @xmath0 taking into account the presence of quintessence fields . let us first present the data . there are only seven lens systems with measured time - delays : b0218 + 357 ( biggs et al . 1999 ; lehr et al . 1999 ; patnaik porcas & browne 1995 ) , q0957 + 561 ( bar - kana et al . 1999 ; kundi et al . 1997 ) , he1104 - 1805 ( courbin , lidman & magain 1998 ; wisotzki , wucknitz , lopez & srensen 1998 ) , pg1115 + 080 ( bar - kana 1997 ; impey et al . 1998 ; schechter et al . 1996 ) , b1600 + 434 ( burud et al . 2000 ; koopmans , de bruyn , xanthopoulos & fassnacht 2000 ) , b1608 + 656 ( fassnacht c. d. et al . 1999 ; koopmans & fassnacht 1999 ) and pks1830 - 211 ( lehr et al . 1999 ; lovell et al . 1998 ; wiklind & combes 1999 ) . due to the image multiplicity , we have in total ten time - delays . the lens model we use , the isothermal lens of mao , witt and keeton ( mao , witt & keeton 2000 ) can not be adapted to q0957 + 561 and b1608 + 656 so we are left with five lens systems and six time - delays , as detailed in table ii . @xmath103 adopting the isothermal model , the relation between the distances of the deflector ( subscript @xmath104 ) and of the source ( @xmath105 ) and the time - delay between two images labelled 1 and 2 in terms of observable quantities is @xmath106 where @xmath107,\ ] ] @xmath108 is the angular distance between one of the two images and the center of the deflector and where @xmath109 to estimate @xmath0 we build the likelihood function @xmath110 , \ ] ] where we separated between quantities that contain theoretical parameters and purely observational quantities by definining the variables @xmath111 and where @xmath112 is the total error on @xmath113 , obtained by standard error propagation ( this error is dominated by the uncertainty on the angular positions @xmath108 ) . then we marginalize over @xmath114 ( defined in the range @xmath115 and @xmath116 respectively ) and produce the marginalized and normalized likelihood @xmath117 that represents the likelihood of @xmath0 ( normalized to unity by the constant @xmath118 ) given any possible perfect fluid quintessence model . we also evaluated @xmath119 and @xmath120 , marginalizing over the other variables , but the likelihoods we obtain are too flat to derive any interesting conclusion on the quintessence parameters . we also estimated the effect of imposing a gaussian prior on @xmath121 with mean 0.3 and standard deviation 0.1@xmath122 we compared the marginalized likelihood with the likelihood in the `` standard '' cases @xmath123 and @xmath8 ( pure cosmological constant ) , and @xmath60 and @xmath9 ( no quintessence ) . table iii summarizes the likelihood results : here 1,2,3@xmath37 stand for a probability of 68,95,99% , respectively ; the first four cases are marginalized over @xmath1 and @xmath5 , and the prior is the above mentioned gaussian on @xmath1 . the strongest effects are obtained in the empty beam case , @xmath41 , because in this case the quintessence term is dominating in the last term of the dr equation . the likelihood is shifted to lower values , in comparison to the two `` standard '' cases ( see fig . 5 ) , with an increase in the variance . for @xmath42 the shift is less evident and there is a degeneracy between marginalized likelihood and no quintessence likelihood . thus the dependency of @xmath0 for @xmath1 and @xmath5 is strongest in the empty beam case . with no prior we obtain @xmath124 @xmath125 already with six time - delays , the effect of a quintessence cosmology on the estimation of @xmath0 is therefore not negligible . a qualitative idea of how the method can perform in the future can be gained assuming that the same six time delays can be estimated with only a 10% error on the variable @xmath126 . in this case , we would get not only a better estimation of @xmath0 but also a substantial removal of the degeneracy with respect to @xmath1 and @xmath5 . this exercise illustrates the potentiality of the method towards a detection of quintessence and a distinction from a pure cosmological constant . the results of the simulation for @xmath41 are summarized in table iv and in figure 6 . if 70% or so of the total matter content is filled by a new component with negative pressure and weak clustering , all the classical deep cosmological probes are affected in some way . here we addressed the question of how the dyer - roeder distance changes when this new component , quintessence , is taken into account . we have shown that , particularly in the case of empty beam , the effect of the quintessence is to move the estimate of @xmath0 to lower values with respect to two standard models , and to increase the spread of the likelihood distribution . this is the first time , to our knowledge , that a full likelihood analysis of the almost entire set of time - delays available is performed . as a byproduct of our analysis , we produced fits for a large range of values of @xmath1 and @xmath5 accurate to within 5%@xmath122 the future prospects seem interesting for the time - delay method : with a not unrealistic increase in accuracy ( or in the number of time delays ) , the quintessence could be detected and distinguished from a pure cosmological constant , thanks to the deepness to which lensing effects are observable . an obvious improvement of our analysis , currently underway , is to investigate the dependence on @xmath128 and on curvature , producing a fully marginalized likelihood for @xmath0 . amendola l. , 2000 , phys . rev . d62 , 043511 , preprint astro - ph/9908440baccigalupi c. , perrotta f. & matarrese s. , 2000 , phys . rev . d61 , 023507 , preprint astro - ph/9906066balbi a. , et al . , 2000 , astro - ph/0005124barber a. , et al . , 2000 , astro - ph/0002437bar - kana r. , 1997 , apj , 489 , 21 , preprint astro - ph/9701068bar - 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ph/0004069	we calculate analytically and numerically the dyer - roeder distance in perfect fluid quintessence models and give an accurate fit to the numerical solutions for all the values of the density parameter and the quintessence equation of state . then we apply our solutions to the estimation of @xmath0 from multiple image time delays and find that the inclusion of quintessence modifies sensibly the likelihood distribution of @xmath0 , generally reducing the best estimate with respect to a pure cosmological constant . marginalizing over the other parameters ( @xmath1 and the quintessence equation of state ) , we obtain @xmath2 km / sec / mpc for an empty beam and @xmath3 km / sec / mpc for a filled beam . we also discuss the future prospects for distinguishing quintessence from a cosmological constant with time delays . epsf.tex -.5 in	7,881	226
the production of heavy quarkonium offers a unique perspective into the process of hadronization , because the creation of the relevant valence partons , the heavy quarks , is essentially perturbative . quarkonium production and decay have been the subject of a vast theoretical literature and of intensive experimental study , in which the effective field theory nonrelativistic qcd ( nrqcd ) @xcite has played a guiding role . nrqcd offers a systematic formalism to separate dynamics at the perturbative mass scale of the heavy quarks from nonperturbative dynamics , through an expansion in relative velocity within the pair forming the bound state . in nrqcd , the description of the relevant nonperturbative dynamics is reduced to the determination of a limited number of qcd matrix elements , accessible from experiment and , in principle , lattice computation . a characteristic feature of the application of nrqcd to production processes is the indispensible role of color octet matrix elements , which describe the nonperturbative transition of quark pairs in adjoint representation into quarkonia through soft gluon emission . an early success of nrqcd was to provide a framework for the striking tevatron run i data on high-@xmath2 heavy quarkonium production @xcite , and it has been extensively applied to heavy quarkonia in both collider and fixed target experiments . a wide - ranging review of theory and experiment for quarkonium production and decay has been given recently in ref . much of the analysis has been based on a factorization formalism proposed in @xcite , which offers a systematic procedure for the application of nrqcd to quarkonium production . it is fair to say , however , that , in contrast to quarkonium decay , fully convincing arguments have not yet been given for nrqcd factorization as applied to high-@xmath2 production processes @xcite . this omission may or may not be related to the current lack of confirmation for its predictions on quarkonium polarization at high @xmath2 @xcite . in this paper , we summarize progress toward the derivation of an appropriate factorization formalism for high-@xmath2 quarkonia , illustrating our considerations with results on infrared emission at next - to - next - to - leading order ( nnlo ) . at nnlo we find infrared divergences that do not fall precisely into the pattern suggested in ref . these divergences may , however , be incorporated into color octet matrix elements by a technical redefinition that makes the latter gauge invariant . it is not clear whether this pattern extends beyond nnlo , and we conclude that nrqcd factorization must be examined further for production processes . in any case , all our results are consistent with the factorization of evolution logarithms in the ratio of momentum transfer to quark mass from nonperturbative matrix elements @xcite . in the results presented below , the relevant infrared divergence is proportional to @xmath3 , where @xmath4 is the relative velocity of the heavy pair in the quarkonium rest frame . the rotational invariance of this result ( in the quarkonium rest frame ) makes it possible to match the long - distance behavior of an arbitrary cross section to an octet matrix element in a manner that does not depend on the directions of energetic final - state gluons . in other words , we may factorize the perturbative long - distance contributions from the short - distance cross section , and replace them with a universal nonperturbative matrix element that has the same perturbative long - distance behavior , just as proposed in @xcite and extended in @xcite . we begin our discussion with a brief review of nrqcd factorization at high transverse momentum . we discuss for definiteness the production of the @xmath5 and related heavy quarkonium states @xmath6 in leptonic or hadronic collisions , @xmath7 . to leading power in @xmath8 , which we assume to be a small parameter , production proceeds through gluon fragmentation . according to conventional factorization theorems @xcite , we have ( keeping only the gluon ) @xmath9 where generally we pick the factorization scale @xmath10 to be of the order of @xmath2 . in this expression , the convolution in the momentum fraction @xmath11 is denoted by @xmath12 , and we have absorbed all information on the initial state into @xmath13 . if we also assume nrqcd factorization , we have in addition to ( [ cofact ] ) , @xmath14+x}(p_t)\ , \langle { \mathcal o}^h_n\rangle\ , , \label{nrfact}\ ] ] where the @xmath15 are nrqcd operators , classified by powers of relative velocity and characterized by the various rotational and color transformation properties of the @xmath16 state @xmath17 $ ] . assuming both ( [ cofact ] ) and ( [ nrfact ] ) to hold , we conclude that the gluon fragmentation function is related to the nrqcd matrix element by @xcite @xmath18}(z,\mu , m_c ) \ , \langle { \mathcal o}^h_n\rangle\ , , \label{combofact}\ ] ] where @xmath19}(z,\mu , m_c)$ ] describes the evolution of an off - shell gluon into a quark pair in state @xmath17 $ ] , including logarithms of @xmath20 . in the following , we will study the fragmentation function itself , concentrating on infrared divergences at nnlo . first , however , we make some observations concerning the gauge transformation properties of nrqcd color octet matrix elements . production operators for state @xmath6 were introduced in ref . @xcite in the form ^h_n(0 ) = ^_n(0 ) ( a^_ha_h ) ^_n(0 ) , [ ondef1 ] where @xmath21 is the creation operator for state @xmath6 , and where @xmath22 and @xmath23 involve products of color and spin matrices , and at higher dimensions of covaraint derivatives . although the heavy ( anti)quark fields ( @xmath24 ) @xmath25 are all at the same space - time point ( here @xmath26 ) , the operator @xmath27 is not truly local , because the operator @xmath28 creates particle @xmath6 for out states , in the far future . in particular , operator - valued gauge transformations do not commute with the product @xmath29 in general . a consequence of nonlocality is that the right - hand side of eq . ( [ ondef1 ] ) is not gauge invariant in perturbation theory unless the individual product @xmath30 and @xmath31 are separately invariant . this is the case when the @xmath22 s specify color singlets , but not when they specify color octets . a related issue arises in the field - theoretic definitions of fragmentation functions @xcite , such as @xmath32 in eq . ( [ cofact ] ) above . it is resolved by supplementing the bi - local products of fields by nonabelian phase operators , or wilson lines : @xmath33 = \exp \left[-ig\int_0^\infty d\lambda\ , l \cdot a(x+\lambda l ) \right]$ ] . in contrast to parton distributions , moments of fragmentation functions do not result in expectation values of local operators @xcite , and the corresponding wilson lines are not guaranteed to cancel identically . for a color octet combination , the gauge field @xmath34 is given in adjoint representation , just as for a gluon fragmentation function , for which the operator @xmath35 multiplies the gauge covariant field strength @xmath36 . to be definite , and for convenience in relating nrqcd operators to the gluon fragmentation function , we will choose @xmath37 as a lightlike vector in the minus directon : @xmath38 . our gauge invariant redefinition of production operators in octet representation is now found by the replacement ( which we refer to as a gauge completion ) , ^h_n(0 ) ^_n , c(0 ) _ l^ [ 0,a]_cb(a^_ha_h ) _ l [ 0,a]_ba ^_n , a(0 ) , [ replace ] where we have exhibited the color indices of the octets . in perturbation theory , the wilson lines generate propagators of the form @xmath39 when they carry gluon momentum @xmath40 , and the gluons couple to the wilson line at vertices @xmath41 , with @xmath42 structure constants . these are precisely the same as the propagators and vertices of the eikonal approximation for the emission and absorption of soft gluons by an energetic gluon of momentum @xmath43 . we will use this correspondence below . it is worth noting that the replacement ( [ replace ] ) is nt really necessary if gauge dependence in the matrix elements and coefficient functions is infrared safe . this appears to be an implicit assumption in the discussion of ref . @xcite . as we shall see , however , although gauge dependence is certainly free of infrared singularities at next - to - leading order , at nnlo this is no longer the case . for this discussion , we study the perturbative expansion of the fragmentation function for @xmath45+x$ ] with @xmath46= c(p/2+q)\bar c(p/2-q)$ ] , a pair with total momentum @xmath47 and relative momentum @xmath48 , always projected onto a color singlet state . we can , of course , further project various spin and orbital angular momentum states @xmath17 $ ] as in eq . ( [ combofact ] ) . beyond lowest order , the computation of these diagrams tests nrqcd factorization , eq . ( [ combofact ] ) , which is verified to the extent that we can absorb , that is match , any infrared divergences into the perturbative expansion of nrqcd matrix elements @xmath49 @xcite . we will be interested in gluons whose energies are much below @xmath50 . for nrqcd factorization to hold these soft gluons must factorize from the higher energy radiation that generates logarithms of @xmath51 . because we need only the infrared structure of the diagrams from gluons of very low momentum , we may suppress overall color and combinatorial factors , as well as momentum factors that depend only on the scale of @xmath50 , including the lo fragmentation of the parent gluon , which is off - shell by @xmath52 , into the quark pair . = 5.5 cm [ fig1 ] . figure 1a shows the lowest order diagram @xmath53 , and figs . 1b , c show typical contributions with single soft gluon emission in the fragmentation function , all in cut diagram notation . both fig . 1b and 1c represent a class of four diagrams , in which gluons are attached to one of the two lines of the quark pair in the amplitude and its complex conjugate . in each diagram we may treat the quarks in eikonal approximation , in effect replacing them with time - like wilson lines , whose propagators and vertices are given by @xmath54 and @xmath55 , respectively , for quarks , with @xmath56 a color generator in fundamental representation and @xmath57 the time - like quark four - velocity . the corresponding approximation for antiquarks simply changes the sign at the vertex . the vertical lines in fig . 1b and 1c represent the quark pair final state , including a projection onto a color singlet configuration . the full fragmentation function is found by cutting the remaining lines in all possible ways . when the gluon is cut in fig . 1b , in particular , the resulting diagrams have an uncancelled infrared pole in dimensional regularization , beginning at order @xmath58 when the gluon momentum @xmath59 . this gluonic contribution , however , which involves only interactions between the pair , may be matched straightforwardly to a color octet matrix element , with which it shares an identical topology @xcite . the divergent part of fig . 1b may be isolated by evaluating the four relevant diagrams in eikonal approximation and then expanding them in @xmath60 . alternately , we may expand the momentum - space integrand , isolating the @xmath3 term before the integration . in @xmath61 dimensions , and in the quark pair rest frame , the latter approach leads readily to an expression in terms of the total and relative momenta @xmath47 ( @xmath62 ) and @xmath48 , respectively , @xmath63 \nonumber\\ & & \hspace{30 mm } \times \ [ q^\nu ( p\cdot k ) - ( q\cdot k ) p^\nu ] \frac{1}{[(p\cdot k)^2]^2 } \nonumber\\ & = & { 16\over 3}\ , { \alpha_s\over \pi}\ , { \vec{q}\ , { } ^2\over p^2}\ , { 1\over -\varepsilon } + \dots\ , , \label{nlopt}\end{aligned}\ ] ] where we have exhibited the infrared pole , regularized for @xmath64 . this infrared pole , along with an appropriate color trace , appears at nlo as a multiplicative factor times the lowest - order fragmentation function , as found , for example , in refs.@xcite . in the integral , the terms in square brackets correspond to the lowest - order vertices for the operator @xmath65 , with @xmath66 the field strength . in the quarkonium rest frame , this corresponds to an electric dipole transition from octet to singlet color states @xcite . an advantage of this expansion is that , because the field strength decouples from scalar polarized lines , all singularities associated with gluon momenta collinear to @xmath67 vanish on a diagram - by - diagram basis , and double poles in @xmath68 are entirely absent in the calculation . by the same token , the integral in ( [ nlopt ] ) is gauge invariant in the polarization sum for the final - state gluon of fig . finally , in fig , 1c , the soft gluon connects the pair to the wilson line . in covariant gauges , this diagram has double and single infrared / collinear poles in dimensional regularization . these poles , however , cancel against analogous contributions from the corresponding virtual diagram . imaginary parts of the virtual diagrams , of course , cancel in the fragmentation function , which is real . in practical terms , the contribution of fig . 1c plus its virtual counterpart contributes solely to the perturbative factor @xmath19}(z,\mu , m_c)$ ] in eq . ( [ combofact ] ) . we note that the infrared behavior of all of the diagrams of fig . 1 is common between the fragmentation function and a generic cross section in which a high-@xmath2 gluon of momentum @xmath37 recoils against the heavy quark pair . indeed , the same cancellation mechanism for the infrared divergences of fig . 1c is referred to specifically in ref . @xcite as an essential step in the justification of nrqcd factorization for production cross sections . to the extent that all the infrared poles of diagrams that do not share the topology of nrqcd matrix elements cancel , the matching of perturbative infrared poles with those of the effective theory is ensured . the result of this cancellation is referred to as topological factorization @xcite . the same considerations that lead to the inclusion of nonabelian phases to the fragmentation function , however , suggest that the cancellation of non - factored soft gluons may be nontrivial , and should be checked beyond nlo . of course , a full nnlo calculation of either cross sections or fragmentation functions would be daunting . fortunately , the analysis of the relevant infrared behavior at nnlo requires only the eikonal approximation , and is therefore a much more manageable , although still extensive , task , to which we now turn . we can readily generalize our nlo discussion to nnlo . once again , because we are interested in gluons whose energies are much less that @xmath50 , we can adopt the eikonal approximation , and neglect all dynamics at the scale of the quark masses . in nrqcd , these approximations are relevant to gluons of order @xmath69 , with @xmath70 the relative velocity , but of course in the perturbative calculation the gluon momentum goes to zero . we note that for octet - singlet transitions at nnlo , we need not include virtual gluon exchange between the quark and antiquark , except for diagrams that are already topologically factorized . as a result , we will not need not consider violations of the eikonal approximation from momentum regions at or below the scale @xmath71 , or the resulting @xmath72 singular behavior . representative diagrams for the fragmentation function are shown in fig . 2 . the full infrared fragmentation function is again generated by taking all allowed cuts of the remaining lines of each such diagram . we are concerned only with diagrams that connect octet to singlet quark states , and which are not topologically factorized , since these are the potential sources of nonfactoring behavior in both the fragmentation function and related cross sections . we recall once more that the familiar argument for nrqcd factorization is based on the conjecture that all infrared regions in these diagrams cancel after this limited sum over cuts @xcite . in fact , we shall see that this is the case at nnlo only if we employ the gauge - completed definitions for nrqcd matrix elements , as in eq . ( [ replace ] ) above . [ fig2 ] as at nlo , we will concern ourselves here with infrared behavior at order @xmath3 , which may be computed by evaluating each diagram in eikonal approximation and then expanding their sum in @xmath73 , or by expanding the diagrams first , as in eq . ( [ nlopt ] ) . noting that the two approaches give the same answer , we summarize the results here , and provide details of the calculations elsewhere @xcite . the individual classes of diagrams in fig . 2a and 2b , for which two gluons are exchanged between the quarks and the wilson line , satisfy the infrared cancellation conjecture of ref . @xcite , by summing over the possible cuts and connections to quark and antiquark lines , as do diagrams that have three gluon - eikonal vertices on the quark pair and one on the wilson line . for the class of diagrams related to fig . 2c , however , with a three - gluon interaction , this cancellation fails . expanding again to second order in the relative momentum @xmath48 , the full contribution from fig . 2c , found by cutting the gluon line @xmath74 and the wilson line can be written by analogy to eq . ( [ nlopt ] ) as @xmath75 \nonumber\\ & & \hspace{40 mm } \times \ [ q^\mu ( p\cdot k_1 ) - ( q\cdot k_1 ) p^\mu ] \ , ~ [ q^\nu ( p\cdot k_1)-(q\cdot k_2 ) p^\nu]~ \nonumber \\ & & \hspace{40 mm } \times \frac{1}{[p\cdot k_1 + i\epsilon]^2~ [ p\cdot k_2 - i\epsilon ] ^2 } \nonumber\\ & & \hspace{40 mm } \times \frac{1}{[k_2 ^ 2 - i \epsilon]~[(k_2- k_1)^2 - i\epsilon]~ [ l\cdot ( k_1 - k_2 ) - i\epsilon ] } \ , , \label{nnlopt}\end{aligned}\ ] ] where @xmath76 $ ] represents the momentum part of the three - gluon coupling . as in eq . ( [ nlopt ] ) , we have suppressed color factors and momentum dependence at the scale @xmath50 . as observed above , the field - strength vertices eliminate collinear singularities on a diagram - by - diagram basis . the leading singularities in ( [ nnlopt ] ) and related diagrams are therefore never worse than @xmath77 . summing over all such contributions , however , we find a noncancelling real infrared pole in the fragmentation function , which may be written in invariant form as @xmath78\ , . \label{gn}\ ] ] in the rest frame of heavy - quarkonium ( @xmath79 ) , this becomes simply @xmath80 where @xmath4 is the relative velocity of the heavy quark pair . we will give the extension of this result to all powers in @xmath70 elsewhere @xcite . ( [ gn1 ] ) shows explicitly the breakdown of the simplest topological factorization of infrared dependence at nnlo . its presence implies that infrared poles would appear in coefficient functions at nnlo and beyond when the factorization is carried out with octet nrqcd matrix elements defined in the conventional manner , eq . ( [ ondef1 ] ) . on the other hand , when defined according to its gauge - completed form ( [ replace ] ) , each octet nrqcd matrix element itself generates precisely the same pole terms given in ( [ gn1 ] ) above . thus , at least at nnlo and to order @xmath3 , nrqcd factorization can accomodate these corrections . we note , however , that at this order the nnlo correction is independent of the direction @xmath81 of the wilson line , while lorentz invariance alone would seem to allow a correction of the form @xmath82 . the presence of poles with this coefficient could indicate problems with factorization , that is , with matching the poles of the fragmentation function to those of the perturbative cross section . these could arise when the integration over @xmath48 is not rotationally invariant , as might happen for polarized final states , and/or final states with gluons of energies of order @xmath50 in the rest frame of @xmath6 , since their directions are arbitrary . we are unable at this stage to rule out the occurance of such terms at higher orders in the strong coupling , or in higher powers of @xmath3 . we have discussed the gauge invariance properties of color octet matrix elements in nrqcd as they appear in fragmentation functions . gauge invariance and factorization require that they include nonabelian phase operators to match otherwise nonfactoring infrared divergences beginning at nnlo . we have shown that at nnlo there are infrared ( not collinear ) poles from fragmentation diagrams that are not topologically equivalent to conventional nrqcd operators . we have observed that rotational invariance in the quarkonium rest frame is an important consistency condition for the possibility of nrqcd factorization in terms of the gauge - completion of nrqcd operators . should rotational invariance fail at higher orders , we anticipate that it will be necessary to complement the nrqcd classification of nonperturbative parameters for quarkonium production with matrix elements of additional operators , involving nonlocal products of the field strengths encountered above . finally , although our explicit calculations are carried out in the eikonal approximation for gluons of energies far below @xmath50 , it is not difficult to verify that these gluons do not interfere with the generation of standard evolution for the @xmath44 fragmentation functions . the arguments for this result will be given elsewhere , along with details of the calculations described above . this work was supported in part by the national science foundation , grants phy-0071027 , phy-0098527 and phy-0354776 , and by the department of energy , grant de - fg02 - 87er40371 . we thank geoff bodwin for emphasizing the importance of this topic , and for many useful discussions . collins and g. sterman , nucl . b185 , 172 ( 1981 ) ; + j.c . collins , d.e . soper and g. sterman , nucl . b261 , 104 ( 1985 ) ; b308 , 833 ( 1988 ) ; + g.t . bodwin , phys . d31 , 2616 ( 1985 ) erratum ibid . d34 , 3932 ( 1986 ) .	we explore the role of soft gluon exchange in heavy quarkonium production at large transverse momentum . we find uncanceled infrared poles at nnlo that are not matched by conventional nrqcd matrix elements . we show , however , that gauge invariance and factorization require that conventional nrqcd production operators be modified to include nonabelian phases or wilson lines . with appropriately modified operators , factorization is restored at nnlo . we also argue that , in the absence of special cancellations , infrared poles at yet higher orders may require the inclusion of additional nonlocal operators , not present in the nrqcd expansion in relative veloctiy . yitp - sb-05 - 01 + * fragmentation , factorization and infrared poles + * in heavy quarkonium production gouranga c. nayak@xmath0 , jian - wei qiu@xmath1 and george sterman@xmath0 _ @xmath0c.n . yang institute for theoretical physics , stony brook university , suny + stony brook , new york 11794 - 3840 , u.s.a . _ _ @xmath1department of physics and astronomy , iowa state university + ames , ia 50011 , u.s.a . _ pacs numbers : 12.38.bx , 12.39.st , 13.87.fh,14.40gx	6,874	401
there are a few facts about the universe we live in that we have now become certain of . one such fact is that our universe is expanding . in 1929 edwin hubble discovered that distant galaxies appear to be receding from us with velocity proportional to their distance ( see fig . [ hubble ] ) : @xmath0 where @xmath1 is the famous hubble constant , whose value is @xcite . this observation marks the discovery that we live in an expanding universe . because the above relation is linear it can be deduced that there is no centre for this expansion ( called hubble expansion ) ; distant galaxies are receding away from any observer , wherever placed , according to the above relation . the universe expansion , as it is perceived today , does not correspond to galaxies travelling through space away from each - other but it is due to an expansion of space itself , between the galaxies , which is responsible for galaxies being pulled away from each - other . hence , according to hubble law above , the expansion of the universe is self - similar ; the three - dimensional equivalent of a photograph magnification . such a uniform expansion suggests that the content of the universe is also uniformly distributed . indeed , galaxy survey observations show that the distribution of galactic clusters and superclusters becomes uniform on scales above 100 mpc , which is about a few percent of the size of the observable universe ( e.g. see fig . [ 2df ] ) . that is to say that structure in the universe is not of fractal form ; one does not find bigger structures the larger the distance scale considered . instead , the largest structures correspond to bubble and filament like matter distributions comprised by galactic superclusters ( see fig . [ 2df ] ) whose characteristic size ( @xmath2 100 mpc ) is much less than the cosmological horizon scale which marks the limit of our observational capability . the above correspond to observational support for the so - called _ cosmological principle _ , which simply states that , on large scales , `` the universe is homogeneous and isotropic '' , i.e. invariant under translations and rotations . this means that there are no special points or directions in the universe , no centre , no edges and no axis of rotation . using the cosmological principle it has been possible to study the global dynamics of the universe as dictated by einstein s general relativity theory . according to general relativity there is no stable spacetime solution which contains uniformly distributed matter and radiation and remains static . hence , general relativity demands that the universe is either expanding or collapsing . this agrees well with the observed hubble expansion . now , going backwards in time , we would expect the universe to be contracting and its energy density increasing . this means that matter in the early universe is predominantly relativistic , with a planckian spectrum of high black - body temperature . furthermore , the universe content is fully ionised with matter being strongly coupled to radiation through thomson scattering . as the universe expands , its density is diluted and its temperature decreases . at some moment , the plasma filling the universe cools down enough to allow neutral atoms to form . when this occurs , the scattering of radiation from matter is drastically reduced , which means that the universe content becomes transparent , allowing radiation to travel free , similarly to what happens at the photosphere ; the visible surface of the sun . at that time radiation is said to `` decouple '' from matter . this released radiation fills all space and survives until the present time . indeed , in 1978 penzias and wilson won the nobel prize because they discovered this so - called cosmic microwave background ( cmb ) radiation . as shown in fig , [ cmb ] , the cmb has an almost perfect blackbody spectrum peaked at microwave wavelengths corresponding to temperature of @xmath3 k. the distribution of the cmb is extremely uniform with minute variations in its temperature at the level . such uniformity is another piece of supporting evidence for the cosmological principle . the existence of the cmb shows that the early universe was hot and dense and a lot different from the universe at present . this fact , along with the observation that the universe is expanding , agrees well with general relativity which suggests that our expanding universe has finite age . this means that , when travelling backwards in time , the reversed expansion is such that all distances shrink in a self - similar manner to zero in finite time . that initial moment of infinite density has been dubbed the big bang ; the onset of time itself . the latest dynamical estimate for the age of the universe is @xmath4 gyrs @xcite , which is in good agreement with estimates of the ages of the oldest astrophysical objects such as , for example , globular star clusters @xcite . hence , cosmologists today operate in the framework of a standard model , which considers that the universe began its existence 14 billion years ago , passed through a hot period of ultra - high density , and has continued to cool down until the present because of the expansion of space . this back - bone history of the universe is called the hot big bang and has been observationally confirmed as far back as the time when the universe was no more than a few seconds old . despite the observational support for the hot big bang the fact is that , fundamentally , the cosmological principle is incompatible with a finite age for the universe . this paradox is the so - called _ horizon problem _ and has to do with the apparent uniformity of the universe over distances which are causally unconnected . for example , the cmb appears to be correlated at regions beyond the causal correlation scale ( the so - called particle horizon ) ; it appears to be in thermal equilibrium and at a preferred reference frame . general relativity does not single out a preferred reference frame , which would imply that doppler shifts in the frequency of the cmb over causally unconnected regions ( with different centre of mass frames ) would result in @xmath5 . in contrast , observations suggest that matter at these regions does lie at the same reference frame when the cmb is emitted and the fractional perturbation of the cmb temperature is much smaller . one can understand the problem schematically by considering the spacetime diagram in fig . [ horizon ] . the slanted lines depict signals which travel with the speed of light , which is the fastest possible propagation through space . along the time axis the present time @xmath6yrs is depicted at the tip of our past light - cone , that is the region of spacetime from which we can receive signals and lies below the large slanted lines in the diagram . the horizontal line corresponds to the time of the decoupling of radiation from matter , when the cmb was emitted at @xmath7yrs . the small triangles correspond to the light - cones at the time of decoupling and designate the regions which are in causal contact when the cmb was emitted ; at @xmath8 an observer can not be affected by events occurring beyond his / her light - cone . because @xmath9 our field of vision today includes a large number of regions which were beyond causal contact at the time of the cmb emission and yet these uncorrelated regions appear to be in thermal equilibrium at the same temperature . + the most compelling mechanism to date which overcomes the horizon problem is the theory of _ cosmic inflation _ originally suggested by guth @xcite and independently by starobinsky @xcite . in a nutshell , cosmic inflation can be defined as a brief period of superluminal expansion of space in the early universe . general relativity allows inflation because the latter does not correspond to displacement of matter or energy through space with velocity faster than lightspeed . instead , it is space itself which is expanding faster than light can travel . indeed , inflationary expansion is readily obtained by general relativity if certain conditions are satisfied . how does inflation overcome the horizon problem ? well , inflation produces correlations beyond the causal horizon by expanding an initially causally connected region to a size large enough to encompass the observable universe . the region which corresponds to the observable universe is supposed to lie within the light - cone before the onset of inflation . being initially causally self - correlated it can become uniform by interacting with itself . after it is inflated to super - horizon distances this original uniformity sets up uniform initial conditions after inflation , even if the original region corresponds to many causally - disconnected regions . this is depicted schematically in fig . [ horizon ] , where during inflation the growth of an initially causally correlated region is shown to be faster than lightspeed so as to engulf the entire field of vision today . hence , we see that , provided it lasts long enough , cosmic inflation can impose the cosmological principle as an initial condition for the hot big bang . however , if the cosmological principle was exact then there would be no structure in the universe ; no galaxies , no stars with planets harbouring intelligent life . all these structures require the uniformity of the universe to be violated . if not , the universe today would be filled only with the cmb radiation and a thinned out gas at temperature @xmath10 , evenly distributed through space . our existence demands that the cosmological principle is only approximately respected . it so happens that cosmic inflation can provide also a small violation of the cosmological principle . this is due to quantum effects during inflation , which , as explained below , can introduce a tiny variation in the density @xmath11 of the universe , called the _ primordial density perturbation _ @xmath12 . + this density perturbation reflects itself onto the cmb through the so - called sachs - wolfe effect @xcite , which describes how the cmb photons become redshifted when crossing regions of density higher than average ( so - called overdensities ) because they lose energy while struggling to exit from the gravitational potential wells of these growing overdensities . hence , variations in the density of the universe cause perturbations in the apparent temperature of the cmb radiation : @xmath13 indeed the observations ( e.g. see fig . [ wmap ] ) suggest that @xmath14 @xcite , which , albeit tiny , turns out to be enough to explain the formation of structures in the universe , such as galaxies and galactic clusters . starting from an initial density perturbation , structure formation proceeds through the process of _ gravitational instability_. the latter amounts to intensifying , in a runaway manner , the contrast between overdense and underdense regions and is based on the fact that overdensities grow more massive by attracting matter from surrounding underdensities , depleting them even further . indeed , numerical simulations have shown that density perturbations of magnitude @xmath15 at @xmath8 suffice to generate the observed structures given 14 billion years of growth . therefore , it seems that inflation can successfully impose both the cosmological principle onto the universe and the deviations from it , which are necessary for structure formation . inflation produces the primordial density perturbation through a process called _ particle production _ that arises from considering the superluminal expansion of space in conjunction with the notion of the quantum vacuum . we are all familiar with the concept of the classical vacuum . a box filled with vacuum is a box which is totally empty of matter and energy , i.e. the energy in the box is . however , in quantum theory , this can not be true and the box has to have . this is due to _ heisenberg s uncertainty principle _ , which , for energy , can be expressed mathematically as @xmath16 the meaning of the above expression is that the energy of a closed system ( such as our box ) can not be precisely determined but it has to fluctuate by an amount @xmath17 , which , in a given time period @xmath18 , has to satisfy the above constraint . the crucial point is that the reduced planck s constant is planck s constant . ] @xmath19 featured above , is small but positive , which means that , for any finite time interval , @xmath17 has to be non - zero . in fact , the smaller the period of time considered the larger the fluctuation of the energy has to be . hence , the uncertainty principle amounts to a controlled violation of energy conservation . this violation manifests itself as the brief appearance of pairs of particles and anti - particles , the reason being that other quantities such as the electric charge or other quantum numbers , are indeed conserved . consequently , the quantum vacuum is not empty but , instead , it is filled with constantly appearing and disappearing pairs of so - called virtual particles and anti - particles ( see fig . [ qv ] ) . the energy of these virtual particles is the so - called vacuum ( or zero - point ) energy . the above may sound like a bunch of flowery ideas , but it so happens that there is experimental proof of the existence of virtual particles . this is the famous casimir experiment , first realised by casimir and polder in 1948 @xcite . consider a pair of parallel conducting plates , not electrically charged and not connected through an electrical circuit . the plates are just standing in empty space ( i.e. in vacuum ) , close to each - other as shown in fig . [ casimir ] . classically there is no force on the plates except from their mutual gravitational attraction , which is extremely weak and can be ignored . now , let us consider the existence of virtual particles in the space between the plates and in particular , the appearance of virtual photons . photons are the quanta of the electromagnetic field . they can be thought of as wave packets of electromagnetic radiation whose energy @xmath20 is related to their wavelength @xmath21 as @xmath22 where @xmath23 is the speed of light in vacuum . the existence of photons has been confirmed by the photo - electric phenomenon , whose particle interpretation awarded einstein his nobel prize in 1921 . now , since the plates are conducting they can not allow a non - zero electric field inside them . hence , virtual photons appearing between the plates can only have a discrete spectrum of energies , corresponding to wavelengths that satisfy the constraint @xmath24 where @xmath25 is a positive integer and @xmath26 is the distance between the plates . the above constraint ensures that the amplitude of the electric field remains zero at the surface of the conducting plates , as depicted in fig . [ casimir ] . however , the virtual photons appearing outside the plates are not constrained by the above relation and can have any wavelength . this means that many more energies of virtual photons are allowed outside the plates compared to the discrete spectrum allowed between the plates . thus , the vacuum energy density in the space between the plates is different from the one outside the plates . this difference ( gradient ) of vacuum energy gives rise to a force @xmath27 , which has been measured and found to be in precise agreement with the predictions of quantum electrodynamics , confirming thereby the existence of virtual photons @xcite . before combining the notion of the quantum vacuum with cosmic inflation we need to briefly discuss black holes . black holes are predicted by general relativity and are thought to exist astrophysically , for example in the centres of galaxies . a black hole is an extremely compact object with a gravitational field which is locally so intense that anything approaching close enough can never escape . indeed , a black hole is surrounded by an _ event horizon _ , which is the surface on which the escape velocity from the black hole is equal to the speed of light . this means that one would need to travel with the speed of light away from the black hole simply to remain on the event horizon . within the horizon the gravitational attraction is so strong that , even when travelling with lightspeed , one can not avoid being pulled inwards . since no matter or energy can travel faster than the speed of light the event horizon forms a boundary on the causal structure of spacetime in the sense that events within the event horizon can not affect events outside it . one can think of the event horizon as a surface which is permeable from one direction only ( from the outside ) . a classical black hole , being enclosed within an event horizon , can only absorb matter and energy which is captured by its pull from its environment . in - falling matter increases the gravitational field of the black hole , which in turn increases the radius of its event horizon as the gravitational pull further extends its effect in the surrounding space . hence , a classical black hole can only increase in mass and size . such was the understanding of black holes until hawking studied them in conjunction with the quantum vacuum . hawking considered the appearance of virtual particle pairs in the vicinity of the event horizon . suppose that one of the particles of the pair falls within the event horizon as shown in fig . the other part of the pair may follow it , but since it is still outside the event horizon , it has a non - zero ( albeit tiny ) probability of escaping from the black hole , whereas , by definition of the event horizon , the particle within it has no chance of escaping . thus , since pairs of virtual particles are constantly appearing from nothing ( through the controlled violation of energy conservation by the uncertainty principle ) in the vicinity of the event horizon , the net effect of the above possibility is that a tiny fraction of virtual particles does escape from the black hole . such particles can not meet their counterparts of the original pair because the latter have fallen within the event horizon and can never escape . therefore , these particles can not annihilate with their partners ; their conserved quantum numbers can not disappear . consequently , the virtual particles which avoided falling within the event horizon ( while their partners did not ) survive and become real particles , which can be detected by distant observers @xcite . stable real particles can survive indefinitely . for stable particles which used to be virtual particles that escaped from the event horizon of a black hole , this means that @xmath18 in the uncertainty relation ( that allowed their existence ) can become very large . thus , we find @xmath28 , i.e. over large time periods , energy conservation has to be restored . since the particles which escaped from the black hole , can be detected by an observer to have a positive energy @xmath29 , with respect to this distant observer , the pair partners of these particles ( the ones which did fall into the black hole ) must have energy @xmath30 , where @xmath31 . hence , from the viewpoint of the distant observer , the black hole appears to be absorbing particles of negative energy which equals in magnitude the positive energy of the particles radiated away having escaped from the event horizon . receiving negative energy the black hole reduces in mass and its event horizon reduces in size , while emitting energy ( equivalent to its mass reduction ) in the form of so - called _ hawking radiation_. hawking found that the emitted radiation has a black - body spectrum corresponding to a characteristic temperature , called the _ hawking temperature _ @xmath32 @xcite . surprisingly , @xmath32 was found to increase the smaller the black hole becomes , i.e. the emission of hawking radiation intensifies as the black hole shrinks . this runaway behaviour ends up in the evaporation of the black hole @xcite . turning this around , the emission of hawking radiation from supermassive astrophysical black holes , such as the ones in the centres of galaxies , is negligible and is overwhelmed by absorption of positive energy from their environment , such as surrounding matter or even the cmb radiation . what does black hole evaporation have to do with cosmic inflation ? well , it so happens that the cosmological horizon ( the range of causal correlations ) during inflation corresponds to an event horizon of an `` inverted '' black hole ; a black hole `` inside - out '' , centred at the observer . this can be understood by considering that , during inflation , matter inside the horizon is being `` sucked out '' by the superluminal expansion of space , in analogy with the fact that nearby matter is being `` sucked in '' by a black hole . in this manner the virtual particle pairs of the quantum vacuum can be pulled outside the horizon before they have a chance to annihilate as shown in fig . [ superlum ] . in this case , being over superhorizon distance apart , they are beyond causal contact and can never find each - other to annihilate . thus , they cease being virtual particles and become real particles instead . this is why this process is called _ particle production _ @xcite . it is important to state here that the existence of an event horizon suffices to have particle production . it is the division of the causal structure of spacetime , enforced by the event horizon , which gives rise to the particle production process ; the separation of the virtual particle pairs . thus , the event horizon , once it exists , can be seen to emit hawking radiation without the need of a black hole or any concentration of mass . during inflation , the event horizon is filled with hawking radiation ( since it corresponds to the outside of the `` inverted '' black hole ) . since , the horizon is centred at the observer and we can put an observer anywhere in space , this means that all space is filled with hawking radiation , i.e. particle production occurs everywhere once we have superluminal expansion . now , in quantum theory , particles correspond to waves with an associated de broglie wavelength . for example , a massive particle of mass @xmath33 is characterised by a wavelength @xmath34 . this dual nature of particles is hard to visualise , although we are all familiar with many `` particle - like '' properties of waves , such as the reflection of an incident sea wave on a concrete wall . it is also easy to accept that waves , like particles , carry energy and momentum , e.g. radio - waves , laser beams or tsunami waves . particles correspond to waves travelling on an otherwise smooth sea , which we call a field ( the particle is part of the field as the sea wave is part of the sea ) , for example photons travel through the sea of the electromagnetic field . virtual particles of the quantum vacuum , therefore , correspond to spontaneously arising ripples on the calm sea of their corresponding fields . this is why they are also referred to as quantum fluctuations of these fields . based on the above , the particle production process during inflation can be also understood as the stretching of quantum fluctuations by the superluminal expansion of space , to superhorizon distances . this stretching transforms the quantum fluctuations into classical perturbations of the fields . one can imagine them as mountains and valleys , corresponding to classical variations of the field in question over superhorizon distances . a variation of the values of fields gives rise to a variation of their energy density . this , therefore , is the way that particle production during inflation produces variations of the density of the universe which generate the primordial density perturbation , that sources the formation of structures in the universe @xcite . schematically , we can write @xmath35+{\sf inflation } \;\rightarrow\ ; \frac{\delta\phi}{\phi}\;\rightarrow\ ; \frac{\delta\rho}{\rho } \;\rightarrow\;{\sf galaxies}\ ] ] where @xmath36 is one of the fields in question , whose typical variation during inflation is given by the hawking temperature @xmath37 . thus , according to this scenario , all structures in the universe , such as galaxies , stars , planets and ultimately ourselves , originated as quantum fluctuations ( like the ones we observe in the lab through the casimir experiment ) during a period of superluminal expansion of space . do we have any observational support for this amazing scenario ? indeed we do . high precision observations of the cmb temperature perturbations , which are due to the primordial density perturbation , have revealed a stunning agreement with the predictions of inflation . [ peaks ] depicts an analysis of these temperature perturbations in spherical harmonics , which follows the theoretical prediction of inflation ( shown by the solid lines ) with remarkable consistency @xcite . it was this kind of analysis that led to the collapse of a rival theory for the generation of the density perturbations ( that of cosmic strings ) , which did not predict a sequence of peaks in the angular spectrum @xcite ( they are clearly observed in fig . [ peaks ] ) . the simplest and most natural realisations of inflation suggest that the hawking temperature @xmath32 remains roughly constant during the inflationary phase of the universe . this means that the variations of the fields , produced by the particle production process , are of the same amplitude even though they may correspond to much different length - scales , because the quantum fluctuations which exit the horizon early are stretched to much larger distances than those which exit the horizon later on . as a result we expect that the spectrum of the density perturbations is almost scale - invariant , i.e. their amplitude is independent of the length - scale considered . indeed , parameterising the scale dependence of the perturbation spectrum on the wavenumber @xmath38 as @xmath39 , we expect that the spectral index would be @xmath40 , where @xmath41 with @xmath42 being the corresponding length - scale . the latest observations suggest that @xmath43 @xcite , which is very close to scale - invariance exactly as inflation predicts . the deviation from exact scale invariance was also expected , and has to do with the fact that the inflationary expansion has to end well before the first second of the universe history . another feature of the observed cmb temperature perturbations which agrees well with inflation is the fact that they appear to be predominantly gaussian . the nature of quantum fluctuations is stochastic ( that is to say random ) which suggests that @xmath44 has a gaussian bell - shaped distribution centred at @xmath32 . thus , inflation naturally predicts gaussian perturbations , as is indeed observed . the most important feature of the observations which supports the inflationary scenario for the generation of the primordial density perturbation is the following . the main peak in the angular power spectrum in fig . [ peaks ] corresponds to the scale of the cosmological horizon at the time @xmath8 of the emission of the cmb . as shown in fig . [ peaks ] , this peak corresponds to the multipole moment with @xmath45 . in the bottom half ( the te part ) of fig . [ peaks ] , we see that there is an anti - correlation feature at multipole moments significantly lower than the one of the main peak , which correspond to superhorizon scales at the time of the emission of the cmb . this means that the density perturbations show correlations beyond the causal horizon , a feature that only the superluminal expansion of an inflationary phase can explain . looking into the particle production process in more depth one has to wonder which kind of fields are the ones whose quantum fluctuations are responsible for the generation of the primordial density perturbation . until recently , all mechanisms considered the use of scalar fields for the density perturbations . scalar fields are hypothetical spin - zero fields , which are characterised by one degree of freedom , that is a single value at a given point in space . scalar fields are ubiquitous in theories beyond the standard model of particle physics , such as supersymmetry ( the scalar partners of the observed fermion fields ) or string theory ( the so - called moduli fields , which parametrise the size and shape of the hypothesised extra dimensions ) . however , it has to be stressed that no scalar field has ever been observed . this means that designing models using unobserved scalar fields undermines their predictability and falsifiability , despite the high precision of the recent cosmological observations . the standard model of particle physics does contain one scalar field , which is the only piece of it that has not been experimentally confirmed . this is the famous higgs field , thought to be responsible for the masses of standard model particles ( which are all the known particles such as electrons , quarks and so on ) . observing the quanta of this field , the higgs bosons , is one of the primary objectives of the large hadron collider ( lhc ) , which is about to become operational in the coming months in cern , geneva . observation of the higgs particles , among other things , will substantiate the claim that scalar fields do exist and are not merely mathematical constructions . however , what if the higgs bosons are not found by the lhc ? do we have an alternative proposal for the generation of the density perturbations from inflation ? can we form the galaxies without scalar fields ? recently , a proposal alternative to scalar fields has been put forward , which uses vector boson fields instead @xcite . these are spin - one fields , characterised by three degrees of freedom corresponding to their magnitude , direction and orientation at every point in space . in contrast to scalar fields , vector bosons are indeed observed . the standard model of particle physics contains four vector boson fields : one is the familiar photon , i.e. the quantum of the electromagnetic field , and then we have the three massive vector bosons of the weak nuclear force - decay , which transforms neutrons to protons by emission of electrons , otherwise known as @xmath46-radiation . ] , which are the so - called z - boson and the w@xmath47-bosons , that have been observed in cern in the 80s . vector bosons were not originally considered as candidates for the particle production process in inflation because of two generic obstacles , which were thought to inhibit their suitability for this role . the major obstacle has to do with the inherent anisotropic nature of vector fields . inflation would homogenise a vector field as it does with everything else in the universe when it solves the horizon problem . now , if a vector field is to generate or even affect the primordial density perturbation it needs to dominate ( or nearly dominate ) the content of the universe at some time so that it can affect the universe expansion rate . however , a uniform vector field ( homogenised by inflation ) is , in general , anisotropic because it picks up a preferred direction in space . if such an anisotropic contribution to the density of the universe became dominant it would result in anisotropic expansion which would , ultimately , be in conflict with the predominant isotropy of the uniform cmb radiation . however , it can be shown that this problem can be overcome if the vector field is undergoing rapid coherent oscillations when it dominates the universe @xcite . the equation of motion for a homogeneous massive ( and abelian ) vector field in the expanding universe is @xmath48 where the dot denotes time derivative , @xmath33 is the mass of the field and @xmath49 is the fractional rate of the universe expansion ; the so - called hubble parameter , whose value today is the hubble constant @xmath1 featured in the hubble law . originally the vector field is taken to be subdominant , i.e. its contribution to the density budget of the universe is negligible . after inflation the universe expansion rate is gradually decreasing so that the value of the hubble parameter @xmath49 diminishes . eventually , we have @xmath50 , which means that one can ignore the second term in the above equation of motion ( the so - called friction term ) rendering the equation similar to the one of a harmonic oscillator . hence , the homogeneous massive vector field eventually undergoes harmonic oscillations , with frequency @xmath33 , much larger than the inverse of the characteristic timescale of the universe expansion , given by @xmath49 . these harmonic oscillations rapidly alternate the orientation of the vector field . as a result , the oscillating massive homogeneous vector field features no net direction and behaves like an isotropic fluid . therefore , it can dominate the universe content without generating an excessive large - scale anisotropy @xcite , which would otherwise be in conflict with the cmb . the second obstacle to the use of vector bosons for the particle production process is more subtle . in the case of scalar fields , particle production during inflation requires the mass of the virtual particles to be small @xmath51 , otherwise they annihilate before being pulled over superhorizon distances by the superluminal expansion . hence , scalar fields need to be light to undergo particle production during inflation . if the same argument is applied to vector fields then a problem arises : light vector boson fields are approximately conformally invariant , which inhibits their particle production . this can be understood as follows . a massless vector field ( e.g. the photon ) is conformally invariant . this suggests that it is unaffected by the expansion of the universe , because it perceives this expansion as a conformal transformation to which it is insensitive . in terms of its virtual particles , this means that they are not pulled outside the horizon during inflation by the superluminal expansion . hence , the quantum fluctuations of a massless vector field are not stretched by the expansion to become classical perturbations . consequently , a light vector field is approximately conformally invariant , so one would expect the particle production process to be suppressed . to overcome this obstacle an explicit breakdown of the vector field conformality is required . fortunately , there are numerous mechanisms in the literature which achieve this , but the disadvantage is that particle production is model - dependent ; it depends on the mechanism considered . on the positive side though , this very fact may allow us to discriminate between mechanisms , through comparison with observations . there are a number of observational characteristics which may signify the contribution of a vector field to the primordial density perturbation . for example , in the above described mechanism which employed an oscillating massive vector field , there is a chance to produce a weak large - scale anisotropy in the universe . the reason is that the coherent oscillations of the massive vector field are not exactly harmonic because their amplitude is gradually decreasing as their energy density is depleted by the universe expansion . because of this , the oscillating vector field is not exactly isotropic , which means that , if it dominates the universe , it may give rise to a weak large - scale anisotropy . it so happens that there is recent tantalising evidence that such a weak large - scale anisotropy does exist in the cmb . indeed , there is a highly improbable correlation found between the quadrupole and octupole moments in the cmb angular spectrum , which reveals the existence or a preferred direction , called the `` axis of evil '' @xcite . plotted along this direction ( instead of the usual galactic coordinates ) low multipoles in the cmb produce a clear pattern , as shown in fig . an oscillating vector field may well explain this feature , which is impossible to account for under the traditional scalar field mechanisms . a potentially clearer signal for the contribution of vector fields to the primordial density perturbation is the appearance of statistical anisotropy in the cmb temperature perturbations . statistical anisotropy amounts to anisotropic patterns arising in the cmb designating special orientation . an example of how such patterns might look like is shown in fig . . one can think of them as similar to patterns in city planning produced by rows of identical houses . statistical anisotropy in the cmb can arise due to anisotropic particle production during inflation @xcite . as mentioned , vector fields have three degrees of freedom and can be analysed in three independent components . each of these components undergoes particle production during inflation if the conformality of the vector field is suitably broken . however , the efficiency of the particle production process may be different for each of these components and depends on the mechanism which breaks the conformal invariance of the vector field . statistical anisotropy is a new observable , which is inherently connected with a vector field contribution to the density perturbations . so far there is no observational detection of statistical anisotropic patterns in the cmb or in the distribution of structures in the universe ( e.g. rows of galaxies ) . however , the observational bounds are rather weak and allow the existence of statistical anisotropy at a level as high as 30% @xcite . the planck satellite , which was launched in may of 2009 by the european space agency , is expected to increase the cmb precision measurements by an order of magnitude and it is highly likely to observe statistical anisotropy . the main mission of the planck satellite is to observe non - gaussian features in the predominantly gaussian cmb temperature perturbations . if a vector field contributes to the primordial density perturbation the non - gaussianity in the cmb can also be anisotropic , in a direction which is correlated to the statistical anisotropy of the perturbation spectrum @xcite . this is a smoking gun for the contribution of vector fields in the formation of structures in the universe . before concluding let us discuss an interesting possible spin - off of the above ideas , which might connect the formation of structures in the universe with cosmic magnetism . observations suggest that the majority of galaxies carry magnetic fields of equipartition strength : @xmath52 g @xcite , meaning that their energy density is comparable to the kinetic energy density of the galaxy . in spirals the magnetic field follows the spiral arms @xcite ( see for example fig . [ pmf ] ) . this means that it is not frozen into the plasma of the galactic disk , which strongly suggests that it is instead rearranged by a magnetic dynamo mechanism , similar to the one that operates in our sun . the magnetic dynamo can also amplify an initial magnetic field up to equipartition value ( when dynamic backreaction stops further amplification ) but it needs a non - zero seed field to feed on . for the sun the seed field is presumed to be the magnetic field of the milky way . for galaxies at the time of formation the seed field needs to be at least as strong as @xmath53 g @xcite with a coherency scale @xmath54pc , corresponding to the largest turbulent eddy . the origin of this seed field remains elusive . suppose that the hypercharge vector boson field of the electroweak theory ( i.e. the unification of electromagnetism with the weak nuclear force ) has , through some mechanism , its conformality broken during inflation , so that it obtains a flat superhorizon spectrum of perturbations . after inflation , the universe cools down due to its expansion . at some moment , electroweak unification breaks down when the temperature of the hot big bang reduces below @xmath55gev , where @xmath56 is the boltzmann constant . at the electroweak phase transition , the hypercharge field is projected onto the photon and the z - boson directions , through the weinberg angle as shown in fig . [ ysplit ] . the z - boson condensate , being massive , rapidly oscillates a few tens of times before decaying to other lighter standard model particles . during these oscillations its superhorizon perturbation spectrum ( which is due to the projection of the original hypercharge spectrum ) might affect the density perturbation in the universe if the density of the oscillating field is comparable to the density of the universe at the time . the photon superhorizon perturbations ( also due to the hypercharge spectrum ) give rise to a magnetic field with scale dependence @xmath57 . due to the high conductivity of the primordial plasma the magnetic field flux is conserved and the magnetic field survives until the epoch of galaxy formation . it is a simple calculation to show that , provided the z - boson has enough density to nearly dominate the universe before its decay , the primordial magnetic field is strong and coherent enough to trigger the galactic dynamo and explain the magnetic fields of the galaxies @xcite . this scenario , if realised , may connect directly cmb observations of statistical anisotropy with galactic magnetism . according to state of the art cosmology , all structures in the universe , including galaxies , stars , planets and ourselves , originated as quantum fluctuations of suitable fields similar to the ones which we observe in the lab with the casimir experiment . these quantum fluctuations were transformed into classical perturbations ( giving rise to the primordial density perturbation ) when stretched to sizes larger than the cosmological horizon ( the range of causal correlations ) during a brief period of superluminal expansion of space , called cosmic inflation . inflation had to take place in the very early stages of the universe history in order to solve one of the fundamental paradoxes of modern cosmology , the so - called horizon problem , i.e. the apparent uniformity of the universe over distances which appear to be beyond causal contact . inflation forces this uniformity onto the universe by expanding an originally causally connected region of space to size large enough to encompass the observable universe at present . inflation also creates the deviations from uniformity which are necessary for the formation of structures , by generating the primordial density perturbation , which leads to structure formation through the process of gravitational instability . recent precise observations of the perturbations of the temperature of cmb radiation , which reflect the primordial density perturbation , confirm both the particle production process through which these perturbations were generated , and also cosmic inflation itself . these observations are the earliest data at hand , since inflation has to take place when the universe is only a tiny fraction of a second old . the precision of cosmological observations has reached the level which demands theoretical model - building to become detailed and rigorous . because of this and also in light of the forthcoming lhc findings it may be necessary to explore alternatives beyond the traditional scalar field hypothesis for the realisation of the particle production process during inflation . one such possibility is considering vector boson fields . massive vector fields can indeed generate the primordial density perturbation without excessive large - scale anisotropy if they undergo coherent oscillations before they dominate the universe . the scenario is characterised by a number of distinct observational signatures . for example it may explain the observed weak large - scale anisotropy already found in the cmb ( `` axis of evil '' ) and it can give rise to statistical anisotropy in the cmb temperature perturbations , which amounts to direction dependent patterns that , at present , are allowed at a level up to 30% and may well be detected by the forthcoming observations of the planck satellite mission . non - gaussian features in the cmb perturbations may also be anisotropic in a manner correlated with the statistical anisotropy in the cmb spectrum . this would be a smoking gun for the contribution of a vector field in the density perturbation of the universe . the author would like to thank the audiences of the 2008 christmas conference of the faculty of science and technology of lancaster university and also the physics department of the university of ioannina , for their feedback . the author also wants to thank his collaborators mindaugas kariauskas , david . h. lyth and yeinzon rodriguez - garcia . this work is based on research funded ( in part ) by the marie curie research and training network `` universenet '' ( mrtn - ct-2006 - 035863 ) and by stfc grant st / g000549/1 . s. cole _ et al . _ [ the 2dfgrs collaboration ] , `` the 2df galaxy redshift survey : power - spectrum analysis of the final dataset and cosmological implications , '' mon . not . * 362 * ( 2005 ) 505 [ arxiv : astro - ph/0501174 ] . b. chaboyer , p. demarque , p. j. kernan and l.m . krauss , astrophys . j. * 494 * ( 1998 ) 96 [ arxiv : astro - ph/9706128 ] ; r.g . gratton , f. fusi pecci , e. carretta , g. clementini , c.e . corsi and m. lattanzi , astrophys . j. * 491 * ( 1997 ) 749 [ arxiv : astro - ph/9704150 ] . casimir and d. polder , phys . * 73 * ( 1948 ) 360 ; h.b.g . casimir , indag . * 10 * ( 1948 ) 261 [ kon . ned . wetensch . * 51 * ( 1948 frpha,65,342 - 344.1987 knawa,100n3 - 4,61 - 63.1997 ) 793 ] . k. dimopoulos , phys . d * 74 * ( 2006 ) 083502 . d. grasso and h.r . rubinstein , phys . * 348 * ( 2001 ) 163 ; m. giovannini , int . j. mod . d * 13 * ( 2004 ) 391 . k. land and j. magueijo , phys . * 95 * ( 2005 ) 071301 [ arxiv : astro - ph/0502237 ] . k. dimopoulos , m. karciauskas , d.h . lyth and y. rodriguez , jcap * 0905 * ( 2009 ) 013 [ arxiv:0809.1055 [ astro - ph ] ] . groeneboom and h.k . eriksen , astrophys . j. * 690 * ( 2009 ) 1807 [ arxiv:0807.2242 [ astro - ph ] ] . m. karciauskas , k. dimopoulos and d.h . lyth , arxiv:0812.0264 [ astro - ph ] . kronberg , rept . * 57 * ( 1994 ) 325 . r. beck , a. brandenburg , d. moss , a. shukurov and d. sokoloff , ann . astrophys . * 34 * ( 1996 ) 155 . a. c. davis , m. lilley and o. tornkvist , phys . d * 60 * ( 1999 ) 021301 . k. dimopoulos , arxiv:0806.4680 [ hep - ph ] .	the fundamental paradox of the incompatibility of the observed large - scale uniformity of the universe with the fact that the age of the universe is finite is overcome by the introduction of an initial a period of superluminal expansion of space , called cosmic inflation . inflation can also produce the small deviations from uniformity needed for the formation of structures in the universe such as galaxies . this is achieved by the conjunction of inflation with the quantum vacuum , through the so - called particle production process . this mechanism is explained and linked with hawking radiation of black holes . the nature of the particles involved is discussed and the case of using massive vector boson fields instead of scalar fields is presented , with emphasis on its distinct observational signatures . finally , a particular implementation of these ideas is included , which can link the formation of galaxies , the standard model vector bosons and the observed galactic magnetic fields . inflation , particle production , primordial density perturbation , cmb radiation , structure formation	11,559	229
to compile the sample , image fields from the vla - first survey ( becker et al . 1995 ) containing components bright and extended enough to judge the source morphologies were inspected by - eye . this gave an initial 100 candidates with extended winged emission ( fig . 1 ; cheung 2006 ) . compared to previously known examples ( e.g. , lal & rao 2006 ) , the new candidates are systematically fainter ( @xmath010@xmath1 ) and more distant ( @xmath2@xmath30.3 ) . new optical spectroscopic observations are identifying many of the fainter , more distant optical hosts . most candidates have clear winged emission and higher resolution vla observations of initially @xmath040 sources have been obtained to confirm the morphological identifications . of the candidates , enough are legitimate x - shaped sources ( conventionally , those with wing to lobe extents of @xmath40.8:1 ) to more than double the number known . lower frequency gmrt observations of selected objects are being pursued to map any spectral structure to estimate the particle ages in the wings to test formation scenarios ( e.g. , dennett - thorpe et al . 2002 ) . we examined the host galaxies of about a dozen new and previously known examples with available sdss images ( 54 sec exposures ) to quantify any asymmetry in the surrounding medium as required by hydrodynamic wing formation models ( e.g. , capetti et al . most of the galaxies are highly elliptical with the minor axes roughly aligned with the wings , consistent with the findings of capetti et al . for a similarly sized sample . however , we found smaller ellipticities ( @xmath5@xmath60.1 ) in at least two examples , 3c192 and b2 0828 + 32 , confirming previous studies of these hosts ( smith & heckman 1989 ; ulrich & r " onnback 1996 ) . round " hosts are not necessarily incompatible with the hydrodynamic picture as observed @xmath5 values can be lowered by projection . this should be investigated more thoroughly with a dedicated host galaxy imaging program . becker , r.h . , white , r.l . , & helfand , d.j . 1995 , apj , 450 , 559 capetti , a. , et al . 2002 , a&a , 394 , 39 cheung , c.c . 2006 , aj , submitted dennett - thorpe , j. et al . 2002 , mnras , 330 , 609 lal , d.v . , & rao , a.p . 2006 , mnras , in press ( astro - ph/0610678 ) merritt , d. , & ekers , r.d . 2002 , science , 297 , 1310 smith , e.p . , & heckman , t.m . 1989 , apjs , 69 , 365 ulrich , m .- h . , & r " onnback , j. 1996 , a&a , 313 , 750	a small number of double - lobed radio galaxies are found with an additional pair of extended low surface brightness ` wings ' of emission giving them a distinctive ` x'-shaped appearance . one popular explanation for the unusual morphologies posits that the central supermassive black hole ( smbh)/accretion disk system underwent a recent realignment ; in a merger scenario , the active lobes mark the post - merger axis of the resultant system ( e.g. , merritt & ekers 2002 ) . however , this and other interpretations are not well tested on the few ( about one dozen ) known examples . in part to remedy this deficiency , a large sample of winged and x - shaped radio sources is being compiled for a systematic study . an initial sample of 100 new candidates is described as well as some of the follow - up work being pursued to test the different scenarios .	873	236
nonlocal conserved charges play the central role in quantum integrability @xcite . for the superstring in @xmath0 their existence was proven in the classical sigma - model in @xcite using the green - schwarz - metsaev - tseytlin formalism , and in @xcite using the pure spinor formalism . the existence of the nonlocal conserved charges at the quantum level was proven in @xcite . the most important feature of the string worldsheet @xmath2-model ( besides the conformal invariance ) is the existence of the brst structure . the physically meaningful constructions should respect the action of the brst operator @xmath3 . the nonlocal conserved charges of @xcite are only brst invariant up to the boundary terms . to be more precise , we need to introduce the transfer matrix which is the generating function of the nonlocal conserved charges . for an open contour @xmath4 connecting points @xmath5 and @xmath6 on the string worldsheet : we define the transfer matrix : @xmath7 = p\exp \left ( -\int^b_a j[z ] \right)\ ] ] where @xmath4 is a contour connecting points @xmath5 and @xmath6 on the string worldsheet , and @xmath8 is a representation of @xmath9 . the currents @xmath10 $ ] depend on the spectral parameter @xmath11 and the representation @xmath8 . the transfer matrix is a function of the _ spectral parameter _ it is the generating function of the conserved charges . it can be also though of as the wilson line operator corresponding to the flat connection @xmath10 $ ] on the string worldsheet . in this paper we will use both expressions : `` transfer matrix '' and `` wilson line '' , and understand them as synonyms . the brst variation of the wilson line @xmath12 $ ] results in the boundary terms . using the notations of @xcite : @xmath13(z ) & = & \left ( { 1\over z } \varepsilon\lambda_3(b ) + z \varepsilon\lambda_1(b ) \right ) t[c_a^b](z ) - \nonumber \\ & & - t[c_a^b](z ) \left ( { 1\over z } \varepsilon\lambda_3(a ) + z \varepsilon\lambda_1(a ) \right ) \end{aligned}\ ] ] this equation is very interesting . nontrivial boundary terms in ( [ boundarytermsinbrst ] ) allow to `` bootstrap '' at least at the classical level the structure of @xmath14@xmath15 matrices , see section 7 of @xcite . on the other hand , these boundary terms present a problem : the nonlocal conserved charges are not physical quantities , at least not in an obvious sense . ( because they are not in the kernel of @xmath3 . ) what should we do with them ? a natural thing to try is to find an operator which when inserted at the endpoint of the wilson line would make it @xmath3-closed . this is somewhat analogous to the open wilson line in qcd . the expression @xmath16 is unphysical , because it is not gauge invariant . the physical quantity is : @xmath17 we may call @xmath18 and @xmath19 `` the plugs '' because they fix `` leaking boundary terms '' in gauge tranformations , or in brst transformations . can we find similar plugs for the wilson line on the string worldsheet in @xmath0 ? in this paper we will report a progress in this direction . remember that the wilson line depends on a choice of representation ; we have to choose a representation @xmath8 of @xmath20 . consider the space of states of the linearized supergravity multiplet in @xmath0 . it splits into the direct sum of infinitely many infinite - dimensional irreducible representations of @xmath20 , each corresponding to a bps state . we will argue that when @xmath8 is one of those infinite - dimensional bps representations , then there is a suitable plug of the ghost number 2 . this is closely related to the vertex operators for the massless states in @xmath0 . in fact we will relate the brst cohomology complex corresponding to the endpoint of the wilson line to the brst complex corresponding to the vertex operators . this is essentially an example of the frobenius reciprocity . the main nontrivial point is the construction of the vertex operators transforming _ strictly covariantly _ under the global supersymmetries of @xmath0 . this is different from flat space where massless vertices transform covariantly only up to brst exact terms . wilson lines in infinite - dimensional representations played an important role in the integrable context in @xcite in the construction of the q - operator . they also played an important role in the ads / cft context in @xcite for the interpretation of the ym feynman diagramms in the string worldsheet theory . consider an infinite - dimensional irreducible representation @xmath21 of @xmath20 . it is natural to ask the question : @xmath22 the results of our paper imply that this answer can be answered directly in terms of the structure of @xmath21 , as a representation of @xmath20 . namely , given @xmath21 , we consider the following complex : @xmath23 defined entirely in terms of @xmath21 see section [ sec : definitionofalgebraiccomplex ] . then , we claim that the multiplicity of @xmath21 in the space of linearized sugra solutions is equal to the dimension of the second cohomology @xmath24 of this complex . ( we only claim this when @xmath21 has `` high enough '' spin on @xmath25 ; see the end of section [ sec : covariantuniversal ] . ) most of the paper is about massless vertex operators in @xmath0 . in section [ sec : verticesingrouptheorylanguage ] we give a geometrical definition of vertex operators using the representation of @xmath0 as the coset space @xmath26 . in section [ sec : definitionofcovariantvertex ] we explain what it means for the vertex to be strictly covariant , and then in sections [ sec : taylorseries ] and [ sec : cohomologicalargument ] prove the existence of such covariant vertices . in section [ sec : flatspacelimit ] we discuss the flat space limit of our construction . ( although it is impossible to construct the strictly covariant vertex in the flat space , but nevertheless the construction in @xmath0 has a well - defined flat space limit , which does transforms strictly covariantly , but only under a subgroup @xmath27 . ) in section [ sec : relationtoendpoint ] we explain how the covariant vertex plugs the endpoint of the wilson line . in section [ sec : applications ] we present some consequences of our construction ; we explain how to prepare the vertex operator depending on the spectral parameter . the algebra of supersymmetries of @xmath0 has a @xmath28 grading : @xmath29 we denote @xmath30 the universal enveloping algebra of @xmath31 . let @xmath32 denote the group element of @xmath30 , _ i.e. _ an element of the group @xmath33 : @xmath34 where @xmath35 , @xmath36 , @xmath37 and @xmath38 . so defined @xmath39 and @xmath40 are coordinates of the super-@xmath0 . the generators of @xmath9 are the same as in @xcite : @xmath41 } \in { \bf g}_{\bar{0}}\ ] ] for a vector space @xmath42 we will denote @xmath43 the dual vector space . in particular , the space of states is denoted @xmath21 and the space of linear functionals on the states is denoted @xmath44 . we will mostly consider the linear functionals which are the values of some supergravity fields ( such as the ramond - ramond field strength ) at a fixed point in @xmath0 . these could be also thought of as non - normalizable elements of @xmath21 , `` delta - functions type of states '' in @xmath21 . for an even vector space @xmath42 we denote @xmath45 the space of antisymmetric tensors . for an odd vector space @xmath45 will stand for symmetric tensors . example : @xmath46 is an odd vector space , and @xmath47 is the dual space . therefore @xmath48 is identified with the fifth order polynomials of some bosonic spinor variable @xmath49 . massless vertex operators in @xmath0 were introduced in @xcite . using the group theory language , we can define the vertex operator as a collection of functions @xmath50 , @xmath51 and @xmath52 of @xmath53 subject to the condition of @xmath54-covariance , which says that for any @xmath55 we should get : @xmath56 here @xmath57 and @xmath58 are the matrix elements of @xmath59 acting on @xmath60 and @xmath61 respectively . for a tensor field @xmath62 we introduce the covariant derivatives : @xmath63 } \varphi_{\alpha_1\ldots\alpha_m\;\dot{\beta}_1\ldots\dot{\beta}_n}(g ) & = & \left.{d\over ds}\right\|_{s=0 } \varphi_{\alpha_1\ldots\alpha_m\;\dot{\beta}_1\ldots\dot{\beta}_n}(e^{-s t^0_{[mn]}}g ) \nonumber\end{aligned}\ ] ] collectively : @xmath64 note that the covariant derivatives @xmath65 for @xmath66 satisfy the condition of @xmath54-covariance , for example : @xmath67 also , the @xmath54-covariance condition ( [ vertexcovariance1 ] ) ( [ vertexcovariance3 ] ) can be formulated as the following explicit expression for the covariant derivative @xmath68}$ ] along @xmath54 : @xmath69 } v_{\beta\gamma}(g ) = \left.{d\over ds}\right\|_{s=0 } v_{\beta\gamma}(e^{-s t^0_{[\mu\nu ] } } g ) = { f_{[\mu\nu]\beta}}^{\beta ' } v_{\beta'\gamma}(g ) + { f_{[\mu\nu]\gamma}}^{\gamma ' } v_{\beta\gamma'}(g)\ ] ] the condition that the vertex operator is @xmath70-closed can be written as follows : @xmath71 where @xmath5 and @xmath72 are defined by these equations . the gauge transformations are : @xmath73 where @xmath74 and @xmath75 are the parameters of the gauge transformations . the brst operator is : @xmath76 our definition of the vertex is slightly weaker than the definition of @xcite . the definition of @xcite requires that : @xmath77 the only nonzero component of the vertex remains @xmath78 . in fact @xmath79 and @xmath80 are always @xmath81-exact . therefore the condition ( [ ghostnumbergauge ] ) can always be satisfied by adding to the vertex something @xmath82-exact . in this sense the components @xmath79 and @xmath80 can always be `` gauged away '' . but we want the _ covariant _ vertex operator . we suspect that it might be impossible to gauge away @xmath79 and @xmath80 in a covariant way . this is the reason why we prefer to allow these components in the definition of the vertex operator . in string theory vertex operators represent states . vertex operators are functions @xmath83 . the global symmetries act on @xmath40 and @xmath84 , and therefore act on vertex operators . on the other hand , the global symmetries act on the space of states . therefore the action of the global symmetry group on states should agree with the action on vertex operators . naively , this would imply that if @xmath85 is a vertex operator corresponding to the state @xmath86 then @xmath87 but in fact this formula , generally speaking , holds only up to brst - trivial corrections ( terms which are @xmath3 of something ) . note that @xmath85 is not defined unambiguously , because we could add to it brst - exact terms and get physically equivalent vertex . we will prove that in @xmath0 it is possible to use this freedom in the definition of @xmath88 and choose @xmath88 so that it transforms covariantly , as in ( [ transformscovariantly ] ) . in our proof we will use the fact that vertices corresponding to supergravity states _ exist_. this was proven in @xcite . given the existence of the vertex , we will prove that it is always possible to correct it by a brst - exact expression , if necessary , to get a _ covariant _ vertex . the first example of the covariant vertex was given in @xcite . it was shown that the _ zero mode _ dilaton vertex is given by the expression ] which is independent of @xmath40 and @xmath84 : @xmath90 this operator plays the central role in @xcite . the corresponding marginal deformation of the action changes the radius of @xmath0 . the radius is invariant under the global symmetries , therefore in this case covariance means invariance ; the vertex ( [ zeromodevertex ] ) is invariant under the global symmetries . this is related to the fact that the action of the pure spinor superstring in @xmath0 is exactly invariant under the global symmetries ( while the action in flat space is invariant only up to adding a total derivative ) . our construction can be considered a generalization of ( [ zeromodevertex ] ) for fields with nontrivial dependence on @xmath40 and @xmath84 . the second example will be presented in the forthcoming paper with o. bedoya , l. bevilqua , and v.o . rivelles ] is : @xmath91 here the indices @xmath92 and @xmath93 enumerate the adjoint representation of @xmath20 . notice that ( [ vbeta ] ) is antisymmetric under the exchange of @xmath92 and @xmath93 . therefore this vertex transforms in the antisymmetric product of two adjoint representations of @xmath20 . this antisymmetric product splits into two irreducible components . the first component is the adjoint representation . but the part of ( [ vbeta ] ) belonging to the adjoint representation is actually @xmath3-exact : @xmath94_c = \nonumber \\ = \varepsilon q_{brst } ( g^{-1 } \varepsilon'(\lambda_3 + \lambda_1 ) g)_c\end{aligned}\ ] ] ( notice that this formula played an important role in section 6 of @xcite . ) the deformation of the action corresponding to ( [ vbeta ] ) follows from the standard descent procedure . let us denote : @xmath95 this is the ghost number 1 cocycle corresponding to the local conserved currents , see also appendix [ sec : ghostnumberone ] . it corresponds to the local conserved currents in the following sense : @xmath96 where @xmath97 is the density of the local conserved charge corresponding to the global symmetries . therefore : @xmath98}(\varepsilon ' ) ) = 2\varepsilon q j_{[a } \lambda_{b]}(\varepsilon')\ ] ] and : @xmath99}(\varepsilon ) ) = -{1\over 2 } \varepsilon q ( j_{[a}\wedge j_{b]})\ ] ] we conclude that for any constant antisymmetric matrix @xmath100 we can infinitesimally deform the worldsheet action as follows : @xmath101}\ ] ] consider for example @xmath102 in the directions of @xmath25 . we get : @xmath103[mn ] } \left ( \int x_{[k } dx_{l ] } \wedge x_{[m } dx_{n ] } + \ldots \right)\ ] ] where @xmath104 describes the embedding of @xmath25 into @xmath105 and dots denote @xmath84-dependent terms . these @xmath84-dependent terms appear because @xmath106 includes @xmath84 . ( [ binsdirections ] ) corresponds to the marginal deformations of the @xmath107 yang - mills known as @xmath89_-deformations _ @xcite , as follows from their quantum numbers . the subspace @xmath108 corresponds to @xmath6 of the following form : @xmath109[mn ] } = \delta^{km } a^{ln } - \delta^{lm } a^{kn } + \delta^{ln } a^{km } - \delta^{kn } a^{lm}\ ] ] where @xmath110 is antisymmetric matrix ; then the corresponding deformation of the lagrangian is a total derivative @xmath111 . the complementary space has real dimension 90 , it corresponds to the representation @xmath112 of @xmath113 . this is the expected quantum numbers of the linearized @xmath89-deformation , cp . section 3.1 of @xcite and references therein . it was observed in @xcite that some of these deformations are obstructed when we pass from the linearized supergravity equations to the nonlinear equations . not all of the deformations ( [ linearizedbeta ] ) can be extended to the solutions of the nonlinear supergravity equations _ as solutions constant in @xmath114 directions _ , but only those which satisfy some nonlinear equations on @xmath100 . if these nonlinear equations are not satisfied , then the nonlinear solutions will have `` resonant terms '' and because of these resonant terms will not be periodic in the global time of @xmath114 . suppose that we are looking at the _ massless _ states transforming in some representation @xmath21 of the global symmetry group @xmath33 . for every state @xmath115 we have the corresponding vertex operator @xmath116 . as we discussed , @xmath116 consists of the components : @xmath117 , @xmath118 and @xmath119 . we write @xmath120 instead of @xmath121 to stress that @xmath120 is a function of @xmath86 . mathematically it would be more appropriate to call it `` a linear operator from the space of states to the space of vertex operators '' . we will call @xmath120 the `` universal vertex '' for the representation @xmath21 because it is a uniform definition of vertex operators for all states in @xmath21 : @xmath122 the global symmetry group @xmath123 acts on both space of states and space of vertex operators . it acts on the space of states by definition , because it is the global symmetry group of the theory . it also acts on the space of vertex operators . the action on the space of vertex operators may seem obvious , but we would like to spell it out explicitly because we feel that some confusion is possible . a vertex operator has components @xmath79 , @xmath78 and @xmath80 which are all functions of the group element @xmath32 , _ i.e. _ @xmath50 , @xmath51 and @xmath52 satisfying the conditions of @xmath54-covariance ( [ vertexcovariance1 ] ) ( [ vertexcovariance3 ] ) . then , the action of the global symmetry transformation @xmath124 is defined as follows : @xmath125 because @xmath126 hits @xmath32 on the right , this action of the global symmetries is manifestly consistent with the conditions of @xmath54-covariance ( [ vertexcovariance1 ] ) ( [ vertexcovariance3 ] ) and also commutes with the covariant derivatives ( [ defcovariantderivatives ] ) . this defines the action of @xmath127 on @xmath116 for any fixed @xmath86 ; the expression : @xmath128 is defined by ( [ globalactiononv ] ) : @xmath129 it is natural to ask , if it is true that ( [ globalactiononuniversalvertex ] ) is equal to this : @xmath130 in other words , if it is true or not that : @xmath131 this is not automatically true . what _ is _ automatically true is this statement : @xmath132 remember that vertex operators are defined modulo brst - exact expressions . the question is , can we choose a representative for @xmath116 in the equivalence class of @xmath133 , `` uniformly in @xmath86 '' , so that ( [ exactcovariance ] ) is true ? the answer to this question is `` no '' in flat space , but `` yes '' in ads . it turns out that in @xmath0 it is possible to choose the vertex operator to be covariant . let us introduce the notation for the action of the global symmetries ( compare to ( [ collectively ] ) ) : @xmath134 the covariant vertex is a superfield _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath135 depending linearly on the state @xmath86 , and such that : 1 . it is annihilated by @xmath70 : @xmath136 and is not @xmath137-exact , and 2 . the action of the global symmetry on @xmath120 as a function of @xmath32 agrees with the action of the global symmetry on the space of states : @xmath138 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let us study vertex operators for states which are not necessarily normalizable . in other words , let us forget about the boundary conditions near the boundary of ads , and study the supergravity states which are not necessarily normalizable . moreover , let us pick a point in @xmath0 and consider the taylor expansion of the supergravity fields around this point . let us not worry about the convergence of the taylor series . just study the supergravity equations , brst cohomology _ etc . _ on formal taylor series . the question of convergence , and the question of the behaviour at spacial infinity , can be studied later . for the study of the taylor series the mathematical notion of the _ coinduced representation _ is useful . let us study the supergravity fields around a point in @xmath0 corresponding to the unit @xmath139 . if we do not insist on convergence , then the space of supergravity fields around a point can be replaced by a more algebraic notion , the so - called _ coinduced representation _ @xcite . for a representation @xmath121 of @xmath140 , we define the _ coinduced representation _ @xmath141 , in the following way : : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the space of linear functions @xmath142 from the universal enveloping algebra @xmath30 to @xmath121 , which satisfy the condition of @xmath54-invariance : @xmath143 is called the _ coinduced representation _ and denoted @xmath141 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the mathematical notation for such functions is : @xmath144 here `` @xmath145 '' means the space of linear maps from @xmath5 to @xmath6 , and the subindex @xmath54 means @xmath54-invariant functions : @xmath146 . the action of @xmath31 on this space is defined by the formula : @xmath147 to summarize : we will now explain that @xmath141 encodes the taylor series of various tensor fields on @xmath0 , where @xmath121 is the representation of @xmath54 corresponding to the type of the tensor . let us start with the trivial representation @xmath148 of @xmath149 . in this case we should get scalar fields on ads . the correspondence between the elements of @xmath150 and the scalar fields on ads goes as follows . given a scalar field @xmath151 , the corresponding element @xmath152 is given by the formula : @xmath153 in this case the @xmath54-invariance condition says that @xmath154 for any @xmath155 , and this is indeed satisfied for @xmath142 defined in ( [ inducedfromscalars ] ) because @xmath156 for any @xmath155 because @xmath157 is well defined on @xmath158 . there is also a map going in the opposite direction . namely , given @xmath142 a linear function from @xmath30 to @xmath159 we define the corresponding scalar field @xmath151 as follows : @xmath160 note that on the right hand side we treat @xmath32 as a _ group element _ is called _ group element _ if it is of the form @xmath161 for some @xmath162 of @xmath30 . we have just explained why for @xmath148 the space @xmath163 encodes the taylor coefficients of the scalar function on ads ; for general @xmath121 a similar construction shows that @xmath163 encodes the taylor coefficients of the tensor field with indices transforming in the representation @xmath121 of @xmath54 . to describe the linearized sugra in @xmath0 we need two pure spinor variables @xmath164 and @xmath165 satisfying the constraints : @xmath166 we will consider various types of vertex operators , which are homogeneous polynomials in @xmath167 and @xmath168 . note that ( [ purespinorconstraints ] ) are invariant under the action of @xmath54 . therefore the polynomials of @xmath167 and @xmath168 form a representation of @xmath54 . we will introduce the notation for such polynomials : 0= 1= 2= 3=to 1 2 we will define the polynomials of @xmath169 by specifying their coefficients , which are elements of @xmath170 ( see section [ sec : notations ] for notations ) . we have to `` discard '' those polynomials which are identically zero because of the pure spinor constraints ( [ purespinorconstraints ] ) . as a trivial example , let us consider the quadratic polynomials of @xmath167 . the coefficients belong to @xmath171 . let us denote @xmath172 the basis vectors of @xmath173 , such that : @xmath174 the space @xmath171 consists of expressions of the form @xmath175 where @xmath176 . as an example of the polynomial which is identically zero , take @xmath177 where @xmath178 is the structure constants defined by @xmath179 . such a polynomial is identically zero because of the pure spinor constraint ( [ purespinorconstraints ] ) : @xmath180 to summarize : @xmath181 where @xmath182 denotes a subspace of @xmath183 corresponding to those polynomials on @xmath60 which vanish identically on @xmath167 because of the pure spinor constraint . we will also introduce : @xmath184 note that @xmath185 and @xmath186 are representations of @xmath54 , but not of @xmath31 . the construction of coinduced representation is used to build the representations of @xmath31 from these spaces . in this section we consider the taylor series of the vertex operator and do not bother about the convergence and the behaviour near the boundary . then the vertex operator can be considered an element of the coinduced representation : @xmath188 we would like to discuss vertex operators `` uniformly '' for all vectors @xmath115 . we will therefore introduce the `` universal '' vertex operator : @xmath189 in other words , we have a linear function on the hilbert space @xmath190 which to every vector @xmath191 associates the corresponding vertex operator : @xmath192 given a state @xmath115 we get @xmath116 an element of @xmath187 . this means , by definition , that for every @xmath86 , the object @xmath116 is a linear map from @xmath30 to @xmath193 satisfying the @xmath54-invariance condition : @xmath194 given such @xmath116 , how do we construct the `` usual '' vertex operator ? as an element of @xmath195 our @xmath116 is a function of @xmath196 with values in @xmath193 . let us evaluate this function on a group element @xmath197 , where @xmath198 . we get @xmath199 an element from @xmath193 , _ i.e. _ a quadratic polynomial in @xmath167 and @xmath168 . the `` usual '' vertex opearator is just the evaluation of this polynomial : @xmath200 the brst complex is : @xmath201 the brst operator acts on the universal vertex @xmath116 in the following way : @xmath202 note that @xmath203 is an element of @xmath204 . in terms of the `` usual '' vertex @xmath205 defined by ( [ usualvertexoperator ] ) we get : @xmath206 [ [ statement - of - covariance ] ] statement of covariance + + + + + + + + + + + + + + + + + + + + + + + note that in eq . ( [ noncovariantvertex ] ) we use the notation @xmath207 rather than @xmath208 . there is no apriori reason why @xmath120 would respect the action of @xmath31 . but in the next section we will see that under some conditions on @xmath21 , it is possible to choose the universal vertex operator which does respect the global symmetry . we will call it the _ covariant _ universal vertex : @xmath209 given eq . ( [ actionofg ] ) this implies : @xmath210 [ [ condition - on - cal - h - sufficiently - high - spin ] ] condition on @xmath21 : sufficiently high spin + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the conditions on @xmath21 are the following . consider @xmath21 as a representation of @xmath211 the symmetry algebra of @xmath25 . as a representation of @xmath113 , @xmath21 is the direct sum of infinitely many finite - dimensional representations of @xmath113 . we request that the minimal value of the quadratic casimir of @xmath113 on @xmath21 be sufficiently high . in this section we will use some facts about the lie algebra cohomology which we learned mostly from @xcite . see chapter 3 6 of @xcite for a very brief summary . the physical states correspond to the cohomology of @xmath3 at the ghost number 2 , therefore : @xmath212 where @xmath213 is the multiplicity of @xmath21 ( how many times @xmath21 enters in the sugra spectrum on @xmath0 ) . we will argue that the second cohomology of the brst operator can be calculated using the covariant subcomplex . in other words , @xmath214 ( notice that @xmath215 is the covariant subcomplex . ) to prove eq . ( [ equalstocovarianth ] ) we rewrite it in the following form : @xmath216 here we have used the fact that for any representation @xmath42 of the lie algebra @xmath31 the zeroth cohomology group @xmath217 equals the space of invariants @xmath218 . in particular , for two representations @xmath219 and @xmath220 , @xmath221 . the idea of the proof of ( [ hpermuted ] ) is to note that the left and the right hand side of ( [ hpermuted ] ) are two different second approximations to calculating the cohomology of the `` total '' differential @xmath222 . therefore the equality of the left hand side and the right hand side follows if we prove that the second approximation is actually exact . let us start by fixing some universal vertex ( not necessarily covariant ) : @xmath223 at this point we do not require that this vertex is covariant ; it is apriori an element of @xmath224 rather than @xmath225 . we will introduce the lie algebra brst operator of @xmath31 . for each generator @xmath226 of @xmath31 we introduce the corresponding ghost @xmath227 , and define : @xmath228 we will consider the action of @xmath229 on expressions polynomial in @xmath227 . the polynomials of @xmath227 are specified by their coefficients ; in degree @xmath230 the coefficients live in @xmath231 . therefore @xmath229 acts on the vertex operator as follows : @xmath232 we will consider the bicomplex with the differential @xmath233 : @xmath234 to prove the existence of the covariant vertex we will consider the spectral sequence computing the cohomology of this bicomplex . there are two ways to construct the spectral sequence . one can first calculate the cohomology of @xmath3 and then consider @xmath229 as a perturbation . the other way is to first calculate the cohomology of @xmath229 and then consider @xmath3 as a perturbation . these two ways of calculating the cohomology of @xmath233 should give the same result . we will see that this implies the existence of the covariant vertex . we will now consider the two methods in turn . [ [ first - q_brst - then - q_lie ] ] first @xmath3 then @xmath229 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the first term of the spectral sequence has : @xmath235 where @xmath236 stands for the @xmath237-ghosts ; an element of @xmath238 is schematically @xmath239 . the differential in the first term is @xmath240 . the second term is : @xmath241 the higher differentials are of the type @xmath242 . [ [ first - q_lie - then - q_brst ] ] first @xmath229 then @xmath3 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the first term is : @xmath243 where @xmath244 . the higher differentials are of the type @xmath245 . [ [ existence - of - the - covariant - vertex ] ] existence of the covariant vertex + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the first page of @xmath246 and @xmath247 ; arrows denote @xmath248 and @xmath249 . the lie algebra ghost number ( the number of @xmath237 s ) increases in the horizonthal direction , while the brst ghost number ( the number of @xmath167 plus the number of @xmath168 ) in the vertical direction . , title="fig:",width=192 ] the first page of @xmath246 and @xmath247 ; arrows denote @xmath248 and @xmath249 . the lie algebra ghost number ( the number of @xmath237 s ) increases in the horizonthal direction , while the brst ghost number ( the number of @xmath167 plus the number of @xmath168 ) in the vertical direction . , title="fig:",width=192 ] the second page of @xmath246 and @xmath247 ; arrows denote @xmath250 and @xmath251 . , title="fig:",width=192 ] the second page of @xmath246 and @xmath247 ; arrows denote @xmath250 and @xmath251 . , title="fig:",width=192 ] the differential @xmath252 of @xmath253 . , width=192 ] first of all , we want to show that @xmath254 . the first observation is that by definition @xmath255 is zero . this is because the vertex is _ covariant up to brst - exact correction _ ( see eq . ( [ uptobrstexact ] ) ) . therefore @xmath256 . also , we will show ( for @xmath21 with large enough spin ) that @xmath257 . this implies that @xmath254 . then we remember the relation between @xmath258 and @xmath259 , which is the following . the space @xmath260 has a filtration , corresponding to the number of the @xmath237-ghosts . namely , @xmath261 consists of expressions containing at least @xmath262 @xmath237-ghosts . then @xmath263 . to summarize : @xmath264 \\[3pt ] e_1^{0,2 } & = & h^2(q_{tot } ) / f^1h^2(q_{tot } ) \label{e1isfactorspace}\end{aligned}\ ] ] on the other hand we will show that @xmath265 ( see eqs . ( [ shapiroforh2 ] ) , ( [ shapiroforh1 ] ) ) and also that @xmath266 ( similar to ( [ shapiroforh1 ] ) ) . this implies that : @xmath267 we are now ready to prove the existence of the covariant vertex . notice that @xmath268 is the space of functions @xmath269 , parametrized by @xmath115 , transforming covariantly under @xmath31 . this means that @xmath270 is the _ covariant subcomplex _ of the brst complex . ( the subspace consisting of the covariant expressions . ) and eq . ( [ h2qtotfrometilde ] ) shows that : @xmath271 now the comparison of ( [ h2qtotfrome ] ) , ( [ e1isfactorspace ] ) and ( [ h2qtotiscovariant ] ) shows that the cohomology of @xmath3 can be calculated using the covariant subcomplex . in fact , if the representation @xmath21 has large enough momentum in @xmath25 , then @xmath272 is zero ( because already @xmath273 and @xmath274 are zero ) . this means that the factor space on the right hand side of ( [ e1isfactorspace ] ) is just @xmath259 . this means that there is a covariant choice of the vertex . in the rest of this section we will prove the required vanishing theorems and explain explicitly how the non - covariant vertex can be modified into the covariant one . since the vertex transforms covariantly up to brst - exact terms , we must have : @xmath275 where @xmath276 note that @xmath277 is @xmath3-closed and has ghost number 1 : @xmath278 we want to argue that there exists @xmath279 such that @xmath280 . more precisely : note that we are free to add to @xmath281 something in the kernel of @xmath3 ; we want to prove that it is possible to use this freedom and choose @xmath281 so that there exists @xmath279 such that @xmath280 . an obstacle to this would be a nonzero @xmath282 where @xmath283 we want to argue that the space @xmath284 on the right hand side is zero . note that the brst cohomology in ghost number 1 corresponds to local conserved charges . but the only conserved charges are the global symmetries @xmath20 , and those transform in the adjoint representation of @xmath31 . this means that on the right hand side of ( [ checkermove ] ) we have : @xmath285 adjustment of the vertex operator . , width=470 ] this cohomology group is zero . indeed , we can compute it using the serre - hochschild spectral sequence of @xmath286 . already the first term of this spectral sequence consists of the following spaces , which are all zero : @xmath287 note that @xmath288 where @xmath289 and @xmath290 . consider the corresponding casimir operators @xmath291 and @xmath292 . for the cohomology to be nonzero , we need both of them zero , but @xmath292 is positive definite at least for @xmath21 with large enough momenta . ( note also that @xmath21 is an infinite dimensional irreducible representation of @xmath31 , so there are no invariants in its tensor product with powers of @xmath31 . ) therefore @xmath280 for some @xmath279 . in other words @xmath293 is zero , and we can proceed with computing @xmath252 . consider @xmath294 . note that @xmath295 has zero pure spinor ghost number , and @xmath296 . since @xmath295 is of ghost number 0 , this implies that @xmath295 is a constant ; it does not contain any @xmath40 or @xmath84 . also , we could have added a constant to @xmath279 without affecting @xmath297 ; therefore @xmath295 by itself is not very well defined by our construction . what is well defined is @xmath295 modulo the image of @xmath229 : @xmath298 \in h^3_{q_{lie}}({\cal h}'\otimes \lambda^{\bullet}{\bf g } ) = h^3({\bf g},{\cal h}')\ ] ] but the lie algebra cohomology group @xmath299 is zero : @xmath300 one can see that it is zero from the serre - hochschild spectral sequence corresponding to @xmath286 . already the first term of this spectral sequence consists of the following spaces , which are all zero : @xmath301 the vanishing of these cohomologies can be proven as follows . note that @xmath302 splits into @xmath289 and @xmath290 . for the cohomology to be nontrivial , both @xmath303 and @xmath304 should be zero . but @xmath305 is positive definite . therefore we can remove @xmath295 by modifying @xmath279 , adding to @xmath279 a constant term @xmath306 so that the modified @xmath307 has @xmath308 . ( note that adding the constant term does not change the image of @xmath309 under @xmath3 . ) is it possible to find such @xmath310 that @xmath311 ? the answer is `` yes '' , because @xmath312 this can be proven using the shapiro s lemma ( proposition 6.8 and theorem 6.9 from @xcite ) : @xmath313 note that @xmath314 . we want to prove that @xmath315 . the space @xmath44 consists of functionals on the space of states . since we work in the vicinity of the fixed point @xmath316 our @xmath44 is generated by the values of various supergravity fields at the point @xmath317 . for example the ramond - ramond field strength @xmath318 and its derivatives . under the action of @xmath319 this space splits into infinitely many finite - dimensional representations . for example @xmath320 lives in @xmath321 where vect is is the vector representation of @xmath319 and index @xmath322 means that the contraction @xmath323 is zero , for the contraction to be zero ; the ramond - ramond 5-form @xmath324 is nonzero in the ads background . ] . it follows from the general theory of lie algebra cohomology that @xmath325 of @xmath319 with coefficients in any finite - dimensional representation is zero . these arguments imply that @xmath310 is in the image of @xmath229 . we can modify @xmath281 by adding to it : @xmath326 then we have : @xmath327 now we use : @xmath328 therefore @xmath329 is in the image of @xmath229 . now the _ modified vertex : _ @xmath330 is covariant . our procedure could perhaps be summarized as follows : @xmath331 in flat space it is impossible to choose a covariant vertex , because of the nontrivial cohomology @xmath332 which represents the nsns 3-form field strength . but one can satisfy a weaker covariance condition . note that in flat space the generators of the lorentz subalgebra @xmath333 of the poincare algebra can not be obtained as commutators of other generators . therefore it is consistent to require the covariance under all translations and supersymmetries , but only some rotations . in particular , it turns out that we can choose a vertex covariant under : @xmath334 this is a subalgebra of the super - poincare algebra : @xmath335 corresponding to the split of the space - time : @xmath336 we will say that the ten spacetime directions split into @xmath337 a - directions and @xmath338 s - directions ( the letters a and s stand for the ads and the sphere ) . let us now explain how the diagramm of fig . [ fig : adjustment ] works in flat space . instead of considering fig . [ fig : adjustment ] literally let us study the similar diagramm for the supersymmetric maxwell field ( rather than supergravity ) . this is a toy model ; the supersymmetric maxwell field in flat space is `` one half of the supergravity field '' . the `` usual '' ( non - covariant ) vertex operator is of the form : @xmath339 } + \ldots\ ] ] where @xmath340 and @xmath341 are the vector potential and the photino . let us first try to understand if it is possible to choose the vertex covariant under the even poincare algebra . the vertex operator ( [ vertexoperator ] ) involves the gauge field @xmath342 . because of the gauge invariance the gauge field is not in one to one correspondence with the physical states . the physical states are described by @xmath343 , not by @xmath342 . to describe @xmath342 in terms of @xmath344 , let us break the translational symmetries by choosing a point @xmath322 in space - time . then we can write , in the vicinity of the chosen point : @xmath345 where @xmath346 and @xmath347 and @xmath348 are the corresponding @xmath349 and lie derivative . for example , @xmath350 and @xmath351 . note that eq . ( [ vicinitygauge ] ) is one particular way to choose a vector potential with the field strength @xmath142 . let us therefore replace @xmath92 with @xmath352 . this breaks the translation symmetry , since the gauge ( [ vicinitygauge ] ) depended on a choice of point @xmath353 . can we restore the translational symmetry ? let us introduce the operator @xmath229 acting on the physical vertex operators in the following way : @xmath354 here the index @xmath355 runs over an infinite set enumerating the basis vectors of @xmath21 , and @xmath356 are the generators of translations @xmath357 . fermionic parameters @xmath358 are the lie - algebraic ghosts of the translation algebra . this operator @xmath229 measures the deviation of the vertex operator from transforming covariantly under the action of the global shift . we observe that @xmath359 is @xmath360 of something : @xmath361 f = \left ( \iota_c { 1\over { \cal l}_e } - \iota_e { 1\over { \cal l}_e } { \cal l}_c { 1\over { \cal l}_e}\right ) f = \nonumber \\ & = & d\left ( { 1\over { \cal l}_e({\cal l}_e+1 ) } \iota_e\iota_c f\right ) \label{descentderham}\end{aligned}\ ] ] let us calculate @xmath229 of this `` something '' : @xmath362 expanding @xmath142 in taylor series around @xmath353 and taking into account that @xmath363 we can see that ( [ equalsconstant ] ) is equal to : @xmath364 we should stress that ( [ equalsconstant ] ) is equal to the _ constant _ ( independent of @xmath40 ) expression ( [ descent ] ) . in other words , the only term in the taylor expansion of ( [ equalsconstant ] ) around the point @xmath353 is the constant term . one can see it , for example , because the lie derivative @xmath348 of ( [ equalsconstant ] ) vanishes . this can be seen from the identity @xmath365 which follows from @xmath363 . notice that if we started with some other point @xmath317 ( not the origin ) , then ( [ descent ] ) would change by @xmath229 of something . for example , an infinitesimal shift by @xmath366 would change ( [ descent ] ) by the @xmath229-exact expression : @xmath367 ( this is a manifestation of the general fact , that a lie algebra acts trivially in its cohomology . ) the @xmath229-cohomology class of : @xmath368 is the obstacle for defining @xmath92 such that @xmath369 in a covariant way . we have so far discussed only the action of shifts . the expression ( [ obstacle ] ) as we defined it represents the cohomology class of the algebra of translations @xmath370 . but we can also think of it as a cocycle of the poincare algebra . indeed , @xmath344 transforms covariantly under rotations and boosts and therefore ( [ obstacle ] ) is closed under the @xmath229 of the full poincare algebra . the @xmath229 of the full poincare algebra is the sum of @xmath371 of translations @xmath370 and @xmath372 of rotations and boosts . expression ( [ obstacle ] ) is in the kernel of @xmath371 by our construction , and more explicitly because @xmath142 is a closed form . but it is also in the kernel of @xmath372 because @xmath142 transforms covariantly under rotations and boosts . another question is whether or not ( [ obstacle ] ) is exact . one can see that this is not exact as a cocycle of the full poincare algebra , in the following way . let @xmath373 stand for the poincare algebra . we have : @xmath374 notice that the space of states of the gauge field contains a proper subspace closed under the action of the poincare algebra . ( in other words , it is not an irreducible representation . ) this subspace consists of those gauge fields which have a constant field strenght : @xmath375 . let us call this subspace @xmath376 : @xmath377 therefore there is a projection @xmath378 this projection naturally acts on the cocycles of @xmath379 with values in @xmath44 , and therefore on the cohomology groups : @xmath380 it is straightforward to see that the projection of ( [ obstacle ] ) to @xmath381 is automatically a nonzero cohomology class . this implies that ( [ obstacle ] ) represents a nontrivial cohomology class in @xmath382 . this is what prevents us from choosing the vertex covariant with respect to the poincare algebra . let us start with introducing some notations . for any vector @xmath384 we denote @xmath385 the vector with the components : @xmath386 also introduce : @xmath387 what happens if we do not require the invariance under the full poincare algebra @xmath373 , but only under the @xmath383 of ( [ psmall ] ) ? then we can restrict ourselves to the subspace of @xmath21 where @xmath305 is a fixed positive number . on this subspace , it is possible to express @xmath342 in terms of @xmath344 in a @xmath388-covariant way . let us choose the covariant gauge : @xmath389 and fix the residual gauge transformations with the additional `` axial '' gauge gauge condition : @xmath390 where @xmath391 is introduced as in ( [ barnotation ] ) . in the gauge ( [ axialgauge ] ) we can express the gauge field @xmath342 in terms of the gauge field strength @xmath343 : @xmath392 now we have two different expressions for the vector potential , eq . ( [ vicinitygauge ] ) and eq . ( [ covariantgauge ] ) . the difference between these two expressions is a gauge transformation . let us use a `` diacritical '' mark to distinguish ( [ covariantgauge ] ) from ( [ vicinitygauge ] ) : @xmath393 note that @xmath394 is @xmath383-covariant : @xmath395 while @xmath92 is not see eq . ( [ descentderham ] ) . also , for every gauge field we can calculate @xmath396 . this is a functional of the gauge field , _ i.e. _ an element of @xmath44 . if we calculate its @xmath229 as a cochain with values in @xmath44 we get : @xmath397 now let us return to eqs . ( [ descentderham ] ) , ( [ equalsconstant ] ) and ( [ descent ] ) . on the subspace @xmath398 the cohomology class of ( [ descent ] ) trivializes : @xmath399 this is analogous to eq . ( [ h3zero ] ) of section [ sec : descent ] . therefore the same arguments as we presented in section [ sec : descent ] should imply that @xmath400 indeed we have : @xmath401 also notice that : @xmath402 this means that the correction of the vector potential : @xmath403 is completely analogous to the correction of the vertex operator described in section [ sec : descent ] . it turns the non - covariant expression @xmath404 into the covariant expression @xmath405 . let us now study the action of the supersymmetry generators . let us consider the part of the @xmath229 involving the supersymmetry generators . we have : @xmath406 q_{brst } & = & \lambda^{\alpha}{\partial\over \partial\theta^{\alpha } } + \lambda^{\alpha}\gamma^{\mu}_{\alpha\beta}\theta^{\beta } { \partial\over \partial x^{\mu}}\end{aligned}\ ] ] here @xmath407 is the supersymmetry transformation acting on the space of states @xmath21 and therefore ( after fixing the gauge ! ) on @xmath342 and @xmath408 . we write only the part of @xmath229 corresponding to the super - translations ; @xmath409 are the bosonic ghosts corresponding to the super - translations . to make @xmath407 act on @xmath92 and @xmath408 we have to choose the gauge . with this notation , let us first of all present @xmath410 acting on the vertex operator ( [ vertexoperator ] ) in the following form : @xmath411 } -{1\over 4 } ( \theta\gamma^{\mu\nu\rho}\theta)(\varepsilon\lambda\gamma_{\rho } \varepsilon'\xi ) \partial_{[\mu}a_{\nu ] } + \nonumber \\ & & + ( \varepsilon\lambda\gamma^{\mu}\varepsilon'\xi)(\psi\gamma_{\mu}\theta ) + ( \varepsilon\lambda\gamma^{\mu}\theta)(\psi\gamma_{\mu}\varepsilon'\xi ) + \ldots = \nonumber \\[10pt ] & = & - { 3\over 2}(\varepsilon\lambda\gamma^{\mu}\theta ) \left ( ( \varepsilon'\xi\gamma_{\mu}\psi ) + { 1\over 2\delta_s}\partial_{\mu } ( \varepsilon'\xi\overline{\gamma}^{\rho}\partial_{\rho}\psi)\right ) - \nonumber \\ & & -{2\over 3 } ( \varepsilon\lambda\gamma^{\rho}\theta)(\varepsilon'\xi\gamma^{\mu\nu}\gamma_{\rho}\theta ) \partial_{[\mu}a_{\nu ] } + \ldots \\ & & + \varepsilon q_{brst } \left ( { 1\over 2 } ( \theta\gamma^{\mu}\varepsilon'\xi)(\psi\gamma_{\mu}\theta ) + { 3\over 2}{1\over 2\delta_s } ( \varepsilon'\xi\overline{\gamma}^{\mu}\partial_{\mu}\psi ) + \right . \nonumber \\ & & \phantom{+ q_{brst}\;\ ; } \left . + ( \theta\gamma^{\mu}\varepsilon'\xi ) a_{\mu } -{1\over 12 } ( \theta\gamma^{\mu\nu\rho}\theta ) ( \theta\gamma_{\rho}\varepsilon'\xi ) \partial_{[\mu}a_{\nu ] } + \ldots \right ) \nonumber\end{aligned}\ ] ] this implies that @xmath412}\end{aligned}\ ] ] therefore we have indeed : @xmath413 on the right hand side @xmath3 is taken of the expression which is @xmath229-exact : @xmath414 then we have : @xmath415 this means that the following vertex operator : @xmath416 transforms covariantly under the odd shifts . this can be also understood as follows : @xmath417 } + \ldots\ ] ] now we recognize what it is : @xmath418 where @xmath419 is the superfield ] related to the maxwell superfield @xmath420 by the chain of transformations : @xmath421 see @xcite for a recent discussion of @xmath422 . in flat space the supergravity fields split into the product of the left and the right component ; the left and right components are essentially free maxwell fields . the bispinor field is defined as follows : @xmath423 where @xmath424 is the field strength superfield of the free maxwell theory ; the @xmath425 component of @xmath422 is the gaugino @xmath426 . this bispinor field is a linear combination of the rr field strengths contracted with the gamma - matrices @xcite : @xmath427 where @xmath428 and @xmath429 are some numeric coefficients which we do not need . the supersymmetry variations of @xmath430 is given by this equation : @xmath431 where @xmath432 is a combination of the left dilatino @xmath433 , and the left gravitino field strength @xmath434}$ ] , and @xmath435 is a combination of the corresponding right fields @xmath436 and @xmath437}$ ] . the @xmath438-covariant vertex in flat space is the product of two expressions of the form ( [ maxwellvertexviaw ] ) : @xmath439 this is brst equivalent to : @xmath440 the construction of section [ sec : cohomologicalargument ] implies that the @xmath33-covariant vertex operator exists in @xmath0 . this construction gives ( [ vertexwithderivatives ] ) when applied in the flat space limit . ( we have demonstrated this for the maxwell field , but the free supergravity vertex in flat space is just the product of `` left '' and `` right '' maxwell vertices . ) therefore both ( [ vertexwithderivatives ] ) and the brst - equivalent ( [ vertexwithoutderivatives ] ) should be the flat space limits of some covariant vertices in @xmath0 . notice that ( [ vertexwithoutderivatives ] ) reduces to ( [ zeromodevertex ] ) in the zero momentum limit , except for the overall normalization factor @xmath441 which becomes singular on the zero mode . however we were not able to write explicit expressions in terms of the supergravity fields in ads space which would explicitly generalize the flat space formulas ( [ vertexwithderivatives ] ) or ( [ vertexwithoutderivatives ] ) . consider the wilson line operator corresponding to a semi - infinite contour going from infinity to some point @xmath6 on the string worldsheet : in some representation @xmath8 of @xmath442 . consider the action of @xmath3 on this operator . if we neglect the contribution of the boundary terms at infinity , then the brst variation is@xcite : @xmath443(z ) = \left ( { 1\over z } \varepsilon\lambda^{\alpha}_3(b)\rho(t^3_{\alpha } ) + z \varepsilon\lambda_1^{\dot{\alpha}}(b)\rho(t^1_{\dot{\alpha } } ) \right ) t[c_{\infty}^b](z)\ ] ] ( see section 2.2 of @xcite and section 7 of @xcite for a discussion of this formula . ) let us fix some vector @xmath408 in the representation @xmath8 `` at infinity '' ; then this expression : @xmath444(z)\psi\ ] ] is a vector in the representation space of @xmath8 . pick a vector @xmath445 in the dual space , and evaluate it on ( [ vectorfrominfinity ] ) : @xmath446(z)\psi\right ) \;\in\ ; { \bf c}\ ] ] this gives a number . consider vectors @xmath445 depending on the pure spinors @xmath447 and the spectral parameter @xmath11 . then eq . ( [ boundarytermssemiinfinite ] ) can be regarded as defining the action of @xmath3 on @xmath445 : @xmath448 this defines the brst complex of the endpoint . the @xmath449-cochains of this complex are elements @xmath450 \ ; , \ ; { \cal p}^n \right)\ ] ] where @xmath451 is defined in section [ sec : purespinors ] , and the differential is ( [ brstcomplexendpoint ] ) . the `` plugs '' which we introduced in section [ sec : theplugs ] are the cohomologies of this complex . unfortunately we do not know a general classification of the cohomologies of this complex for a general representation @xmath8 . we will now consider the special case where @xmath8 is the representation of @xmath20 on the space of states @xmath21 of the bps multiplet . in this case we will relate the brst complex of the endpoint ( [ brstcomplexendpoint ] ) to the brst complex of covariant supergravity vertices . let us remember the general structure of the covariant vertex from section [ sec : covariantuniversal ] : @xmath452 here @xmath453 for the supergravity vertex , but we want to consider the whole brst complex so we keep @xmath449 . let us evaluate @xmath120 on the unit of the group : @xmath454 this defines a correspondence between covariant vertices @xmath120 and vectors in @xmath455 : @xmath456 notice that @xmath120 is a function of @xmath457 while @xmath445 is essentially its value at @xmath458 . nevertheless , the correspondence ( [ evaluationatzero ] ) is a one - to - one correspondence between the elements of @xmath459 and the elements of @xmath455 . indeed , the symmetry under @xmath460 : @xmath461 allows to relate @xmath199 to @xmath462 , see eq . ( [ actionofgoncovariant ] ) . in other words , if we know the value of the covariant vertex at the point @xmath458 _ for all states _ @xmath86 , then because of the global symmetry we know the covariant vertex everywhere ( for arbitrary @xmath40 and @xmath84 ) . this construction is an example of the _ frobenius reciprocity _ : which is true for any representation @xmath42 of @xmath149 ; in our case @xmath463 . to summarize , given the covariant vertex @xmath120 , we define @xmath464 by saying that the value of @xmath445 on @xmath115 is : @xmath465 note that both the left hand side and the right hand side of ( [ vectorfromcovariantvertex ] ) are elements of @xmath451 _ i.e. _ polynomials of @xmath167 and @xmath168 . we can evaluate them on @xmath169 : @xmath466 this is a quadratic polynomial in @xmath167 and @xmath168 . ( [ actionofgoncovariant ] ) and ( [ qonuniversal ] ) imply that the action of @xmath3 on the covariant vertex corresponds to the following action on @xmath445 : @xmath467 this formula for @xmath3 can be interpreted in the following way . the space @xmath468 is obviously a representation of @xmath9 , just because @xmath21 is by definition a representation of @xmath31 . ( the @xmath193 part just `` goes along for the ride '' . ) let us denote this representation @xmath8 ( the action of @xmath469 on @xmath445 is @xmath470 ) . then ( [ brstoncovariantvertex ] ) implies that the action of @xmath82 on @xmath471 corresponds to the action of the nilpotent operator @xmath472 on @xmath445 defined by this formula : @xmath473 this is identical to @xmath472 of ( [ brstcomplexendpoint ] ) at @xmath474 . we conclude that : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath445 represents a cohomology class @xmath475 of the following complex : + @xmath23 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in other words , the brst complex on covariant massless vertices ( independent of derivatives ) is equivalent to the endpoint complex @xmath476 . we have demonstrated that the complex ( [ zequalsone ] ) is equivalent to ( [ brstcomplexendpoint ] ) at @xmath474 . but in fact ( [ brstcomplexendpoint ] ) at @xmath474 is equivalent to ( [ brstcomplexendpoint ] ) at @xmath477 by rescaling of @xmath167 and @xmath168 . in other words , the map @xmath478 is the equivalence of the complex ( [ brstcomplexendpoint ] ) at @xmath474 and the same complex at @xmath477 . there is a relation between the endpoint cohomology and the cohomology of the positive - frequency part of the loop algebra of @xmath479 . consider the algebra formed by the positive frequency @xmath28-twisted loops with values in @xmath479 . we will denote is @xmath480 . the cohomology complex is generated by the ghosts @xmath481 , where @xmath482 and @xmath92 the enumerates the adjoint representation of @xmath479 . we have @xmath483 , @xmath484 , @xmath485 , @xmath486}$ ] , @xmath487 , _ etc . _ the `` energy '' operator @xmath488 counts the lower indices , for example : @xmath489 } = -5 c_{-1}^{\alpha } c_{-4}^{[mn]}\ ] ] notice that @xmath488 is a symmetry of the cohomology complex . another symmetry is the @xmath237-ghost number ( the number of letters @xmath237 ) . let @xmath490 denote the cohomology group with @xmath491 and ghost number @xmath262 . the first cohomology group @xmath492 is generated by @xmath493 ( and therefore has @xmath494 ) . some of other nontrivial cohomology groups are : @xmath495 generally speaking , @xmath496 is generated by the expressions of the form : @xmath497 but @xmath496 is not all of the cohomology , for example there is nontrivial with the character of @xcite ] @xmath498 . the pure spinor cohomology should be identified with the part of the @xmath480 cohomology with `` energy '' equal to minus the ghost number , _ i.e. _ @xmath496 . in flat infinite space massless vertex operators have the form : @xmath499 where @xmath262 is some polynomial of @xmath169 and @xmath84 . we can write @xmath500 where @xmath501 therefore there is a generalization of ( [ vertexinflatspace ] ) : @xmath502 this generalization is only well defined in flat space on those worldsheets which do not have handles , or in toroidal compactifications with appropriate integrality conditions on @xmath503 and @xmath504 . we can formally consider ( [ flatspacegeneralizedvertex ] ) , for example on an infinite worldsheet without handles , if we neglect boundary effects . we will now argue that there is a partial analogue of ( [ flatspacegeneralizedvertex ] ) in @xmath0 . given a state @xmath115 we can prepare a nonlocal @xmath11-dependent covariant vertex operator , in the following way . consider the transfer matrix @xmath505 from infinity to the point @xmath506 on the worldsheet . let us fix a vector @xmath507 , and consider : @xmath508 so defined @xmath509 is analogous to : @xmath510 in particular for @xmath474 we get @xmath511 and therefore : @xmath512 this formula gives us back the covariant vertex for the state @xmath86 if we identify : @xmath513 for any @xmath514 we define @xmath515 as a non - normalizable vector in the space of states , characterized by the formula : @xmath516 where @xmath517 is the hermitean scalar product in @xmath21 . note that @xmath515 strictly speaking does not belong to @xmath21 because it is not normalizable . for example , one dimensional quantum mechanics has @xmath518 the space of square integrable functions of one variable , with the norm @xmath519 . the dual space @xmath44 is the space of generalized functions ; if @xmath514 is defined by the formula @xmath520 then @xmath515 is a delta - function @xmath521 . using these notations we can define the two - point vertex operator : @xmath522 however , we conjecture that this 2-point vertex operator is in fact brst exact . indeed , although we have not checked it explicitly , it should be true that the derivative of @xmath523 with respect to @xmath524 is @xmath3-exact . therefore , up to brst - exact terms this vertex is independent of @xmath524 and @xmath525 . on the other hand , when @xmath526 we get a local vertex operator of the ghost number 4 . there is no @xmath20-invariant cohomology at the ghost number 4 . this implies that ( [ twopointvertex ] ) is brst exact . in this paper we introduced a family of @xmath11-dependent vertex operators ( [ zdependentvertex ] ) parametrized by a choice of the bps representation of @xmath20 . schematically , these vertex operators have a form : @xmath527 \right ) \right\|\psi^{\infty}\right\rangle\ ] ] this expression is strictly speaking not brst invariant , because of the boundary term at infinity . indeed we have put @xmath528 an arbitrary vector from @xmath21 , and this is generally speaking not a valid plug . we assume that we can neglect this boundary term because it is at infinity ] . we can consider @xmath509 locally near the point @xmath506 . notice that @xmath529 is a @xmath169-dependent vector in the dual space to @xmath21 . ( in fact @xmath529 depends on @xmath530 and @xmath531 through @xmath532 . ) we can also think of @xmath529 as an element of @xmath21 , but then we have to remember that it is not normalizable ; it is a @xmath533-function type of state , rather than a proper wave packet . note that for a fixed @xmath169 , our @xmath534 is a _ fixed _ vector in @xmath44 . in other words , for every bps representation @xmath21 we have a map , which takes a pair of pure spinors and transforms them into a vector in the space of bps states : 0= @xmath535 it would be interesting to describe this map explicitly . the non - normalizable vector in @xmath21 on the right hand side of ( [ frompstovectorinh ] ) is obviously not invariant under @xmath20 ( it belongs to an irreducible representation ) . but it transforms covariantly under @xmath536 , in the sense that the action of @xmath319 on the right hand side of ( [ frompstovectorinh ] ) agrees with the action of @xmath319 on the left hand side of ( [ frompstovectorinh ] ) . another way of thinking about @xmath537 is in terms of the cohomology of the operator : @xmath538 acting on the bps representation @xmath21 ( more precisely , the @xmath149-invariant tensor product of @xmath44 with the space of polynomials of @xmath447 ) . our results imply that the second cohomology of this operator is nontrivial , represented by the cocycle ( [ frompstovectorinh ] ) . notice that this provides a purely representation - theoretic characterization of the linearized sugra spectrum on @xmath0 . indeed , the question of the existence of the excitation transforming in the representation @xmath21 is reduced to the calculation of the cohomology of the operator ( [ lambdatinconclusions ] ) , which is defined in terms of the generators @xmath539 of the representation @xmath21 . there is also another example of a plug , a plug of the ghost number 1 . consider the wilson line in the adjoint representation . the cohomology of ( [ lambdatinconclusions ] ) in the adjoint representation is nontrivial and is represented by : @xmath540 this is obviously a @xmath169-dependent vector in the adjoint representation , of the ghost number @xmath541 . one can verify that this is annihilated by ( [ lambdatinconclusions ] ) ; note the relative minus sign of the second term in ( [ adjointrepresentative ] ) . therefore we can take ( [ adjointrepresentative ] ) as a plug , and consider : @xmath542 \right ) \psi^{(\infty ) } \right)\ ] ] at @xmath474 the corresponding integrated vertex operator is the density of the local conserved charge @xmath543 . we will prove in appendix [ sec : ghostnumberone ] that ( [ adjointrepresentative ] ) is the only example of the endpoint cohomology at ghost number 1 . in particular , there is no nontrivial cohomology for representations other than the adjoint . with these notations the 2-point vertex ( [ twopointvertex ] ) reads : @xmath544 \right ) \right\| \mbox{plug}(\tau_1^+,\tau_1 ^ - ) \right\rangle \nonumber\end{aligned}\ ] ] ( but as we discussed at the end of section [ sec : twopoint ] this must be brst exact . ) i want to thank y. aisaka , n.j . berkovits , v. serganova and b.c . vallilo for many useful discussions . this research was supported by the sherman fairchild fellowship and in part by the rfbr grant no . 06 - 02 - 17383 and in part by the russian grant for the support of the scientific schools nsh-8065.2006.2 . part of this work was done during the workshop `` non - perturbative methods in strongly - coupled gauge theories '' , at the galileo galilei institute for theoretical physics in florence . i would like to thank the organizers of this workshop for their hospitality . another part of this work was done during the monsoon workshop in tifr , mumbai . i want to thank the organizers and the staff members at tifr for their hospitality . another part of this work was done during my stay at the ift so paulo ; i want to thank n.j . berkovits for his hospitality . another part of this work was done during the workshop `` fundamental aspects of superstring theory '' at kitp santa barbara ; i want to thank the organizers of the workshop for their hospitality . in this section we will prove that the only cohomology at ghost number 1 are the global @xmath20 conserved charges . the conserved charges are the descendants of the cohomology classes of the ghost number 1 . the `` standard '' local conserved charges correspond to the global symmetries @xmath33 . they descend from the following operator : @xmath545 in other words , we have the following cohomology class of the ghost number one in the adjoint representation of @xmath31 : @xmath546 in this section we will prove that there are no nontrivial cohomology classes of the ghost number 1 in representations other than the adjoint . we will write a representative of the cohomology class in the following way : @xmath549 the condition of @xmath54-invariance says that @xmath550 and @xmath551 should define intertwining operators of @xmath54 : @xmath552 in other words : @xmath553 } v_{\alpha } = { f_{[\rho\sigma]\alpha}}^{\beta}v_{\beta } \label{firstcovariancecondition}\\ & & t^0_{[\rho\sigma ] } v_{\dot{\alpha } } = { f_{[\rho\sigma]\dot{\alpha}}}^{\dot{\beta } } v_{\dot{\beta } } \label{secondcovariancecondition}\end{aligned}\ ] ] the conditions of @xmath54-covariance . the conditions for being annihilated by @xmath81 are : @xmath554 for example , the class ( [ adjointclass ] ) is represented by : @xmath555 we consider the solutions of ( [ conditionthatqannihilates ] ) trivial if they are of the form : @xmath556 where @xmath557}\phi=0 $ ] . _ theorem : _ nontrivial solutions to eqs . ( [ firstidentity ] [ thirdidentity ] ) exist only when @xmath547 is the adjoint representation of @xmath31 , and are given by ( [ onlysolution ] ) up to adding a trivial solution . there are no other nontrivial solutions . [ [ cohomology - classes - at - ghost - number-1-consequences - of - the - defining - equations ] ] cohomology classes at ghost number 1 : consequences of the defining equations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ acting on ( [ firstidentity ] ) by @xmath558 we get : @xmath559 } t^0_{[\rho\sigma ] } v_{\beta } - t^3_{\alpha}t^1_{\dot{\beta}}v_{\beta } \right ) + ( \alpha\leftrightarrow\beta ) = { f_{\alpha\beta}}^{\mu } t^1_{\dot{\beta } } a_{\mu}\ ] ] this with eqs . ( [ firstcovariancecondition ] ) and ( [ thirdidentity ] ) implies : @xmath559 } { f_{[\rho\sigma]\beta}}^{\gamma } v_{\gamma } + ( \alpha\leftrightarrow\beta)\right ) + { f_{\alpha\beta}}^{\mu}t^2_{\mu } \tilde{v}_{\dot{\beta } } = { f_{\alpha\beta}}^{\mu}t^1_{\dot{\beta } } a_{\mu}\ ] ] this with the jacobi identity for @xmath560 implies : @xmath561 similarly we have : @xmath562 let us act on ( [ fv ] ) by @xmath563 : @xmath564 this and ( [ thirdidentity ] ) implies : @xmath565 let us denote : @xmath566 . we have : @xmath567 } = 0\ ] ] note that the gauge transformation @xmath568 where @xmath74 is @xmath149-invariant leads to : @xmath569 therefore we should think of @xmath570 as an element of @xmath571 . but this cohomology group is zero because @xmath572 ( notice that the serre - hochschild spectral sequence for @xmath286 has @xmath573 and @xmath250 acts from @xmath574 to @xmath575 ) . therefore we should be able to gauge away @xmath570 . let us therefore assume : @xmath576 note that this equation and ( [ firstrelationofas ] ) implies : @xmath577 therefore we can rewrite ( [ fv ] ) and ( [ ftv ] ) as follows : @xmath578 let us define @xmath579 by the following equation : @xmath580 eq . ( [ antisymmetryoff ] ) implies that @xmath581 is antisymmetric : @xmath582 . we get : @xmath583 } t^0_{[\rho\sigma ] } a_{\nu } + t^2_{\mu } t^2_{\lambda } a_{\nu } = \nonumber \\ & = & { f_{\lambda [ \mu \|}}^{[\rho\sigma ] } { f_{[\rho\sigma]\|\nu]}}^{\kappa } a_{\kappa } - t^2_{[\mu } t^2_{\nu ] } a_{\lambda } = \nonumber \\ & = & -{1\over 2 } { f_{\mu\nu}}^{[\rho\sigma ] } { f_{[\rho\sigma]\lambda}}^{\kappa } a_{\kappa } - { 1\over 2 } { f_{\mu\nu}}^{[\rho\sigma ] } { f_{[\rho\sigma]\lambda}}^{\kappa } a_{\kappa } = \nonumber \\ & = & - { f_{\mu\nu}}^{[\rho\sigma ] } { f_{[\rho\sigma]\lambda}}^{\kappa } a_{\kappa } \label{t2onf}\end{aligned}\ ] ] this implies , first of all , that @xmath579 is zero when @xmath584 is tangent to @xmath25 and @xmath585 is tangent to @xmath114 . therefore @xmath579 is completely encoded in terms of @xmath586}$ ] which is defined by this equation : @xmath587 } g_{[\rho\sigma]}\ ] ] eq . ( [ t2onf ] ) implies for @xmath588}$ ] : @xmath589 } = { f_{\lambda[\mu\nu]}}^{\kappa } a_{\kappa}\ ] ] this means that the linear space formed by @xmath590 and @xmath586}$ ] is closed under the action of @xmath302 , and is in fact the adjoint representation of @xmath302 ( where @xmath590 corresponds to @xmath591 and @xmath586}$ ] corresponds to @xmath592}$ ] ) . this is already close to what we wanted to prove . but we have to also tame the expressions of this form : @xmath593 } \;\ ; , \;\;\ ; etc.\ ] ] for this purpose , let us use ( [ firstfv ] ) and ( [ secondfv ] ) in this expression : @xmath594 on the other hand , the combination @xmath595 can be calculated using the @xmath54-invariance . this implies : @xmath596 we will also use this : @xmath597 } t^0_{[\rho\sigma ] } t_{\delta}^3 a_{\nu } - { 1\over c } { f_{\mu}}^{\gamma\delta } t^3_{\gamma } t^1_{\dot{\alpha } } t^3_{\delta } a_{\nu } = \nonumber \\ & & = { 1\over c}(fff)(t^3 a ) \label{threestructureconstants}\end{aligned}\ ] ] here we used the schematic notation @xmath598 for a product of three structure constants with some indices contracted , and @xmath237 is determined from @xmath599 . the subspace of @xmath547 generated by acting on @xmath5 and @xmath127 by finitely many @xmath600 and @xmath601 is finite - dimensional . indeed , using ( [ reducea ] ) and ( [ threestructureconstants ] ) we can prove that it is generated as a linear space by expressions of the form : @xmath602 } a_{\mu}\;,\;\ ; t^3_{[\alpha_1 } \cdots t^3_{\alpha_k ] } a_{\mu } \;\;\ ; ( k\geq 0 ) \label{subspacegeneratorsa } \\ & \mbox{and } & g_{[\mu\nu ] } \label{subspacegeneratorsg}\end{aligned}\ ] ] where the square brackets stand for the antisymmetrization of the indices ( for example @xmath603 } a_{\mu}$ ] stands for @xmath604 ) . ( [ reducea ] ) and ( [ threestructureconstants ] ) imply that this subspace is closed under the action of @xmath31 . because @xmath547 is assumed to be irreducible , we conclude that @xmath547 is generated by ( [ subspacegeneratorsa]),([subspacegeneratorsg ] ) . because of the antisymmetrization of the indices of @xmath600 and @xmath601 there are only finitely many linearly independent expressions of the form ( [ subspacegeneratorsa ] ) . this proves that @xmath547 is a finite - dimensional space . _ one can see that for any element @xmath445 of @xmath190 ( _ i.e. _ a finite linear combination of expressions of the form ( [ subspacegeneratorsa ] ) ) there is a number @xmath262 such that for any @xmath609 and any @xmath610 elements @xmath611 of @xmath302 we get : @xmath612 indeed , let us consider for example acting by @xmath613 on expressions of the form @xmath614 } a_{\mu}$ ] . let us define the degree of such an expression by the following formula : @xmath615 } a_{\mu } = \mbox{deg } t^1_{[\dot{\alpha}_1 } \cdots t^1_{\dot{\alpha}_k ] } a_{\mu } = k\ ] ] more precisely , we introduce a filtration of @xmath190 saying that @xmath616 consists of all the elements of @xmath21 which can be written as linear combinations of the expressions of the form ( [ subspacegeneratorsa ] ) of the degree less or equal @xmath617 . using eqs . ( [ reducea ] ) and ( [ threestructureconstants ] ) we derive that for @xmath618 : @xmath619 this implies ( [ eventuallyreachl ] ) . because of the assumption that @xmath547 is completely reducible as a representation of @xmath302 , eq . ( [ eventuallyreachl ] ) implies that @xmath608 . we conclude that @xmath547 is in fact generated by expressions ( [ smallsetofgenerators ] ) . note that the linear space generated by ( [ smallsetofgenerators ] ) consists of the even subspace generated by @xmath590 and @xmath586}$ ] , and odd subspace generated by @xmath620 . the even subspace is the same as in the adjoint representation . therefore the odd space should be also the same . we have argued that the subspace generated by ( [ subspacegeneratorsa ] ) in fact coincides with @xmath547 , based on @xmath547 being an irreducible representation of @xmath31 . this requirement can be replaced with the requirement that @xmath621 . suppose that ( [ subspacegeneratorsa ] ) generate a smaller subspace @xmath622 . let us denote @xmath445 the projection of @xmath121 on @xmath623 : @xmath624 then ( [ secondidentity ] ) implies : @xmath625 these equations imply that @xmath626 actually form the spinor representation of @xmath302 : @xmath627 we will now explain , using ( [ geveninvariance1 ] ) and ( [ geveninvariance2 ] ) , that @xmath445 can be gauged away if @xmath628 . and this cohomology group vanishes if @xmath629 vanishes . more generally , let us consider the subcomplex of the brst complex covariant under @xmath302 . this means consider the chains which satisfy : @xmath630 on this complex the brst operator coincides with the lie cohomology @xmath81-operator in the relative complex @xmath631 . the relative cohomology is related to @xmath632 by the serre - hochschild spectral sequence . namely @xmath633 the differential @xmath634 acts from @xmath635 to @xmath636 . in particular : @xmath637 is related to @xmath638 . @xmath639 this means that @xmath640 can not cancel with anything and therefore @xmath641 implies that @xmath642 . therefore , if @xmath629 vanishes , then @xmath643 should also vanish . a. mikhailov and s. schafer - nameki , _ algebra of transfer - matrices and yang - baxter equations on the string worldsheet in ads(5 ) x s(5 ) _ , _ nucl . phys . _ * b802 * ( 2008 ) 139 , [ http://xxx.lanl.gov/abs/0712.4278[0712.4278 ] ] . v. v. bazhanov , s. l. lukyanov , and a. b. zamolodchikov , _ integrable structure of conformal field theory ii . q- operator and ddv equation _ , _ commun . _ * 190 * ( 1997 ) 247278 , [ http://xxx.lanl.gov/abs/hep-th/9604044[hep-th/9604044 ] ] . r. g. leigh and m. j. strassler , _ exactly marginal operators and duality in four - dimensional n=1 supersymmetric gauge theory _ , _ nucl . * b447 * ( 1995 ) 95136 , [ http://xxx.lanl.gov/abs/hep-th/9503121[hep-th/9503121 ] ] . o. aharony , b. kol , and s. yankielowicz , _ on exactly marginal deformations of n = 4 sym and type iib supergravity on ads(5 ) x s*5 _ , _ jhep _ 06 * ( 2002 ) 039 , [ http://xxx.lanl.gov/abs/hep-th/0205090 [ hep - th/0205090 ] ] . n. berkovits and p. s. howe , _ ten - dimensional supergravity constraints from the pure spinor formalism for the superstring _ , _ nucl . _ * b635 * ( 2002 ) 75105 , [ http://xxx.lanl.gov/abs/hep-th/0112160 [ hep - th/0112160 ] ] . j. kinney , j. m. maldacena , s. minwalla , and s. raju , _ an index for 4 dimensional super conformal theories _ , * 275 * ( 2007 ) 209254 , [ http://xxx.lanl.gov/abs/hep-th/0510251 [ hep - th/0510251 ] ] . j. bhattacharya , s. bhattacharyya , s. minwalla , and s. raju , _ indices for superconformal field theories in 3,5 and 6 dimensions _ , _ jhep _ * 02 * ( 2008 ) 064 , [ http://xxx.lanl.gov/abs/0801.1435[0801.1435 ] ] .	the worldsheet sigma - model of the superstring in @xmath0 has a one - parameter family of flat connections parametrized by the spectral parameter . the corresponding wilson line is not brst invariant for an open contour , because the brst transformation leads to boundary terms . these boundary terms define a cohomological complex associated to the endpoint of the contour . we study the cohomology of this complex for wilson lines in some infinite - dimensional representations . we find that for these representations the cohomology is nontrivial at the ghost number 2 . this implies that it is possible to define a brst invariant open wilson line . the central point in the construction is the existence of massless vertex operators transforming exactly covariantly under the action of the global symmetry group . in flat space massless vertices transform covariantly up to adding brst exact terms , but in ads it is possible to define vertices so that they transform exactly covariantly . calt-68 2722 * @xmath1 + @xmath1 + symmetries of massless vertex operators in @xmath0 * + + +	29,630	291
owing to recent progress in nanotechnology and materials science , nano- and micro - mechanics @xcite have emerged as subjects of great interest due to their potential use in demonstrating macroscopic quantum phenomena , and possible applications in precision measurements , detecting gravitational waves , building filters , signal amplification , as well as switches and logic gates . in particular , on - chip single- or few - phonon devices are ideal candidates for hybrid quantum information processing , due to the ability of phonons to interact and rapidly switch between optical fields and microwave fields @xcite . fabrication of high - frequency mechanical resonators @xcite , demonstration of coherent phonon coupling between nanomechanical resonators @xcite , ground - state cooling @xcite , optomechanics ( in microtoroids @xcite , microspheres @xcite , microdisks @xcite , microring @xcite , photonic crystals @xcite , doubly- or singly - clamped cantilevers @xcite , and membranes @xcite ) have opened new directions @xcite and provided new tools to control and manipulate phonons in on - chip devices . one possible obstacle to further develop this field is the ability to control the flow of phonons , allowing transport in one direction but not the opposite direction @xcite , i.e. , nonreciprocal phonon transport . there have been several attempts to fabricate nonreciprocal devices for phonons @xcite , but these are almost exclusively based on asymmetric linear structures which indeed can not break lorentz reciprocity : a static linear structure can not break reciprocity @xcite . these proposed linear structures do obey the reflection - transmission reciprocity and thus can not be considered as phonon diodes " . diode - like behavior was observed in these linear acoustic structures because the input - output channels were not properly switched @xcite . nonreciprocal phonon transmission inevitably requires magneto - acoustic materials , strong nonlinearity , or a time - dependent modulation of the parameters of a structure . although already demonstrated in optics @xcite , the time - dependent modulation of acoustic parameters of a phononic structure has not been probed yet . magneto - acoustic materials require high magnetic fields to operate and have been studied @xcite ; however , a magnetic - free nonreciprocal device is critical for building on - chip and small - scale phononic processors and circuits . nonlinearity - based nonreciprocity seems to be the most viable approach for creating micro- or nano - scale nonreciprocal devices for controlling and manipulating phonons . recently , there have been several reports on nonlinear mechanical structures and materials @xcite . however , the weak nonlinearity of those acoustic / phononic materials hinders progress in this direction due to the high input powers required to observe the nonlinear effects @xcite . in order to circumvent this problem , coupling a weakly nonlinear structure to an auxiliary system , such as a quantum bit @xcite , has been proposed to engineer effective giant mechanical nonlinearities . in order to achieve the required nonlinearity for nonreciprocal phonon transport and to study nonlinear phononics , here we introduce a new method based on parity - time ( @xmath0 ) symmetry @xcite , which has attracted much attention recently due to their interesting and generally counter - intuitive physics @xcite . parity - time symmetry and its breaking ( broken @xmath0-phase ) have been demonstrated in various physical systems @xcite , such as optical waveguides @xcite , microcavities @xcite , and electrical circuits @xcite . however , mechanical @xmath0-symmetric systems have only been considered quite recently @xcite . in our proposed mechanical @xmath0 symmetric system , a lossy mechanical resonator ( passive resonator ) which has a weak mechanical nonlinearity is coupled to a mechanical resonator with mechanical gain ( active resonator ) that balances the loss of the passive resonator . the active resonator here works as a dynamical amplifier . in the vicinity of the @xmath0-phase transition , the weak nonlinearity is first distributed between the mechanically - coupled resonators and then significantly enhanced due to the localization of the mechanical supermodes in the active resonator . in this way , the effective nonlinear kerr coefficient is increased by over three orders of magnitude . this strong nonlinearity , localized in the active resonator , blocks the phonon transport from the active resonator to the lossy resonator but permits the transport in the opposite direction . for the experimental realization of the proposed nonlinearity - based phonon diode , we provide a system in which a mechanical beam with weak mechanical nonlinearity is coupled to another mechanical beam with gain . we show that this micro - scale system can be switched from a bidirectional transport regime to a unidirectional transport regime , and vice versa , by properly adjusting the detuning between the mechanical frequency of the resonators and the frequency of the driving phononic field , or by varying the amplitude of the input phononic field . the system we consider here consists of two mechanical resonators , one of which has mechanical loss ( passive resonator ) and weak nonlinearity , and the other has mechanical gain ( active resonator ) but no nonlinearity ( see fig . [ fig of mechanical pt ] ) . the mechanical coupling between the resonators is linear and it gives rise to the mechanical supermodes @xmath1 with complex eigenfrequencies @xmath2 given by @xmath3 here @xmath4 is the mechanical frequency of the solitary mechanical resonators ( i.e. , both resonators are degenerate ) , @xmath5 where @xmath6 and @xmath7 denote , respectively , the damping rate of the lossy mechanical resonator and the gain rate of the active mechanical resonator , and @xmath8 is the coupling strength between the mechanical modes . when @xmath9 , the system is in the @xmath0-symmetric regime , and the supermodes are non - degenerate with @xmath10 and have the same damping rate @xmath11 ( see figs . [ fig of amplifying mechanical nonlinearity]a and [ fig of amplifying mechanical nonlinearity]b ) . however , when @xmath12 , the system is in the broken-@xmath0-symmetric regime , the supermodes are frequency - degenerate with @xmath13 ( see figs . [ fig of amplifying mechanical nonlinearity]a and [ fig of amplifying mechanical nonlinearity]b ) and have different damping rates @xmath14 at @xmath15 , the two supermodes are degenerate with the same damping rate , indicating a transition between the pt - symmetric regime and the broken - pt - symmetry regime . this point is generally referred to as the @xmath0-transition point . it is seen that the two supermodes will be lossless in the @xmath0-symmetric regime if the gain and loss are well - balanced , such that @xmath16 . let us assume that the passive resonator is made from a nonlinear acoustic material @xcite with a small nonlinear kerr coefficient @xmath17 . this nonlinearity mediates a cross - kerr interaction between the two mechanical supermodes , which leads to the effective nonlinear coefficients @xmath18 and @xmath19 , in the broken- and unbroken-@xmath0 regimes @xcite : @xmath20 clearly , the effective nonlinear coefficients are significantly enhanced in the vicinity of the phase transition point @xmath15 . moreover , if the gain and loss are well - balanced , i.e. , @xmath16 , the supermodes become almost lossless . this observation is one of the key contributions of this paper . namely , operating the system of two coupled mechanical resonators in the vicinity of the phase transition point will significantly enhance the existing very weak nonlinearity with an extremely small loss rate . using the parameter values of @xmath21 , @xmath22 , and @xmath23 , we show in fig . [ fig of amplifying mechanical nonlinearity ] the evolution of the eigenfrequencies of the system and of the nonlinear coefficient as a function of @xmath24 . the transition from the broken- to the unbroken-@xmath0 symmetric regime and vice versa , as the mechanical coupling strength is varied , is seen in fig . [ fig of amplifying mechanical nonlinearity]a and [ fig of amplifying mechanical nonlinearity]b and it is reflected in the bifurcations of the supermode frequencies and damping rates . moreover , the enhancement of the nonlinearity in the vicinity of the @xmath0-phase transition point is seen in fig . [ fig of amplifying mechanical nonlinearity]c . we find that the nonlinear coefficient is enhanced by more than three orders of magnitude in the vicinity of the transition point . more interestingly , in the broken-@xmath0 regime , the mechanical energy of the coupled system is localized in the active resonator , which leads to a nonlinear mechanical mode with strong self - kerr nonlinearity localized in the active mechanical resonator . this can be interpreted intuitively as follows . the initial weak mechanical nonlinearity is transferred from the passive resonator to the active resonator and it is enhanced by field localization in the broken-@xmath0 regime . owing to the presence of the mechanical gain , the active resonator then enjoys an almost lossless mechanical mode with a giant nonlinearity ( see fig . [ fig of nonlinearity transfer ] ) . finally , we would like to consider how the mechanical nonlinearity will affect the @xmath0-symmetric structure of the system . generally speaking , a strong nonlinearity will shift the transition point of a @xmath0-symmetric system or even destroy the @xmath0 symmetry of such a system @xcite . however , in our case , we start from a system in which a gain resonator is coupled to a lossy resonator with very weak kerr nonlinearity , and thus we can omit the shift of the @xmath0-transition point induced by such a weak nonlinearity . although we generate a strong nonlinearity in the vicinity of the @xmath0-transition point , this is an effective nonlinearity induced in the supermode picture and thus will not affect the supermodes and the @xmath0-transition point of the system . here we investigate the effect of the enhanced mechanical nonlinearity on the phonon transport in the coupled system . we find that the localized strong mechanical nonlinearity leads to unidirectional phonon transport from the passive resonator to the active resonator and blocks phonon transport in the opposite direction ( i.e. , phonon transport from the active to the passive resonator is prevented ) . the transport is almost lossless due to the gain - loss balance of the system . when this system is operated in the vicinity of the @xmath0-phase transition point , the unidirectional phonon transport is possible within a region given by @xcite @xmath25,\ ] ] where @xmath26 is the detuning between the input ( driving ) field frequency @xmath27 and the resonance frequency @xmath4 of the mechanical resonators . additionally , in order to observe the unidirectional phonon transport , the amplitude of the input field should satisfy @xmath28,\ ] ] implying that the intensity of the input field required for unidirectional transport is inversely proportional to the strength of the mechanical nonlinearity @xmath18 . since the strength of the mechanical nonlinearity can be enhanced by more than three orders of magnitude by breaking the @xmath0 symmetry , the threshold of the input - field intensity for observing unidirectional phonon transport can be decreased by at least three orders of magnitude , allowing a low - threshold phonon diode operation . to show unidirectional phonon transport in the broken-@xmath0 regime , let us first fix the amplitude of the input field and vary the detuning @xmath29 . we compare the amplitude transmittance @xmath30 and @xmath31 the former , @xmath32 , denotes the transmission from the passive to the active resonator , that is , the system is driven by a phononic input field @xmath33 of frequency @xmath27 at the passive resonator side and the output @xmath34 is measured at the active resonator side . however , the latter , @xmath35 , denotes the amplitude transmittance from the active resonator to the passive resonator when the system is driven by the field @xmath36 of frequency @xmath27 at the active resonator side and the output @xmath37 is measured at the passive side . the nonlinearity in the system manifests as a bistability and hysteresis in the power transmittance , @xmath38 and @xmath39 obtained as the detuning @xmath29 is up - scanned from smaller to larger detuning and down - scanned from larger to smaller detuning ( see fig . [ fig of unidirectional phonon transport]a ) . we find that during the down - scan , both of the transmittances @xmath40 and @xmath41 stay at the lower branch with values close to zero until @xmath42 , after which they bifurcate from each other only slightly and then jump to the stable points at the upper branch of their respective trajectories ( see fig . [ fig of unidirectional phonon transport]a ) . further decreasing the detuning leads to an increase in @xmath40 , but a decrease in @xmath41 . this implies that there is no unidirectional phonon transport with the parameter values used in the numerical simulations . instead , when the detuning is below a critical value , the phonon transport is bidirectional ; whereas when it is above that critical value there is no phonon transport . during the up - scan , however , after a short stay on the stable state , i.e. , a regime in which there is no bistability and hysteresis in the transmittance , ( during which @xmath43 decreases and @xmath41 increases with growing detuning ) , both of the transmittances follow the upper branches of their trajectories , during which a linear increase in @xmath41 and a slow - rate decrease in @xmath43 are observed ( see fig . [ fig of unidirectional phonon transport]a ) . this behavior continues until @xmath44 for @xmath41 , where it jumps to the lower branch of its trajectory , and becomes zero as the detuning is increased ( see fig . [ fig of unidirectional phonon transport]a ) . this implies that phonon transport from the active mechanical resonator to the passive one is prevented if the detuning is set to @xmath45 . the transmittance @xmath40 stays at its upper branch with a value close to one until @xmath46 , where it jumps to its lower branch and becomes zero . thus , for @xmath47 , phonon transport from the passive to the active resonator is prevented . clearly , in the detuning region @xmath48 , the transmittance @xmath40 is close to one whereas @xmath41 is close to zero in this detuning region phonon transport from the active mechanical resonator to the passive one is forbidden , whereas phonon transport from the passive mechanical resonator to active one is allowed with almost no loss . thus , we conclude that phonon transmission is non - reciprocal in this detuning region , and the rectification is @xmath49 db within the nonreciprocal transport region ( see fig . [ fig of unidirectional phonon transport]a ) . for detuning values smaller than the lower bound of this region , phonon transport is bidirectional . for detuning values larger than the upper bound of the region , phonon transport is not possible . note that our phonon diode should work only when the disturbance and perturbation of the system parameters are not too strong . in fact , within the unidirectional phonon transport window shown in fig . [ fig of unidirectional phonon transport]a , the transmittance @xmath40 has two different branches of metastable values . when we increase the detuning @xmath29 within this unidirectional phonon transport window , @xmath40 will stay in the upper stable branch if we do not severely disturb the system and the phonon diode should operate properly . however , if the disturbance is too strong , @xmath40 will jump from the upper branch to the lower branch and stay in this stable lower branch , without rectification . alternatively , we can fix the detuning and vary the amplitude of the input field to show the nonlinearity - induced bistability and hysteresis . a nonreciprocal phonon transport region is seen when the amplitude of the input field is up - scanned ( see fig . [ fig of unidirectional phonon transport]b ) . the nonreciprocal transport region disappears when the amplitude of the input field is down - scanned . within the nonreciprocal transport region , when the input is varied at fixed detuning ( see fig . [ fig of unidirectional phonon transport]b ) , the rectification is @xmath49 db . similarly , in this case , due to the metastability of the transmittance @xmath40 , the disturbance - induced perturbation of the system parameters may not be too strong otherwise our design of phonon diode will be invalid . the unidirectional phonon transport enabled by the @xmath0-breaking - induced strong mechanical nonlinearity can be used to fabricate lossless phonon diodes in on - chip systems . this may have many applications , such as single - phonon transistors and routers , on - chip quantum switches , and information - processing components . one possible way to realize the proposed phonon diode is to use coupled beams and cantilevers ( see fig . [ fig of experimental setup]a ) . phonon lasing , and hence an active mechanical resonator , has been experimentally realized in an electromechanical beam @xcite . elastically - coupled nano beams and cantilevers , by which the mechanical supermodes can be generated , have also been shown in various experiments @xcite , in which the two mechanical resonators can be independently driven @xcite . thus our proposal is within the reach of current experimental techniques of nano - micro - electromechanical systems . let us now consider the design of the phonon diode system shown in fig . [ fig of experimental setup ] in which a lossy vibrating beam with damping rate @xmath6 and a weak kerr nonlinearity @xcite of strength @xmath17 is elastically coupled to another vibrating beam with gain @xmath7 @xcite . the frequencies of the two beams are both @xmath4 and the mechanical coupling strength is @xmath8 . in fig . [ fig of the power spectrum ] , we present the numerical results performed with the system parameters : @xmath50 khz , @xmath51 khz , @xmath52 khz , @xmath53 khz , @xmath54 khz , and @xmath55 khz . here , we fix the detuning @xmath29 and change the amplitude of the input field . there is a @xmath56 db background noise which includes the combined effect of the thermal noise on the mechanical resonators , the electrical noises induced by the measurement apparatus and other possible sources of noise . the results shown in fig . [ fig of the power spectrum ] for the phonon diode agree well with the general model discussed in the previous section . when the amplitude of the input field is increased , it is clearly seen that there is a nonreciprocal region in which phonon transport from the active beam to the passive beam is almost completely suppressed ( see fig . [ fig of the power spectrum]b(ii ) ) , but phonon transport from the passive beam to the active beam is allowed ( see fig . [ fig of the power spectrum]a(ii ) ) . a rectification ratio of about @xmath57 db is obtained . when the amplitude of the phonon excitation is larger than the upper bound of the unidirectional phonon transport region , the transport is bidirectional . in this case , the phonons can freely move from the active beam to the passive beam and vice versa ( see figs . [ fig of the power spectrum]a(i ) and [ fig of the power spectrum]b(i ) ) . finally , for amplitudes of the phonon excitation smaller than the lower bound of the region , no phonon transport can take place between the resonators ( see figs . [ fig of the power spectrum]a(iii ) and [ fig of the power spectrum]b(iii ) ) . these are the result of hysteresis ( see fig . [ fig of unidirectional phonon transport]b ) caused by the strong mechanical nonlinearity . we have proposed a method to generate ultra - strong mechanical nonlinearity with a very low - loss rate using a @xmath0-symmetric mechanical structure in which a mechanical resonator with gain but no nonlinearity is coupled to a lossy ( i.e. , passive mechanical loss and no gain ) mechanical resonator with very weak nonlinearity . we have showed that the weak mechanical nonlinearity is redistributed in the supermodes of the coupled mechanical system and is enhanced ( by more than three orders of magnitude ) when the mechanical @xmath0 system enters the broken-@xmath0 regime . moreover , owing to the presence of the mechanical gain in one of the resonators to compensate the mechanical loss of the other resonator , the effective mechanical damping rate is decreased in the @xmath0-symmetric system . using experimentally accessible parameter values , we identified the regimes where unidirectional phonon transport is possible from the passive to active resonator but not in the opposite direction . we then proposed an experimentally - realizable system where a mechanical beam with passive loss and weak nonlinearity is coupled to another beam which acts like an active mechanical resonator . a possible bottleneck for this design to achieve a phonon diode operated in ambient condition is whether the mechanical gain observed with the mechanical beams in a controlled environment and at low temperatures @xcite could also be obtained in ambient - temperature conditions . a possible way to overcome this problem , and to realize phonon diodes in ambient conditions , is to use a hybrid system composed of a gain optomechanical resonator and an nonlinear electrically - driven mechanical beam @xcite , where the coupling between them is achieved via the evanescent optical field of the optomechanical resonator @xcite . the mechanical gain of the optomechanical resonators can be provided at ambient conditions by , e.g. , the optomechanical dynamical instability in the blue detuning regime @xcite , which has been demonstrated in optomechanical resonators in various experiments @xcite . since creating strongly - nonlinear mechanical or acoustic materials remains challenging , we believe that the proposed system and the developed approach provide a suitable platform for investigating nonlinear phononics and can be used as a building block to design more complex hybrid optomechanical or electromechanical information processors . we envision that @xmath0 mechanical systems will open a new route for designing functional phononic systems with nonreciprocal phonon responses . + * acknowledgments * jz is supported by the nsfc under grant nos . 61174084 , 61134008 . yxl is supported by the nsfc under grant nos . 10975080 , 61025022 , 91321208 . yxl and jz are supported by the national basic research program of china ( 973 program ) under grant no . 2014cb921401 , the tsinghua university initiative scientific research program , and the tsinghua national laboratory for information science and technology ( tnlist ) cross - discipline foundation . ly and sko are supported by aro grant no . w911nf-12 - 1 - 0026 and the nsfc under grant no . 61328502 . is supported by the riken ithes project , muri center for dynamic magneto - optics via the afosr award number fa9550 - 14 - 1 - 0040 , and grant - in - aid for scientific research ( a ) . * author contributions * jz , bp , sko contributed equally to this work . ly , fn , sko , yxl supervised the project . in order to prove the enhancement of mechanical nonlinearity in the broken-@xmath0-symmetric regime , let us consider a system of two coupled mechanical resonators , in which one of the resonators has mechanical gain ( active resonator ) and thus a positive damping rate @xmath7 and the second mechanical resonator has a passive mechanical loss ( passive resonator ) with loss rate @xmath6 . the resonators have the same mechanical frequency @xmath4 , and the annihilation operators for their mechanical modes are denoted as @xmath58 and @xmath59 , respectively , for the active and passive resonators . moreover , the passive mechanical resonator has a weak mechanical kerr - nonlinearity denoted by @xmath17 . the hamiltonian describing these coupled mechanical resonators can be written as @xmath60 where @xmath8 is the coupling strength between the mechanical modes of the resonators . generally , the nonlinear kerr term in eq . ( [ two coupled mechanical resonators ] ) will shift the boundary between the @xmath0 symmetric regime and the broken-@xmath0 regime . however , in our model , the kerr nonlinearity denoted by @xmath17 is very weak , and we can omit the nonlinearity - induced shift of this boundary . to find the boundary of @xmath0 transition , we consider the first three terms in eq . ( [ two coupled mechanical resonators ] ) @xmath61 which can be written as @xmath62 this hamiltonian can be diagonalized as @xmath63 where the transformation matrix @xmath64 is defined by @xmath65 \\ g_{mm } & \left[\left(\omega_--\omega_0\right)-i\left(\gamma_--\gamma_l\right)\right ] \\ \end{array}\right)}{\sqrt{\left(\omega_{\pm}-\omega_0\right)^2+\left(\gamma_{\pm}-\gamma_l\right)^2+g_{mm}^2}}.\ ] ] consequently , we have @xmath66 as the mechanical supermodes formed by the coupling of the resonators . these supermodes @xmath1 are characterized by the eigenfrequencies @xmath67 and damping rates @xmath68 . for this mechanical @xmath0 symmetric system , there are two different regimes ( see fig . [ fig of eigenfrequencies of the pt system ] ) : \(i ) @xmath0 symmetric regime where @xmath69 and the two supermodes @xmath70 and @xmath71 are nondegenerate in their resonance frequencies ( i.e. , real part of their complex eigenfrequencies ) given by @xmath72 the damping rates of the supermodes ( i.e. , linewidths of the resonances ; imaginary part of their complex eigenfrequencies ) are the same and equal to @xmath73 \(ii ) broken @xmath0-symmetry regime where @xmath74 the two supermodes @xmath70 and @xmath71 are degenerate in their resonance frequencies @xmath75 and their damping rates are different : @xmath76 now let us consider the nonlinear kerr term in eq . ( [ two coupled mechanical resonators ] ) . using eq . ( [ supermodes ] ) , we find @xmath77 by substituting the above equation into the last term on the right hand side of eq . ( [ two coupled mechanical resonators ] ) and dropping the non - resonant terms , we can rewrite the nonlinear kerr term of eq . ( [ two coupled mechanical resonators ] ) as @xmath78 the self - kerr terms @xmath79 and @xmath80 only lead to a frequency - shift of the two supermodes and thus are less important . the cross - kerr term @xmath81 is more important and leads to the redistribution of the nonlinear effect among the two supermodes . from eqs . ( [ condition for pt symmetry])-([damping rate in the pt symmetric regime ] ) , the nonlinear coefficient @xmath82 can be represented in the broken-@xmath0 regime as @xmath18 , and in the @xmath0 symmetric regime as @xmath19 @xmath83 as was observed in photonic experiments @xcite , in the broken-@xmath0 regime the two supermodes @xmath1 are degenerate and the field is localized in the gain resonator , and thus the field @xmath59 is much smaller than @xmath58 . therefore , we can omit the terms related to @xmath59 in the expressions of the supermodes @xmath1 and we have @xmath84 subsequently , we find that the cross - kerr term given in eq . ( [ the cross - kerr term ] ) can induce a self - kerr effect in the gain resonator @xmath85 clearly , when @xmath86 ( in the vicinity of the spontaneous @xmath0-symmetry breaking point : the @xmath0-phase transition point ) , this self - kerr nonlinearity is greatly enhanced . let us now present a detailed analysis for finding the unidirectional phonon transport region near the @xmath0-transition point . in this case , the gain - loss balance between the active resonator , with annihilation operator @xmath58 , and the passive resonator , with annihilation operator @xmath59 , decreases the effective damping rates of the two modes . in the vicinity of the @xmath0-phase transition point ( i.e. , @xmath87 ) , the effective damping rates of the two modes is given by @xmath88 . the coupling between the two mechanical resonators also leads to the transfer of mechanical kerr nonlinearity from the passive resonator to the active resonator , and this mechanical nonlinearity is strongly enhanced near the @xmath0-transition point ( i.e. , @xmath87 ) . hereafter , we will denote this enhanced mechanical kerr nonlinearity coefficient as @xmath18 . let us first consider the phonon transport from the passive resonator to the active resonator . here the phononic field in the passive resonator is generated via an phononic input field with strength @xmath89 and frequency @xmath27 . using the standard input - output formalism @xcite , the output field of the active mechanical resonator is found as @xmath90 , which shows that the output field is proportional to the intracavity field @xmath58 , if we omit the vacuum fluctuations in the input field . thus the transmission from passive to active resonator is given by @xmath91 where @xmath92 represents the steady - state value of the intracavity phonon number in the active resonator . from the steady - state solution of the equations of motion for the coupled mechanical resonator system , we find that @xmath92 satisfies @xmath93 where @xmath94 the algebraic equation ( [ stationary equation of ng ] ) has three or one root depending on the system parameters , and one of the roots is unstable if the algebraic equation ( [ stationary equation of ng ] ) has three roots . when we increase the detuning @xmath95 , such that @xmath96 or equivalently , @xmath97 the system enters the bistable regime . in fact , when @xmath98 , the algebraic equation has three branches of solutions . however , two branches of solutions disappear when @xmath99 ( see ref . @xcite and the supplementary materials of ref . @xcite ) . in this case , the transmittance of the photon transport @xmath100 changes suddenly from a high value to a low value . noting that @xmath101 near the @xmath0 breaking point , the critical detuning @xmath102 can be approximately estimated to be @xmath103 let us now consider the phonon transport from the active mechanical resonator to the passive one . the driving field with strength @xmath89 and frequency @xmath27 is fed into the gain resonator in this case . following the same discussion and approach as for the previous case , it can be shown that a bistability - induced phase transition occurs when the detuning @xmath95 satisfies @xmath104 or equivalently , @xmath105 near the @xmath0-breaking point , @xmath106 , and thus @xmath107 can be approximately estimated to be @xmath108 combing eqs . ( [ upper bound of the unidirectional phonon transport window ] ) and ( [ lower bound of the unidirectional phonon transport window ] ) , we find that when the detuning @xmath29 is within the following region @xmath109=\left[\frac{g_{mm}^2\omega_0}{\omega_0 ^ 2+\chi^2},\frac{g_{mm}^2}{\omega_0-\sqrt{3}\chi}\right],\ ] ] it is possible to observe the unidirectional phonon transport , i.e. , the phonon transport from the passive resonator to the active resonator is allowed , whereas the phonon transport from the active resonator to the passive resonator is blocked . figure . [ fig of unidirectional phonon transport window]a shows the transmittance functions @xmath110 and @xmath111 as a function of the detuning @xmath29 . it is ( as explained in the main text ) clear that there is a unidirectional phonon transport region when the detuning is up - scanned from smaller to larger detuning . we also show in fig . [ fig of unidirectional phonon transport window]b the rectification ratios for up - scanning and down - scanning the detuning @xmath29 . similar to our previous discussions , a non - reciprocal region can be observed for the up - scanning process , while it disappears for the down - scanning process , and a high rectification - ratio , larger than @xmath57 db , can be obtained within the nonreciprocal region . up to this point , we do not consider the amplitude of the input field . let us assume that the detuning @xmath29 is fixed and is within the detuning region given by eq . ( [ unidirectional phonon transport window ] ) . we then vary the amplitude of the input field to show the bistability and the hysteresis in the transmittance functions . let us first assume that @xmath112 . if we consider the phonon transport from the passive resonator to the active resonator , we can obtain an algebraic equation similar to that given in eq . ( [ stationary equation of ng ] ) . the bistable transition point corresponds to the stationary points of the function @xmath113 by setting @xmath114 , the stationary point of @xmath115 can be found as @xmath116\left(3\tilde{\mu}^2\right)^{-1}.\ ] ] the upper bound of the unidirectional phonon transport region is given by @xmath117.\end{aligned}\ ] ] near the @xmath0-transition point , we have @xmath118 , and thus it can be approximately estimated that @xmath119 let us now consider the case of phonon transport from the active resonator to the passive resonator when the amplitude of the input field is varied and the detuning is kept fixed . in this case , we obtain @xmath120 where @xmath121 similar to eq . ( [ function for the upper bound of the input intensity window ] ) , the bistable transition point can be found by calculating the stationary points of the function @xmath122 which leads to @xmath123\left(3\tilde{\tilde{\mu}}^2\right)^{-1}.\ ] ] the lower bound of the unidirectional phonon transport region is then given by @xmath124.\end{aligned}\ ] ] near the @xmath0-transition point , we have @xmath118 , and it can be approximately estimated that @xmath125 we thus conclude that nonreciprocal phonon transport takes place if the amplitude of the input is within the region @xmath126.\ ] ] similarly , when @xmath127 , the nonreciprocal region for the amplitude of the input field can be written as @xmath128.\ ] ] in fig . [ fig of unidirectional phonon transport window]c , we present the transmittances as a function of the amplitude of the input field when the detuning is kept fixed within the unidirectional transport region given in eq . ( [ unidirectional phonon transport window ] ) . we see that the lower stable branches of the bistable curves shown in fig . [ fig of unidirectional phonon transport window]c ( the parts of the bistable curves before the bistable transitions occur ) increase when we increase the intensity of the input field . this decreases the rectification , as shown in fig . [ fig of unidirectional phonon transport window]d . in order to check the performance of the proposed system as an isolator for phonons , we study the system considering that phonons are injected in the system in both directions , that is simultaneously at the passive and active resonator sides . if the system exhibits unidirectional phonon transport under this condition , then the proposed system can be used as an isolator . the equations of motion of the system for this case can be written as @xmath129 where the last terms on the right - hand - sides of eq . ( [ dynamical equation for phonon isolator ] ) denote the input fields . the steady - state solution of eq . ( [ dynamical equation for phonon isolator ] ) leads to @xmath130 where @xmath131 let us first fix @xmath132 , and vary the detuning @xmath133 . in this case , the bistable transitions for both directions occur when the detuning @xmath29 satisfies @xmath134 when the detuning is up - scanned from smaller to larger detuning values , the bistable transition occurs for @xmath135 when the detuning @xmath29 is down - scanned from larger to smaller detuning values , the bistable transition occur at @xmath136 the transmittances presented in fig . [ fig of bistable curves for phonon isolator]a clearly show the bistable operation . a close look at fig . [ fig of bistable curves for phonon isolator]a reveals that the transition from the bistable region to the stable trajectories takes place at the same points for both directions . we can not find a detuning region within which transport in one direction is allowed and the transport in the other direction is prevented . thus , we conclude that when phonons are injected simultaneously at both input ports , we can not see a unidirectional operation . consequently , it is impossible to use this system as an isolator for phonons . let us now fix @xmath137 , @xmath138 , @xmath139 , and vary @xmath140 , to check the possibility of providing a phonon isolator . the bistable transition point is just the stationary points of the function @xmath141 by setting @xmath142 , we find @xmath143\left(3\bar{\mu}^2\right)^{-1},\\ \bar{n}_{g2}^&=&\left[2\bar{\mu}\bar{\omega}-\sqrt{4\bar{\mu}^2\bar{\omega}^2 - 3\bar{\mu}^2\left(\bar{\gamma}^2+\bar{\omega}^2\right)}\right]\left(3\bar{\mu}^2\right)^{-1},\end{aligned}\ ] ] the bistable transition occurs at latexmath:[\[\label{input intensity for up - scan } when the amplitude of the input field @xmath89 is up - scanned and for @xmath145 ) . for this case too , we do not see a unidirectional phonon transport region if we feed the inputs at the active and passive resonators sides simultaneously . thus we conclude that although the proposed system can be used as phonon diode allowing nonreciprocal phonon transport , it can not function as an isolator for phonons . r. w. andrews , r. w. peterson , t. p. purdy , k. cicak , r. w. simmonds , c. a. regal , and k. w. lehnert , _ bidirectidonal and efficient conversion between microwave and optical light _ , nature phys . 10 * , 321 - 326 ( 2014 ) . a. d. oconnell , m. hofheinz , m. ansmann , r. c. bialczak , m. lenander , e. lucero , m. neeley , d. sank , h. wang , m. weides , and j. wenner , j. m. martinis , and a. n. cleland , _ quantum ground state and single - phonon control of a mechanical resonator _ , nature * 464 * , 697 - 703 ( 2010 ) . q. lin , j. rosenberg , d. chang , r. camacho , m. eichenfield , k. j. vahala , and o. painter , _ coherent mixing of mechanical excitations in nano - 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75 ( 2008 ) . z. lin , h. ramezani , t. eichelkraut , t. kottos , h. cao , and d. n. christodoulides , _ unidirectional invisibility induced by @xmath0-symmetric periodic structures _ , phys . lett . * 106 * , 213901 ( 2011 ) . a. guo , g. j. salamo , d. duchesne , r. morandotti , m. volatier - ravat , v. aimez , g. a. siviloglou , and d n. christodoulides , _ observation of @xmath0-symmetry breaking in complex optical potentials _ , phys . lett . * 103 * , 093902 ( 2009 ) . l. feng , y .- l . xu , w. s. fegadolli , m .- h . lu , j.e.b . oliveira , v. r. almeida , y .- f . chen , and a. scherer , _ experimental demonstration of a unidirectional reflectionless parity - time metamaterial at optical frequencies _ , nature mater . * 12 * , 108 - 113 ( 2012 ) . s. bittner , b. dietz , u. gnther , h. l. harney , m. miski - oglu , a. richter , and f. schfer , _ @xmath0 symmetry and spontaneous symmetry breaking in a microwave billiard _ , phys . lett . * 108 * , 024101 ( 2012 ) . n. bender , s. factor , j. d. bodyfelt , h. ramezani , d. n. christodoulides , f. m. ellis , and t. kottos , _ observation of asymmetric transport in structures with active nonlinearities _ , phys . * 10 * , 234101 ( 2013 ) . b. peng , s. k. ozdemir , f. c. lei , f. monifi , m. gianfreda , g. l. long , s. h. fan , f. nori , c. m. bender , and l. yang , _ parity - time - symmetric whispering - gallery microcavities _ , nature phys . * 10 * , 394 - 398 ( 2014 ) . l. chang , x. jiang , s. hua , c. yang , j. wen , l. jiang , g. li , g. wang , and m. xiao , _ parity - time symmetry and variable optical isolation in active - passive - coupled microresonators _ , nature photon . * 8 * , 524 - 529 ( 2014 ) . e. gil - santos , d. ramos , a. jana , m. calleja , a. raman , and j. tamayo , _ mass sensing based on deterministic and stochastic responses of elastically coupled nanocantilevers _ , nano lett . * 9 * , 4122 - 4127 ( 2009 ) . g. anetsberger , o. arcizet , q. p. unterreithmeier , r. riviere , a. schliesser , e. m. weig , j. p. kotthaus , and t. j. kippenberg , _ near - field cavity optomechanics with nanomechanical oscillators _ , nature phys . * 5 * , 909 - 914 ( 2009 ) .	nonreciprocal devices that permit wave transmission in only one direction are indispensible in many fields of science including , e.g. , electronics , optics , acoustics , and thermodynamics . manipulating phonons using such nonreciprocal devices may have a range of applications such as phonon diodes , transistors , switches , etc . one way of achieving nonreciprocal phononic devices is to use materials with strong nonlinear response to phonons . however , it is not easy to obtain the required strong mechanical nonlinearity , especially for few - phonon situations . here , we present a general mechanism to amplify nonlinearity using @xmath0-symmetric structures , and show that an on - chip micro - scale phonon diode can be fabricated using a @xmath0-symmetric mechanical system , in which a lossy mechanical - resonator with very weak mechanical nonlinearity is coupled to a mechanical resonator with mechanical gain but no mechanical nonlinearity . when this coupled system transits from the @xmath0-symmetric regime to the broken-@xmath0-symmetric regime , the mechanical nonlinearity is transferred from the lossy resonator to the one with gain , and the effective nonlinearity of the system is significantly enhanced . this enhanced mechanical nonlinearity is almost lossless because of the gain - loss balance induced by the @xmath0-symmetric structure . such an enhanced lossless mechanical nonlinearity is then used to control the direction of phonon propagation , and can greatly decrease ( by over three orders of magnitude ) the threshold of the input - field intensity necessary to observe the unidirectional phonon transport . we propose an experimentally realizable lossless low - threshold phonon diode of this type . our study opens up new perspectives for constructing on - chip few - phonon devices and hybrid phonon - photon components .	12,811	475
a groupoid is a small category whose arrows are all invertible . more precisely we have the following definition . a * groupoid * over a set @xmath0 is a set of * arrows * @xmath1 along with a * target map * @xmath2 , a * source map * @xmath3 , a * identity section * @xmath4 which is a injective function , a * partially defined operation * ( or product ) @xmath5 on @xmath1 , which is a function : @xmath6 and a * inversion map * @xmath7 , @xmath8 . these are the structure maps of the groupoid . they satisfy several identities . 1 . for any @xmath9 we have @xmath10 2 . then for any @xmath11 we have also @xmath12 . this allows us to write the expression @xmath13 and to state that the operation @xmath5 is associative : @xmath14 3 . for any @xmath15 the identity section satisfies @xmath16 and @xmath17 4 . the inversion map is an involution : @xmath18 . for any @xmath15 we have @xmath19 and @xmath20 and @xmath21 [ defgroid ] an equivalent definition of a groupoid emphasizes the fact that a groupoid may be defined only in terms of its arrows . a groupoid is a set @xmath1 with two operations @xmath22 , @xmath23 , which satisfy a number of properties . with the notations @xmath24 , @xmath25 , these properties are : for any @xmath26 1 . if @xmath27 and @xmath28 then @xmath29 and @xmath30 and we have @xmath31 , 2 . @xmath32 and @xmath33 , 3 . if @xmath27 then @xmath34 and @xmath35 . [ defgroid2 ] starting with the definition [ defgroid2 ] , we can reconstruct the objects from definition [ defgroid ] . the set @xmath36 is formed by all products @xmath37 , @xmath38 . for any @xmath38 we let @xmath39 and @xmath40 . the identity section is just the identity function on @xmath0 . a groupoid is denoted either by @xmath41 , or by @xmath42 . in the second case we shall use the notation @xmath43 . in most of this paper we shall simply denote a groupoid @xmath42 by @xmath1 . the transformation @xmath44 is a * morphism of groupoids * defined from @xmath45 to @xmath46 is a pair of maps : @xmath47 and @xmath48 which commutes with the structure , that is : @xmath49 , @xmath50 , @xmath51 , @xmath52 and @xmath53 is a morphism of operations , from the operation @xmath5 to operation @xmath54 . a * hausdorff topological groupoid * is a groupoid @xmath1 which is also a hausdorff topological space , such that inversion is continuous and the multiplication is continuous with respect to the topology on @xmath55 induced by the product topology on @xmath56 . we denote by @xmath57 the * difference function * : @xmath58 we shall consider the convergence of nets @xmath59 of arrows , with @xmath60 a parameter in a directed set @xmath61 . in this paper the most encountered directed set @xmath61 will be @xmath62 . a * normed groupoid * @xmath63 is a groupoid @xmath41 with a * norm * function @xmath64 , such that : 1 . @xmath65 if and only if there is a @xmath66 with @xmath67 , 2 . for any @xmath9 we have @xmath68 , 3 . for any @xmath15 we have @xmath69 . a * norm * @xmath70 * is separable * if it satisfies the property : 1 . if there is a net @xmath71 such that for any @xmath72 @xmath73 , @xmath74 and @xmath75 then @xmath76 . [ defnorm ] let @xmath77 be a normed groupoid and @xmath78 its difference function . the norm @xmath70 composed with the function @xmath78 gives a new function @xmath79 : @xmath80 which induces a distance on the space @xmath81 , for any @xmath82 : @xmath83 the * metric groupoid * @xmath84 associated to the normed groupoid @xmath1 is the following metric groupoid : 1 . the objects of @xmath84 are the metric spaces @xmath85 , with @xmath86 ; 2 . the arrows are right translations @xmath87 3 . the multiplication of arrows is the composition of functions ; 4 . the norm is defined by : @xmath88 . [ dmgru ] remark that arrows in the metric groupoid @xmath84 are isometries . it is also clear that @xmath84 is isomorphic with @xmath1 by the morphism @xmath89 . the * @xmath90-double groupoid * @xmath91 associated to @xmath1 is another way to assembly the metric spaces @xmath92 , @xmath86 , into a groupoid . the definition of this groupoid is : 1 . the arrows are @xmath93 ; 2 . the composition of arrows is : @xmath94 , the inverse is @xmath95 , therefore as a groupoid @xmath91 is just the union of trivial groupoids over @xmath92 , @xmath86 ; 3 . it follows that @xmath96 and the induced @xmath90 and @xmath97 maps are : @xmath98 and @xmath99 , for any @xmath100 with @xmath101 ; 4 . the norm is the function @xmath79 . [ ddoub ] this groupoid has the property that @xmath78 is a morphism of normed groupoids . finally , suppose that for any @xmath102 there is @xmath15 such that @xmath103 and @xmath104 . then any separable norm @xmath70 on @xmath1 induces a distance on @xmath105 , by the formula : @xmath106 if the groupoid is not connected by arrows then @xmath107 may take the value @xmath108 and the space @xmath0 decomposes into a disjoint union of metric spaces . any norm @xmath70 on a groupoid @xmath1 induces three notions of convergence on the set of arrows @xmath1 . a net of arrows @xmath109 * simply converges * to the arrow @xmath38 ( we write @xmath110 ) if : 1 . for any @xmath111 there are elements @xmath112 such that @xmath113 , 2 . we have @xmath114 and @xmath115 . a net of arrows @xmath59 * left - converges * to the arrow @xmath38 ( we write @xmath116 ) if for all @xmath117 we have @xmath118 and moreover @xmath119 . a net of arrows @xmath59 * right - converges * to the arrow @xmath38 ( we write @xmath120 ) if for all @xmath117 we have @xmath121 and moreover @xmath122 . [ defconv ] it is clear that if @xmath116 or @xmath120 then @xmath123 . right - convergence of @xmath124 to @xmath125 is just convergence of @xmath124 to @xmath125 in the distance @xmath126 , that is @xmath127 . left - convergence of @xmath124 to @xmath125 is just convergence of @xmath128 to @xmath129 in the distance @xmath130 , that is @xmath131 . [ propconv ] let @xmath1 be a groupoid with a norm @xmath70 . 1 . if @xmath116 and @xmath132 then @xmath133 . if @xmath120 and @xmath134 then @xmath133 . 2 . the following are equivalent : 1 . @xmath1 is a hausdorff topological groupoid with respect to the topology induced by the simple convergence , 2 . @xmath70 is a separable norm , 3 . for any net @xmath109 , if @xmath135 and @xmath136 then @xmath133 . 4 . for any net @xmath109 , if @xmath137 and @xmath132 then @xmath133 . [ [ proof . ] ] proof . + + + + + + \(i ) we prove only the first part of the conclusion . we can write @xmath138 , therefore @xmath139 the right hand side of this inequality is arbitrarily small , so @xmath140 , which implies @xmath133 . \(ii ) remark that the structure maps are continuous with respect to the topology induced by the simple convergence . we need only to prove the uniqueness of limits . @xmath141 4 . is trivial . in order to prove that 4.@xmath141 3 . , consider an arbitrary net @xmath109 such that @xmath135 and @xmath136 . this means that there exist nets @xmath142 such that @xmath143 , @xmath144 and @xmath145 . let @xmath146 and @xmath147 . we have then @xmath148 and @xmath149 . then @xmath150 and @xmath151 . we deduce that @xmath133 . 1.@xmath152 3 . is trivial . @xmath141 2 . we finish the proof by showing that 2 . @xmath141 3 . by a reasoning made previously , it is enough to prove that : if @xmath153 and @xmath154 then @xmath133 . because @xmath70 is separable it follows that @xmath155 and @xmath156 . we have then @xmath157 , therefore @xmath158 the norm @xmath70 induces a left invariant distance on the vertex group of all arrows @xmath159 such that @xmath160 . this distance is obviously continuous with respect to the simple convergence in the group . the net @xmath161 simply converges to @xmath162 by the continuity of the multiplication ( indeed , @xmath163 simply converges to @xmath162 ) . therefore @xmath164 . it follows that @xmath165 is arbitrarily small , therefore @xmath133 . @xmath166 instead of a norm we may use families of seminorms . a * family of seminorms * on a groupoid @xmath1 is a family @xmath167 of functions @xmath168 with the properties : 1 . for any @xmath66 and @xmath169 we have @xmath170 ; if @xmath171 for any @xmath169 then there is @xmath66 such that @xmath67 , 2 . for any @xmath169 and @xmath172 we have @xmath173 , 3 . for any @xmath169 and @xmath15 we have @xmath174 . a groupoid @xmath1 endowed with a family of seminorms @xmath167 is called a * seminormed groupoid. a family of seminorms @xmath167 is separable if it satisfies the property : 1 . if there is a net @xmath175 such that for any @xmath72 @xmath73 , @xmath74 and for any @xmath169 we have @xmath176 then @xmath76 . [ defseminormedg ] families of morphisms induce families of seminorms . let @xmath1 be a groupoid and @xmath177 be a normed groupoid . a @xmath177 family of morphisms * is a set @xmath178 of morphisms from @xmath1 to @xmath179 such that for any @xmath15 there is @xmath180 with @xmath181 . the following proposition has a straightforward proof which we omit . let @xmath1 be a groupoid , @xmath177 be a normed groupoid and @xmath178 a @xmath177 family of morphisms . then the set @xmath182 is a family of seminorms . definition [ defconv ] can be modified for the case of families of seminorms . let @xmath183 be a semi - normed groupoid . a net of arrows @xmath59 * simply converges * to the arrow @xmath38 ( we write @xmath123 ) if : 1 . for any @xmath184 there are elements @xmath112 such that @xmath143 , 2 . for any @xmath169 we have @xmath185 . a net of arrows @xmath59 * left - converges * to the arrow @xmath38 ( we write @xmath116 ) if for all @xmath117 we have @xmath118 and moreover for any @xmath169 we have @xmath186 . a net of arrows @xmath59 * right - converges * to the arrow @xmath38 ( we write @xmath120 ) if for all @xmath117 we have @xmath121 and moreover for any @xmath169 we have @xmath187 . [ defconv2 ] with these slight modifications , the proposition [ propconv ] still holds true . this is visible from the examination of its proof . let us finally remark that if @xmath188 is a seminormed groupoid , where @xmath178 is a @xmath177 family of morphisms , then a net @xmath189 converges ( simply , left or right ) to @xmath38 if and only if for any @xmath180 the net @xmath190 respectively converges in @xmath177 . we shall use right - convergence , according to definition [ defconv ] , but left - convergence or simple convergence could also be used . in relation to this see for example the remark [ rema1 ] . let @xmath1 be a normed groupoid with a separable norm . a net @xmath191 of functions @xmath192 * uniformly converges on bounded sets * to the function @xmath193 ( in the sense of the left convergence ) if : 1 . for any @xmath194 and @xmath195 we have @xmath196 , 2 . for any @xmath197 there is @xmath198 such that for any @xmath199 , @xmath200 and any @xmath195 with @xmath201 , @xmath202 , we have : @xmath203 in the case of a groupoid @xmath1 with a separable family of seminorms @xmath167 , the definition of uniform convergence is the same , excepting the modification of ( ii ) above into : for any @xmath197 and any seminorm @xmath169 there is @xmath204 such that for any @xmath199 , @xmath200 and any @xmath195 with @xmath205 , @xmath206 , we have : @xmath207 similarly , in a normed groupoid with a separable norm @xmath70 , the uniform convergence on bounded sets of a net of functions @xmath208 to @xmath209 means that for any @xmath197 there is @xmath198 such that for any @xmath199 , @xmath200 and any @xmath210 with @xmath202 we have : @xmath211 . sometimes it is interesting to work with normed categories , instead of normed groupoids . briefly said , a normed category is a small category endowed with an involutionary antimorphism ( an `` inverse '' ) and with a norm function defined on arrows . a * normed category * @xmath63 is a small category @xmath212 with inverses : the set of objects is @xmath213 , the set of arrows is identified with @xmath212 ; there are two functions @xmath214 , named * source * and * target * , a * multiplication * @xmath215 ( notation : @xmath216 or @xmath217 ) , where @xmath218 a * inversion map * @xmath219 ( notation @xmath220 ) and a * norm * @xmath221 , which satisfy the following conditions : 1 . @xmath222 , @xmath223 , for any @xmath224 , 2 . the multiplication is associative @xmath225 3 . the inverse is an involution and a antimorphism : @xmath226 , @xmath227 . 4 . @xmath65 if and only if there is @xmath228 such that @xmath229 , 5 . for any @xmath230 we have @xmath68 , 6 . for any @xmath231 we have @xmath69 . let @xmath232 be defined by @xmath233 , @xmath234 . the * norm * @xmath70 * is separable * if it satisfies the property : 1 . if there is a net @xmath235 such that for any @xmath72 @xmath73 , @xmath74 and @xmath75 then @xmath76 . [ defcatnorm ] seminormed categories are defined further , by making a slight modification of definition [ defseminormedg ] . let @xmath212 be a category with inverses ( which satisfies ( i)(iv ) definition [ defcatnorm ] ) . let @xmath236 . a * family of seminorms * on a category with inverses @xmath212 is a family @xmath167 of functions @xmath237 with the properties : 1 . for any @xmath66 and @xmath169 we have @xmath238 ; if @xmath171 for any @xmath169 then @xmath239 , 2 . for any @xmath169 and @xmath240 we have @xmath173 , 3 . for any @xmath169 and @xmath231 we have @xmath174 . a category @xmath212 with inverses endowed with a family of seminorms @xmath167 is called a * seminormed category. a family of seminorms @xmath167 is separable if it satisfies the property : 1 . if there is a net @xmath241 such that for any @xmath117 @xmath73 , @xmath74 and for any @xmath169 we have @xmath176 then @xmath76 . [ defseminormedc ] all considerations made before concerning convergence for normed or seminormed separable groupoids , extend without effort to normed categories , or to seminormed separable categories . a little bit of care is needed though : everywhere we should replace source and target maps ( of groupoids ) by the ( algebraically defined ) @xmath242 and @xmath243 . we give several examples of normed groupoids which will be of interest later in this paper . let @xmath244 be a metric space . we form the normed trivial groupoid * @xmath245 over @xmath0 : 1 . the set of arrows is @xmath246 and the multiplication is @xmath247 therefore we have @xmath248 , @xmath249 , @xmath250 , @xmath251 . the norm is just the distance function @xmath252 . it is easy to see that if @xmath253 is a normed trivial groupoid over @xmath0 then @xmath244 is a metric space . the metric groupoid @xmath254 can be described as the groupoid with objects pointed metric spaces @xmath255 , @xmath66 , arrows @xmath256 , @xmath257 , and norm @xmath258 . the @xmath90-double groupoid @xmath259 can be described as the groupoid with arrows @xmath260 , composition @xmath261 , inverse @xmath262 and norm @xmath263 . remark first that @xmath70 is a separable norm , according to definition [ defnorm ] ( iv ) . indeed , for any @xmath264 there is only one arrow @xmath265 such that @xmath266 , @xmath267 , namely the arrow @xmath268 . any net @xmath109 with @xmath269 , @xmath270 is the constant net @xmath271 . if @xmath272 then @xmath273 , therefore @xmath274 . we deduce from proposition [ propconv ] that we have only one interesting notion of convergence , which is simple convergence . in the particular case of normed trivial groupoids the definition [ defconv ] of simple convergence becomes : a net @xmath275 simply converges to @xmath276 if we have @xmath277 that is if the nets @xmath278 converge respectively to @xmath279 . indeed this is coming from the fact that for any @xmath72 there are unique @xmath280 such that @xmath281 . these are @xmath282 and @xmath283 . let @xmath0 be a non empty set , let @xmath284 be a metric spaces and @xmath285 its associated normed trivial groupoid . any function @xmath286 induces a morphism @xmath287 from the trivial groupoid @xmath288 to @xmath289 by @xmath290 . any family @xmath291 of functions from @xmath0 to @xmath292 with the separation property : for any @xmath293 @xmath294 there is @xmath295 with @xmath296 , gives us a @xmath297 family of morphisms , which in turn induces a family of seminorms on @xmath288 . we can construct normed groupoids starting from definition [ ddoub ] of @xmath90-double groupoids . let @xmath245 be a groupoid and @xmath298 the associated @xmath90-double groupoid . then for any @xmath299 and for any @xmath300 with @xmath301 we have @xmath302 conversely , suppose that @xmath1 is a groupoid and that for any @xmath86 we have a distance @xmath303 . if ( [ leftdis ] ) is true for any @xmath304 and for any @xmath300 with @xmath305 then @xmath306 define a norm on @xmath1 such that @xmath307 is the associated @xmath90-double groupoid . [ pdoub ] therefore any normed groupoid @xmath245 can be seen as the bundle of metric spaces @xmath308 , such that ( a ) each fiber @xmath81 has a distance @xmath309 , and ( b ) the distances @xmath310 are right invariant with respect to the groupoid composition , in the sense of relation ( [ leftdis ] ) . [ [ proof.-1 ] ] proof . + + + + + + for the first implication remark that @xmath311 . moreover let @xmath312 , @xmath313 . then @xmath314 , therefore @xmath315 for the converse implication , we have to prove that if @xmath314 then @xmath316 , with @xmath70 defined as in the formulation of the proposition . this is easy : let @xmath317 , then @xmath318 , @xmath319 and ( [ pdoub ] ) implies the desired equality . the verification that @xmath70 is indeed a norm on @xmath1 is straightforward , as well as the fact that @xmath79 is the induced norm on @xmath320 . @xmath321 let @xmath1 be a group with neutral element @xmath322 , which acts from the left on the space @xmath0 . associated with this is the * action groupoid * @xmath323 over @xmath0 . the action groupoid is defined as : the set of arrows is @xmath324 and the multiplication is @xmath325 therefore @xmath326 , @xmath327 , @xmath328 , @xmath329 , as a particular case of definition [ defnorm ] , a * normed action groupoid * is an action groupoid @xmath323 endowed with a norm function @xmath330 with the properties : 1 . @xmath331 if and only if @xmath332 , 2 . @xmath333 , 3 . @xmath334 . remark that the norm function is no longer a distance function . in the case of a free action ( if @xmath335 for some @xmath66 then @xmath336 ) we may obtain a norm function from a distance function on @xmath0 . indeed , let @xmath337 be a distance . define then @xmath338 by @xmath339 then @xmath340 is a normed action groupoid . the associated @xmath90-double groupoid can be seen as @xmath341 , with composition @xmath342 and inverse @xmath343 . for any @xmath66 we have a distance @xmath344 , defined by @xmath345 conversely , according to proposition [ pdoub ] and relation ( [ leftdis ] ) , a norm on a action groupoid can be constructed from a function @xmath346 which associates to any @xmath66 a distance @xmath310 on g , such that for any @xmath66 and @xmath347 we have @xmath348 in this case we can define the norm on the action groupoid by @xmath349 . a particular case is @xmath350 , when a normed action groupoid is just a group endowed with a right invariant distance . the norm @xmath70 is separable if the following condition is satisfied : for any @xmath293 and any net @xmath351 with the property @xmath352 for all @xmath353 , if @xmath354 then @xmath76 . let @xmath355 and @xmath356 be two groupoids . we denote by @xmath357 the groupoid which has as objects sub - groupoids of @xmath356 and invertible morphisms between sub - groupoids of @xmath356 as arrows . a groupoid action of @xmath1 on @xmath356 is just a morphism @xmath53 of groupoids from @xmath1 to @xmath357 . in fewer words , for any @xmath15 let @xmath358 be the associated morphism of sub - groupoids , defined from the sub - groupoid denoted by @xmath359 to the sub - groupoid denoted by @xmath360 . for any @xmath361 we use the notation @xmath362 . compositions in @xmath1 and in @xmath356 are denoted multiplicatively . let @xmath363 be the set @xmath364 the action of @xmath1 on @xmath356 satisfies the following conditions : 1 . for any @xmath365 such that @xmath366 we have @xmath367 , @xmath368 and for any @xmath369 we have @xmath370 ; 2 . for any @xmath300 and @xmath371 we have @xmath372 . any groupoid action induces a groupoid structure on @xmath363 , by the composition law @xmath373 . at a closer look we may notice an example of a groupoid action in proposition [ pdoub ] . indeed , let @xmath1 be a groupoid and @xmath374 the associated @xmath90-double groupoid . then @xmath1 acts on @xmath374 by @xmath375 , for any @xmath300 and any @xmath376 such that @xmath377 . therefore @xmath378 and the associated action groupoid is @xmath379 with multiplication defined by @xmath380 relation ( [ leftdis ] ) in proposition [ pdoub ] tells that @xmath1 acts on the normed groupoid @xmath381 by isometries . in general , the action groupoid induced by the action of a groupoid @xmath1 on a normed groupoid @xmath356 _ by isometries _ may be an object as interesting as a normed groupoid . the references used are glickenstein @xcite and lott @xcite section 5 . a smooth tale groupoid @xmath1 is riemannian if there is a riemannian metric on @xmath382 such that the germs of the arrows ( arrows are seen as local diffeomeorphisms on @xmath382 ) are isometries . a riemannian groupoid is not a normed groupoid . a good reference for this section is @xcite chapters 5 and 6 . much more details could be found , for instance , in @xcite . let @xmath244 be a a complete , separable metric space ( polish space ) . the class of borel probabilities on @xmath0 is denoted by @xmath383 . if @xmath384 are polish spaces then @xmath385 is also a polish space , with the distance @xmath70 defined by @xmath386 . if @xmath387 is a borel map and @xmath388 then the push - forward of @xmath389 through @xmath390 is the measure @xmath391 defined by : for any @xmath392 @xmath393 in particular , the projections @xmath394 , @xmath395 , @xmath396 , define push - forwards from @xmath397 to @xmath398 . let @xmath0 , @xmath292 be polish spaces . a measure - valued map @xmath399 is a borel map if for any borel set @xmath400 the function @xmath401 is borel . the following is the disintegration theorem @xcite theorem 5.3.1 ( see also references therein ) . ( in @xcite the theorem is formulated for radon spaces , here we stay in the frame of polish spaces ) . let @xmath402 be two polish spaces , @xmath403 a borel map and @xmath404 , @xmath405 . then there exists a borel measure - valued function @xmath406 , which is @xmath407-a.e . uniquely determined , such that : 1 . for @xmath407-a.e . @xmath408 we have @xmath409 , 2 . for every borel map @xmath410 $ ] @xmath411 [ disthm ] in the particular case @xmath412 , @xmath413 and @xmath414 , @xmath415 , the disintegration theorem implies that for any @xmath416 with marginals @xmath417 there exist borel measure - valued maps @xmath418 , @xmath389-a.e . uniquely determined , and @xmath419 , @xmath407-a.e . uniquely determined , such that for any borel function @xmath420 $ ] we have : @xmath421 @xmath422 for any two probability measures @xmath389 , @xmath407 on @xmath0 the set of all transport plans between @xmath389 and @xmath407 is defined as : @xmath423 the category @xmath424 of transport plans on @xmath0 is the category with inverses which has as objects the elements of @xmath383 and the class of arrows between @xmath425 is @xmath426 . the composition of arrows is given by composition of transport plans , defined further . let @xmath427 and @xmath428 . by the disintegration theorem [ disthm ] , there are @xmath407-a.e . uniquely defined measure - valued maps @xmath429 such that for any borel function @xmath420 $ ] we have : @xmath430 @xmath431 then the composition @xmath432 is defined by : for any borel function @xmath420 $ ] @xmath433 finally , this category has a contravariant `` inverse '' functor , which associates to each arrow @xmath434 the arrow @xmath435 defined by : for any borel function @xmath420 $ ] @xmath436 we denote by @xmath437 the class of 1-lipschitz maps from @xmath0 to @xmath438 . the category @xmath439 is a separable seminormed category , with the family @xmath167 of seminorms : for any @xmath440 and any @xmath441 @xmath442 @xmath439 it is also a normed category , with the norm : @xmath443 the following relation between the norm @xmath70 in the transport category and the seminorms @xmath444 , is related to kantorovich formulation of the transport problem and the dual of this problem : for any @xmath445 and any @xmath441 we have @xmath446 the topology of @xmath424 with the norm @xmath70 is called `` strong topology '' , while the topology of @xmath424 as a seminormed groupoid is called the `` weak topology '' . for any pair formed by a borel function @xmath447 and by a measure @xmath448 , there is an associated transport plan @xmath449 . if @xmath450 is representable as a measure @xmath451 then we say that @xmath452 is induced by a transport map . the category @xmath453 is the subcategory of @xmath424 with objects elements of @xmath383 and arrows transport plans induced by transport maps . several facts deserve to be mentioned . notice that if @xmath454 then @xmath455 @xmath389-a.e . in @xmath0 . also , in the category @xmath453 the composition of arrows is the following : @xmath456 finally , the sub - category @xmath457 of @xmath453 with objects elements of @xmath383 and arrows @xmath451 such that there exists a borel function @xmath159 with @xmath458 , is a groupoid . let @xmath245 be a normed groupoid with a separable norm . a deformation of @xmath245 is basically a `` local action '' of a commutative group @xmath459 on @xmath1 which satisfies several properties . @xmath460 is a commutative group endowed with a group morphism @xmath461 to the multiplicative group of positive real numbers . this morphism induces a invariant topological filter over @xmath459 ( a end of @xmath459 ) . further we shall write @xmath462 for @xmath353 converging to this end , and meaning that @xmath463 . the neutral element of @xmath459 is denoted by @xmath322 . to any @xmath199 is associated a transformation @xmath464 , which may be called a dilatation , dilation , homothety or contraction . for the precise properties of the domains and codomains of @xmath465 for @xmath199 see the subsection [ bordet ] . for the moment is sufficient to know that for any @xmath199 we have @xmath466 and @xmath467 . basically the domain and codomain of @xmath465 are neighbourhoods of @xmath0 . moreover , these sets are chosen so that various compositions of transformations @xmath465 are well defined . in the formulation of properties of deformations we shall use a uniform convergence on bounded sets . we explain further what uniform convergence on bounded sets means in the case of nets of functions indexed with the directed net the group @xmath459 ( ordered such that limits are taken in the sense @xmath468 . a * dilation of a separated normed groupoid * @xmath245 is a map assigning to any @xmath199 a transformation @xmath469 which satisfies the following : 1 . for any @xmath199 @xmath470 . moreover @xmath471 is an action of @xmath459 on @xmath1 , that is for any @xmath472 we have @xmath473 , @xmath474 and @xmath475 . 2 . for any @xmath86 and any @xmath199 we have @xmath476 . moreover the transformation @xmath477 contracts @xmath478 to @xmath105 uniformly on bounded sets , which means that the net @xmath479 converges to the constant function @xmath480 , uniformly on bounded sets , in the sense of definition [ defcon ] . moreover the domains and codomains @xmath478 , @xmath481 satisfy the conditions from definition [ dax0 ] , section [ bordet ] . a * dilation of a separated normed or seminormed category * is defined in the same way as a dilation of a normed groupoid , only that @xmath90 and @xmath97 functions are no longer the source and arrow functions , but @xmath482 and @xmath483 . [ ddefor ] the dilation @xmath484 of @xmath245 induces a right - invariant dilation of the normed groupoid @xmath485 . the proof of the following proposition is straightforward and we do not write it . for any @xmath199 we define @xmath486 on @xmath487 , given by : @xmath488 this is a deformation of the normed groupoid @xmath489 is a normed groupoid and moreover @xmath78 is a morphism of normed groupoids ( that is a norm preserving morphism of groupoids ) , which commutes with dilations in the sense : for any @xmath490 @xmath491 . [ pdefindd ] let @xmath492 and @xmath493 be two dilations of the normed groupoids @xmath494 , @xmath495 respectively . then @xmath496 is a morphism of dilations if : @xmath53 is a morphism of groupoids , it preserves the norms ( it is a isometry ) and it commutes with dilations ( it is `` * linear * '' ) . dilations are locally defined . this is explained in the following definition , which should be seen as the axiom a0 of dilations . the domains and codomains of a dilation of @xmath245 satisfy the following axiom a0 : 1 . for any @xmath199 @xmath466 and @xmath497 , 2 . for any bounded set @xmath498 there are @xmath499 such that for any @xmath199 , @xmath500 : @xmath501 @xmath502 3 . for any bounded set @xmath498 there are @xmath503 and @xmath504 $ ] such that for any @xmath199 , @xmath505 and any @xmath506 we have : @xmath507 [ dax0 ] concerning ( iii ) definition [ dax0 ] , the first part of a1 definition [ ddefor ] implies that @xmath508 is well defined for any @xmath509 such that @xmath510 . the purpose of this section is to define several deformations of normed groupoids , such that the diagram from figure [ figure2 ] becomes a commutative diagram of morphisms of dilations . let us consider a triple @xmath511 with @xmath245 a normed groupoid and @xmath484 a dilation . for any @xmath512 there are two normed induced groupoids , such that the arrows in the diagram ( [ figure2 ] ) are morphisms . as dilatations are not globally defined and they are used to transport groupoid operations , it follows that the transported objects ( operation , norms , ... ) are not globally defined . therefore the induced groupoids are not groupoids , but `` local '' groupoids , in a sense which is clear in the context . [ rkdom ] the deformation @xmath513 is equal to @xmath1 as a set and its operations , norm and dilation are transported by the map @xmath514 ( with the precautions concerning the domains of definition of the transported objects mentioned in remark [ rkdom ] ) . the deformation @xmath515 is equal to @xmath91 as a groupoid and its norm and dilation are transported by the map @xmath516 . [ defdia ] more precisely , the deformation @xmath513 is described by : 1 . @xmath517 as a set , @xmath518 and @xmath519 , which follow from the computations using a1 , a2 definition [ ddefor ] : @xmath520 also @xmath521 . the composition operation and inverse are @xmath522 these are well defined ( at least locally ) because of the axiom a0 definition [ dax0 ] . notice that @xmath523 from the diagram [ figure2 ] appears as the difference function associated with the operation @xmath524 , defined as @xmath525 3 . the norm @xmath526 is defined as : @xmath527 4 . we may transport the dilation @xmath484 of @xmath245 into a dilation @xmath528 of @xmath529 , but from the commutativity of @xmath459 we get that @xmath530 therefore it is the same dilation . the deformation @xmath515 is described by : 1 . @xmath531 as a groupoid ; remark that this is compatible with the transport of operations using the map @xmath532 ( because this map is an endomorphism of the groupoid @xmath91 ) , 2 . with respect to the relation ( [ dfird ] ) , notice that @xmath533 and @xmath523 as represented in figure [ figure2 ] is a morphism of groupoids , 3 . the norm @xmath534 is defined as : @xmath535 and it is easy to check that @xmath523 is also a isometry . we transport the dilation @xmath536 of @xmath537 into a dilation @xmath538 @xmath539 the commutativity of the diagram [ figure2 ] is clear now . at the core of the introduction of dilations lies the fact that we can construct group operations from them . more precisely we are able to construct , by using compositions of dilations and the groupoid operation , approximately associative operations which shall lead us eventually to group operations in the tangent groupoid of a dilation . let @xmath511 be a dilation and @xmath540 the associated dilation of the @xmath90-double groupoid . further we shall be interested only in the properties of the following map . for any @xmath86 and any @xmath199 we define the * dilatation * : @xmath541 [ defdilc ] the domain of definition of @xmath542 is in fact only a subset of @xmath543 , according to the axiom 0 explained in definition [ dax0 ] section [ bordet ] . this map comes from the definition ( [ rtransdel ] ) of the dilation @xmath544 , namely @xmath545 for any @xmath199 with @xmath546 sufficiently small we can define @xmath547 ( as in figure [ figure2 ] ) from ( a subset of ) @xmath548 to @xmath1 . remark that @xmath549 , therefore the following composition is well defined : @xmath550 then @xmath551 . related to the function @xmath552 is the following @xmath553 the following expression makes sense too , for any pair of elements @xmath554 from ( a subset of ) @xmath555 : @xmath556 \label{sumdef}\ ] ] it is also true that @xmath557 . these three functions are interesting operations . the function @xmath558 is an approximate difference operation , @xmath559 is an approximate inverse and @xmath560 is an approximate sum operation . a look at the figure [ figure1 ] will help . there is graphically explained how @xmath561 is constructed . .,scaledwidth=50.0% ] let us imagine that we are looking at a figure in the euclidean plane . then @xmath562 is just a homothety , @xmath159 , @xmath563 are vectors with the same origin @xmath101 , @xmath564 is the difference of vectors @xmath565 ( or @xmath566 , it s the same as long as we are in a commutative world ) . in the euclidean plane , as @xmath546 goes to @xmath480 , the `` vector '' @xmath567 slides towards @xmath568 and @xmath561 is obtained from @xmath567 by composition with the vector @xmath569 . thus @xmath570 has the meaning of a approximate difference of vectors @xmath159 , @xmath563 . the functions @xmath558 , @xmath559 and @xmath560 can be expressed in terms of dilatations introduced in definition [ defdilc ] . indeed , let us define , for any triple @xmath347 with @xmath571 , and such that @xmath572 are sufficiently small , the following approximate difference function with three arguments : @xmath573 the approximate inverse function with two arguments : @xmath574 and the following approximate sum function with three arguments : @xmath575 we have then : @xmath576 we are in the framework of emergent algebras and idempotent right quasigroups , as introduced in @xcite . we recall here the definition of a idempotent right quasigroup and induced operations . an idempotent right quasigroup ( irq ) is a set @xmath0 endowed with two operations @xmath577 and @xmath578 , which satisfy the following axioms : for any @xmath579 1 . @xmath580 2 . @xmath581 we use these operations to define the sum , difference and inverse operations of the irq : for any @xmath582 1 . the difference operation is @xmath583 . by fixing the first variable @xmath584 we obtain the difference operation based at @xmath584 : @xmath585 . the sum operation is @xmath586 . by fixing the first variable @xmath584 we obtain the sum operation based at @xmath584 : @xmath587 . the inverse operation is @xmath588 . by fixing the first variable @xmath584 we obtain the inverse operator based at @xmath584 : @xmath589 . for any @xmath590 we define also the following operations : 1 . @xmath591 , @xmath592 , 2 . for any @xmath593 let @xmath594 and @xmath595 , 3 . for any @xmath596 let @xmath597 and @xmath598 . [ dplay ] for any @xmath599 the triple @xmath600 is a irq . we denote the difference , sum and inverse operations of @xmath600 by the same symbols as the ones used for @xmath601 , with a subscript `` @xmath602 '' . for any @xmath199 and for any @xmath66 we can define a irq operation on @xmath81 by @xmath603 . we have then : @xmath604 by computation it follows that @xmath605 . the approximate difference , sum and inverse operations are exactly the ones introduced in the preceding section . in @xcite we introduced idempotent right quasigroups and then iterates of the operations indexed by a parameter @xmath606 . this was done in order to simplify the notations mostly . here , in the presence of the group @xmath459 , we might define a @xmath459-irq . let @xmath459 be a commutative group . a @xmath459-idempotent right quasigroup is a set @xmath0 with a function @xmath607 such that @xmath608 is a irq and moreover for any @xmath472 and any @xmath293 we have @xmath609 [ defgammairq ] it is then obvious that if @xmath610 is a irq then @xmath611 is a @xmath612-irq ( we define @xmath613 ) . the following is a slight modification of proposition 3.4 and point ( k ) proposition 3.5 @xcite , for the case of @xmath459-irqs ( the proof of this proposition is almost identical , with obvious modifications , with the proof of the original proposition ) . in any irq @xmath614 be a @xmath459-irq . then we have the relations : 1 . @xmath615 2 . @xmath616 3 . @xmath617 4 . @xmath618 5 . @xmath619 6 . @xmath620 7 . @xmath621 8 . for any @xmath622 and any @xmath623 we have the distributivity property : @xmath624 [ pplay ] later we shall apply this proposition for the irq @xmath92 with the operations induced by dilatations @xmath465 . as @xmath625 the components of the deformations indexed by @xmath389 from the diagram [ figure2 ] ( namely the operation , norm and respective dilation maps ) may converge in the sense of section [ secdefconv ] to the components of another normed groupoid with dilations . 1 . there is a function @xmath626 which is the limit @xmath627 uniformly on bounded sets in the sense of definition [ defcon ] . moreover the convergence with respect to @xmath628 is the same as the convergence with respect to @xmath629 and in particular @xmath630 implies @xmath631 . 2 . there is a dilation @xmath632 of the normed groupoid @xmath633 such that for any @xmath199 the transformation @xmath634 converges uniformly on bounded sets to @xmath635 . indeed , these properties of the function @xmath628 come from the following observation . let us define on @xmath91 the function : @xmath641 then for any @xmath86 the function @xmath70 ( with two arguments ) gives a distance on the set @xmath81 . in the case of a @xmath484-structure the axiom a3 can be written as : @xmath642 uniformly on bounded sets . this gives properties ( b ) , ( c ) above from a passage to the limit of the properties of the distance @xmath70 . for any @xmath86 the restriction of the norm @xmath629 on the trivial groupoid @xmath643 gives a distance on the space @xmath81 . the dilatation @xmath644 has the property : for any @xmath199 and @xmath86 @xmath645 therefore we can define @xmath646 from ( a subset of ) @xmath647 to @xmath647 by : @xmath648 suppose that @xmath511 is a gw @xmath484-structure . then for any @xmath86 the triple @xmath649 is a dilatation structure , with @xmath484 defined by ( [ firstcomright ] ) and @xmath650 restrictioned to @xmath81 . the proof is just a translation of the definition [ defgdsweak ] in terms of metric spaces , using the equivalence between metric spaces and normed trivial groupoids . at the end we obtain definition [ defweakstrong ] of dilatation structures on metric spaces , given further . for simplicity we shall list the axioms of a dilatation structure @xmath651 without concerning about domains and codomains of dilatations . for the full definition of dilatation structure , as well as for their main properties and examples , see @xcite , @xcite , @xcite . the notion appeared from my efforts to understand the last section of the paper @xcite ( see also @xcite , @xcite , @xcite , @xcite ) . 1 . in the following definition [ defweakstrong ] we are no longer asking the metric space @xmath244 to be locally compact . also , uniform convergence in compact sets is replaced by uniform convergence in bounded sets . 2 . because of the modifications explained at ( a ) , we have to ask explicitly that the uniformities induced by @xmath652 and @xmath70 are the same . 3 . finally , dilatation structures in the sense of the following definition [ defweakstrong ] are a bit stronger than dilatation structures in the sense introduced and studied in @xcite , @xcite , namely we ask for the existence of a `` limit dilatation '' , see the last axiom . this limit exists for strong dilatation structures , but not for dilatation structures in the sense introduced in @xcite , @xcite . 1 . for any point @xmath66 the function @xmath484 induces an action @xmath656 , where @xmath657 is the collection of all continuous , with continuous inverse transformations @xmath658 such that @xmath659 . the function @xmath660 is continuous . moreover , it can be continuously extended to @xmath661 by @xmath662 and the limit @xmath663 is uniform with respect to @xmath664 in bounded set . 3 . there is @xmath665 such that for any @xmath584 there exists a function @xmath666 , defined for any @xmath667 in the closed ball ( in distance d ) @xmath668 , such that @xmath669 uniformly with respect to @xmath584 in bounded set . moreover the uniformity induced by @xmath670 is the same as the uniformity induced by @xmath70 , in particular @xmath671 implies @xmath672 . ( for metric spaces ) the following limit exists : @xmath673 for any @xmath512 , uniformly with respect to @xmath674 in bounded sets . in axiom a2 we may alternatively put that the limit is uniform with respect to @xmath678 . similarly , we may ask in axiom a4weak ( for metric spaces ) that the limit is uniform with respect to @xmath679 , @xmath680 . 1 . if @xmath244 is locally compact then the function @xmath681 is continuous as an uniform limit of continuous functions on a compact set . if @xmath244 is also separable then from the existence of the limit @xmath652 and from axiom a1 we obtain the fact that @xmath681 and @xmath70 induce the same uniformities . 2 . by definition @xmath652 is symmetric and satisfies the triangle inequality , but it can be a degenerated distance function : there might exist @xmath682 such that @xmath683.but the end of axiom a2 eliminates this possibility . let @xmath684 be a dilatation structure , @xmath66 , and let @xmath685 then the net of metric spaces @xmath686 converges in the gromov - hausdorff sense to the metric space @xmath687 . moreover this metric space is a metric cone , in the following sense : for any @xmath512 such that @xmath688 we have @xmath689 and @xmath690 the first part of the proposition is just a reformulation of axiom a3 , without the condition of uniform convergence . for the second part remark that @xmath691 and also that @xmath692 therefore if we pass to the limit with @xmath462 in these two relations we get the desired conclusion . @xmath321 let @xmath651 be a dilatation structure . the translation groupoid @xmath693 has as objects the distances @xmath694 for all @xmath66 and all @xmath199 . the arrows are of the form @xmath695 , with @xmath696 composition of arrows is composition of functions . [ transgrou ] @xmath693 is a groupoid . moreover , arrows are isometries , in the sense that for any @xmath199 and @xmath697 the arrow @xmath698 is an isometry from the source to the range : for ant @xmath699 we have : @xmath700 use proposition [ pplay ] . for example , the fact that composition of arrows is well defined is equivalent with ( e ) from the mentioned proposition . invertibility of arrows is ( a ) , and so on . the isometry claim follows from a straight computation . @xmath321 a * groupoid strong @xmath484-structure * ( or a gs @xmath484-structure ) is a triple @xmath701 such that @xmath484 is a map assigning to any @xmath199 a transformation @xmath469 which satisfies the axioms a1 , a2 from definition [ defgdsweak ] and the following axioms a3mod and a4 : m. buliga , dilatation structures in sub - riemannian geometry , ( 2007 ) , contemporary geometry and topology and related topics , cluj - napoca , cluj university press ( 2008 ) , 89 - 105 , http://arxiv.org/abs/0708.4298	we study normed groupoids with dilations and their induced deformations . addtoresetfiguresection @figureh , t addtoresettablebsection @tableh , t addtoresetequationsection [ section ] [ thm]proposition [ thm]lemma [ thm]corollary [ thm]definition [ thm]remark [ section ]	16,129	103
quantized control problems have been an active research topic in the past two decades . discrete - level actuators / sensors and digital communication channels are typical in practical control systems , and they yield quantized signals in feedback loops . quantization errors lead to poor system performance and even loss of stability . therefore , various control techniques to explicitly take quantization into account have been proposed , as surveyed in @xcite . on the other hand , switched system models are widely used as a mathematical framework to represent both continuous and discrete dynamics . for example , such models are applied to dc - dc converters @xcite and to car engines @xcite . stability and stabilization of switched systems have also been extensively studied ; see , e.g. , the survey @xcite , the book @xcite , and many references therein . in view of the practical importance of both research areas and common technical tools to study them , the extension of quantized control to switched systems has recently received increasing attention . there is by now a stream of papers on control with limited information for discrete - time markovian jump systems @xcite . moreover , our previous work @xcite has analyzed the stability of sampled - data switched systems with static quantizers . in this paper , we study the stabilization of continuous - time switched linear systems with quantized output feedback . our objective is to solve the following problem : given a switched system and a controller , design a quantizer to achieve asymptotic stability of the closed - loop system . we assume that the information of the currently active plant mode is available to the controller and the quantizer . extending the quantizer in @xcite for the non - switched case to the switched case , we propose a lyapunov - based update rule of the quantizer under a slow - switching assumption of average dwell - time type @xcite . the difficulty of quantized control for switched systems is that a mode switch changes the state trajectories and saturates the quantizer . in the non - switched case @xcite , in order to avoid quantizer saturation , the quantizer is updated so that the state trajectories always belong to certain invariant regions defined by level sets of a lyapunov function . however , for switched systems , these invariant regions are dependent on the modes . hence the state may not belong to such regions after a switch . to keep the state in the invariant regions , we here adjust the quantizer at every switching time , which prevent quantizer saturation . the same philosophy of emphasizing the importance of quantizer updates after switching has been proposed in @xcite for sampled - data switched systems with quantized state feedback . subsequently , related works were presented for the output feedback case @xcite and for the case with bounded disturbances @xcite . the crucial difference lies in the fact that these works use the quantizer based on @xcite and investigates propagation of reachable sets for capturing the measurement . this approach also aims to avoid quantizer saturation , but it is fundamentally disparate from our lyapunov - based approach . this paper is organized as follows . in section ii , we present the main result , theorem [ thm : stability_theorem ] , after explaining the components of the closed - loop system . section iii gives the update rule of the quantizer and is devoted to the proof of the convergence of the state to the origin . in section iv , we discuss lyapunov stability . we present a numerical example in section v and finally conclude this paper in section vi . the present paper is based on the conference paper @xcite . here we extend the conference version by addressing state jumps at switching times . we also made structural improvements in this version . _ notation : _ let @xmath0 and @xmath1 denote the smallest and the largest eigenvalue of @xmath2 . let @xmath3 denote the transpose of @xmath4 . the euclidean norm of @xmath5 is denoted by @xmath6 . the euclidean induced norm of @xmath4 is defined by @xmath7 . for a piecewise continuous function @xmath8 , its left - sided limit at @xmath9 is denoted by @xmath10 . for a finite index set @xmath11 , let @xmath12 be a right - continuous and piecewise constant function . we call @xmath13 a _ switching signal _ and the discontinuities of @xmath13 _ switching times_. let us denote by @xmath14 the number of discontinuities of @xmath13 on the interval @xmath15 $ ] . let @xmath16 be switching times , and consider a switched linear system @xmath17 with the jump @xmath18 where @xmath19 is the state , @xmath20 is the control input , and @xmath21 is the output . assumptions on the switched system are as follows . [ ass : system ] _ for every @xmath22 , @xmath23 is stabilizable and @xmath24 is observable . we choose @xmath25 and @xmath26 so that @xmath27 and @xmath28 are hurwitz . _ furthermore , the switching signal @xmath13 has an average dwell time @xcite , i.e. , there exist @xmath29 and @xmath30 such that @xmath31 we need observability rather than detectability , because we reconstruct the state by using the observability gramian . in this paper , we use the following class of quantizers proposed in @xcite . let @xmath32 be a finite subset of @xmath33 . a quantizer is a piecewise constant function @xmath34 . this implies geometrically that @xmath35 is divided into a finite number of the quantization regions @xmath36 @xmath37 . for the quantizer @xmath38 , there exist positive numbers @xmath39 and @xmath40 with @xmath41 such that @xmath42 the former condition gives an upper bound of the quantization error when the quantizer does not saturate . the latter is used for the detection of quantizer saturation . we place the following assumption on the behavior of the quantizer near the origin . this assumption is used for lyapunov stability of the closed - loop system . [ @xcite ] [ ass : near origin ] _ there exists @xmath43 such that @xmath44 for every @xmath45 with @xmath46 . _ we use quantizers with the following adjustable parameter @xmath47 : @xmath48 in , @xmath49 is regarded as a `` zoom '' variable , and @xmath50 is the data on @xmath51 transmitted to the controller at time @xmath52 . we need to change @xmath49 to obtain accurate information of @xmath53 . the reader can refer to @xcite for further discussions . the quantized output @xmath54 may chatter on boundaries among quantization regions . hence if we generate the input @xmath55 by @xmath54 , the solutions of must be interpreted in the sense of filippov @xcite . however , this generalization does not affect our lyapunov - based analysis as in @xcite , because we will use a single quadratic lyapunov function between switching times . similarly to @xcite , we construct the following dynamic output feedback law based on the standard luenberger observers : @xmath56 where @xmath57 is the state estimate . the estimate also jumps at each switching times @xmath58 : @xmath59 then the closed - loop system is given by @xmath60 if we define @xmath61 and @xmath62 by @xmath63 then we rewrite in the form @xmath64 the state @xmath61 of the closed - loop system jumps at each switching time @xmath58 : @xmath65 where @xmath66 we see from assumption [ ass : system ] that @xmath67 is hurwitz for each @xmath22 . for every positive - definite matrix @xmath68 , there exist a positive - definite matrix @xmath69 such that @xmath70 we define @xmath71 , @xmath72 , @xmath73 , and @xmath74 by @xmath75 \underline \lambda_q : = { \displaystyle \min_{p \in \mathcal{p } } \lambda_{\min}(q_p ) } , \quad c_{\max } : = { \displaystyle \max_{p \in \mathcal{p}}\\|c_p\\|}. \end{array}\end{aligned}\ ] ] fig . [ fig : cssqof ] shows the closed - loop system we consider . by adjusting the `` zoom '' parameter @xmath49 , we can achieve global asymptotic stability of the closed - loop system . this result is a natural extension of theorem 5 in @xcite to switched systems . _ [ thm : stability_theorem ] define @xmath76 by @xmath77 and let @xmath39 be large enough to satisfy @xmath78 if the average dwell time @xmath79 in is larger than a certain value , then there exists a right - continuous and piecewise - constant function @xmath49 such that the closed - loop system has the following two properties for every @xmath80 and every @xmath81 : _ _ ( i ) convergence to the origin : _ @xmath82 . _ ( ii ) lyapunov stability : _ to every @xmath83 , there corresponds @xmath84 such that @xmath85 we shall prove convergence to the origin and lyapunov stability in sections iii and iv , respectively . we also present an update rule of the `` zoom '' parameter @xmath49 in section 3 . the sufficient condition on @xmath79 is given by in theorem [ lem : convergence_to_origin ] below . define @xmath86 and @xmath87 by @xmath88 we split the proof into two stages : the `` zooming - out '' and `` zooming - in '' stages . since the initial state @xmath89 is unknown to the quantizer , we have to capture the state @xmath61 of the closed - loop system by `` zooming out '' , i.e. , increasing the `` zoom '' parameter @xmath49 . we first see that @xmath61 can be captured if we have a time - interval with a given length that has no switches . [ lem : capture_non_switched ] _ consider the closed - loop system . set the control input @xmath90 . choose @xmath91 , and define @xmath92 and the observability gramian @xmath93 assume that there exists @xmath94 such that we can observe @xmath95 for all @xmath96 . let the `` zoom '' parameter @xmath49 be piecewise continuous and monotone increasing in @xmath97 . if we set the state estimate @xmath98 at @xmath99 by @xmath100 and if we choose @xmath101 so that @xmath102 then @xmath103 . _ since no switch occurs by , we can easily obtain this result by extending theorem 5 in @xcite for the non - switched case . we therefore omit the proof ; see also the conference version @xcite . it follows from theorem [ lem : capture_non_switched ] that in order to capture the state @xmath61 , it is enough to show the existence of @xmath94 satisfying and for all @xmath104 . to this end , we use the following lemma on average dwell time @xmath79 : [ lem : adt_upperbound ] _ fix an initial time @xmath105 . suppose that @xmath13 satisfies the average dwell - time assumption . let @xmath106 . if we choose @xmath107 so that @xmath108 then there exists @xmath109 $ ] such that @xmath110 . _ let us denote the switching times by @xmath16 , and fix @xmath107 . suppose that @xmath111 for all @xmath109 $ ] . then we have @xmath112 indeed , if @xmath113 for some @xmath114 and if we let @xmath115 be the smallest such integer , then we obtain @xmath116 and @xmath117 . this contradicts with @xmath118 $ ] . thus we have . from , we see that for @xmath119 , @xmath120 it follows from that @xmath121 therefore @xmath122 satisfies the following inequality : @xmath123 since @xmath124 was arbitrary , is equivalent to @xmath125 thus we have shown that if holds for all @xmath109 $ ] , then @xmath126 satisfies . the contraposition of this statement gives a desired result . [ thm : non_switch_interval ] _ consider the closed - loop system with average dwell - time property . set the control input @xmath90 . fix @xmath127 , @xmath128 , and @xmath106 . increase @xmath49 in the following way : @xmath129 for @xmath130 , @xmath131 for @xmath132 and @xmath133 . then there exists @xmath94 such that and hold for all @xmath104 . _ if @xmath134 switches occur in the interval @xmath135 $ ] , then we have @xmath136 since @xmath137 , it follows from that @xmath138 clearly , this inequality holds in the case when no switches occur . since shows that @xmath139 and since the growth rate of @xmath140 is larger than that of @xmath141 , there exists @xmath142 such that @xmath143 in conjunction with , this implies that holds for every @xmath144 . let @xmath122 be an integer satisfying . then lemma [ lem : adt_upperbound ] guarantees the existence of @xmath145 $ ] such that holds for every @xmath96 . this completes the proof . it follows from theorems [ lem : capture_non_switched ] and [ thm : non_switch_interval ] that if we update the `` zoom '' parameter @xmath49 as in and if we set the state estimate @xmath98 by , then the state @xmath61 of the closed - loop system can be captured . [ rem : brief_remark_for_lyap ] if the initial state @xmath89 is sufficiently small , then @xmath146 in is zero . in this situation , we can capture @xmath61 by @xmath147 for all switching signal with average dwell - time property . we use this fact for the proof of lyapunov stability ; see section 4 . next we drive the state @xmath61 of the closed - loop system to the origin by `` zooming - in '' , i.e. , decreasing the `` zoom '' parameter @xmath49 . since @xmath49 increases at each switching time during this stage , the term `` zooming - in stage '' may be misleading . however , @xmath49 decreases overall under a certain average dwell - time assumption , so we use the term `` zooming - in '' as in @xcite . let us first consider a fixed `` zoom '' parameter @xmath49 . the following lemma shows that if no switches occur , then the state trajectories move from a large level set to a small level set of the lyapunov function @xmath148 in a finite time that is independent of the mode @xmath149 : [ lem : fix_zoom_parameter ] _ define @xmath67 and @xmath150 as in and , respectively . fix @xmath22 , and consider the non - switched system @xmath151 choose @xmath152 . if @xmath39 satisfies @xmath153 where @xmath71 , @xmath72 @xmath74 , and @xmath76 are defined by and , then the following two level sets of the lyapunov function @xmath154 are invariant regions for every trajectory of : @xmath155 furthermore , if @xmath156 for all @xmath157 $ ] , then @xmath158 for every @xmath22 . hence if @xmath159 satisfies @xmath160 then every trajectory of with an initial state @xmath161 satisfies @xmath162 _ since the mode @xmath22 is fixed , this lemma is a trivial extension of lemma 5 in @xcite for single - modal systems . we therefore omit its proof ; see also the conference version @xcite . using lemma [ lem : fix_zoom_parameter ] , we obtain an update rule of the `` zoom '' parameter @xmath49 to drive the state @xmath61 to the origin . [ lem : convergence_to_origin ] _ consider the system under the same assumptions as in lemma [ lem : fix_zoom_parameter ] . assume that @xmath163 . for each @xmath164 with @xmath165 , the positive definite matrices @xmath166 and @xmath167 in the lyapunov equation satisfy @xmath168 for some @xmath169 . define @xmath170 and @xmath171 by @xmath172 @xmath173 fix @xmath174 so that is satisfied , and set the `` zoom '' parameter @xmath175 for all @xmath176 and @xmath177 $ ] in the following way : if no switches occur in the interval @xmath178 $ ] , then @xmath179 otherwise , @xmath180 where @xmath181 are the switching times in the interval @xmath178 $ ] . then @xmath182 for all @xmath183 . furthermore , if @xmath79 satisfies @xmath184 then @xmath185 . _ to prove that @xmath182 for all @xmath183 , it is enough to show that if @xmath186 , then @xmath187 let us first investigate the case without switching on the interval @xmath188 $ ] . we see from lemma [ lem : fix_zoom_parameter ] that @xmath182 for all @xmath189 and that @xmath190 . since @xmath191 , a routine calculation shows that @xmath192 . we now study the switched case . let @xmath193 be the switching times in the interval @xmath188 $ ] . let us define @xmath194 for simplicity of notation . lemma [ lem : fix_zoom_parameter ] implies that @xmath195 ( @xmath196 ) are invariant sets for all @xmath197 , @xmath198 . moreover , by , if @xmath199 , then @xmath200 ( @xmath196 ) for all @xmath201 . hence @xmath202 leads to @xmath203 to obtain @xmath204 we show that @xmath205 . assume , to reach a contradiction , that @xmath206 since @xmath207 is an invariant region for all @xmath208 , we also have @xmath209 define a lyapunov function @xmath148 for each @xmath22 . since a filippov solution is ( absolutely ) continuous , @xmath210 exists for each @xmath211 . from , we obtain @xmath212 on the other hand , since @xmath213 for all @xmath214 $ ] , gives @xmath215 and hence we have from @xmath216 that @xmath217 if we repeat this process and use , then @xmath218 which contradicts . thus we obtain @xmath219 and hence holds . from and , we derive the desired result , because @xmath220 . finally , since @xmath221 , gives @xmath222 for every @xmath223 and @xmath224 . if @xmath225 , that is , if the average dwell time @xmath79 satisfies , then @xmath226 . since @xmath182 for all @xmath183 , we obtain @xmath227 . ( a ) we can compute @xmath228 by linear matrix inequalities . moreover , if the jump matrix @xmath229 in is invertible , then lemma 13 of @xcite gives an explicit formula for @xmath228 . ( b ) the proposed method is sensitive to the time - delay of the switching signal at the `` zooming - in '' stage . if the switching signal is delayed , a mode mismatch occurs between the plant and the controller . here we do not proceed along this line to avoid technical issues . see also @xcite for the stabilization of asynchronous switched systems with time - delays . ( c ) we have updated the `` zoom '' parameter @xmath49 at each switching time in the `` zooming - in '' stage . if we would not , switching could lead to instability of the closed - loop system . in fact , since the state @xmath61 may not belong to the invariant region @xmath230 without adjusting @xmath49 , the quantizer may saturate . ( d ) similarly , `` pre - emptively '' multiplying @xmath49 at time @xmath231 by @xmath232 does not work , either . this is because such an adjustment does not make @xmath230 invariant for the state trajectories . for example , consider the situation where the state @xmath61 belongs to @xmath233 at @xmath234 due to this pre - emptively adjustment . then @xmath61 does not converge to the origin . let @xmath235 be a switching time . since @xmath236 may not be a subset of @xmath237 , it follows that @xmath61 does not belong to the invariant region @xmath230 at @xmath238 in general . let us denote by @xmath239 the open ball with center at the origin and radius @xmath240 in @xmath241 . in what follows , we use the same letters as in the previous section and assume that the average dwell time @xmath79 satisfies . the proof consists of three steps : 1 . obtain an upper bound of the time @xmath242 at which the quantization process transitions from the `` zoom - out '' stage to the `` zoom - in '' stage . 2 . show that there exists a time @xmath243 such that the state @xmath61 satisfies @xmath244 for all @xmath245 . 3 . set @xmath84 so that if @xmath246 , then @xmath244 for all @xmath247 . we break the proof of lyapunov stability into the above three steps . \1 ) let @xmath126 satisfy and let @xmath84 be small enough to satisfy @xmath248 we see from the state bound that @xmath249 for @xmath250 $ ] from assumption [ ass : near origin ] . as we mentioned in remark [ rem : brief_remark_for_lyap ] briefly , lemma [ lem : adt_upperbound ] implies that the time @xmath242 , at which the stage changes from `` zooming - out '' to `` zooming - in '' , satisfies @xmath251 for every switching signal with the average dwell - time assumption . \2 ) fix @xmath252 . by , @xmath253 , and hence we see from that @xmath254 achieving @xmath186 can be chosen so that @xmath255 where @xmath256 is defined by @xmath257 note that @xmath256 is independent of switching signals . let @xmath258 be the smallest integer satisfying @xmath259 define @xmath260 . since @xmath221 and @xmath225 , and give @xmath261 for all @xmath262 and @xmath224 . since @xmath263 satisfies , it follows that that @xmath264 lies in @xmath239 for all @xmath245 . recall that @xmath265 and that @xmath264 is an invariant region for all @xmath183 from theorem [ lem : convergence_to_origin ] . thus we have @xmath266 \3 ) define @xmath267 since @xmath268 , it follows from , and that @xmath269 for all @xmath270 $ ] . set @xmath84 so that @xmath271 since @xmath272 , by , , , and , assumption [ ass : near origin ] gives @xmath273 in the interval @xmath274 $ ] , so @xmath275 and @xmath276 in the same interval . combining this with , we obtain @xmath277 for all @xmath247 . thus @xmath278 from and , we see that lyapunov stability can be achieved . consider the continuous - time switched system with the following two modes : @xmath280 with jump matrices @xmath281 . as the feedback gain and the observer gain of each mode , we take @xmath282 let @xmath38 be a uniform - type quantizer with parameters @xmath283 , @xmath284 the parameters @xmath285 in the `` zooming - out '' stage are @xmath286 , @xmath287 , and @xmath288 . also , define @xmath289 and @xmath290 in and @xmath291 in by @xmath292 , @xmath293 @xmath294 , where @xmath295 means a diagonal matrix whose diagonal elements starting in the upper left corner are @xmath296 . then we obtain @xmath297 in , @xmath298 in , @xmath299 in , and @xmath300 in . figure [ fig : cont_simulation ] ( a ) and ( b ) show that the euclidean norm of the state @xmath301 and the estimate @xmath98 , and the `` zoom '' parameter @xmath49 , respectively , with initial condition @xmath302^{\top}$ ] and @xmath303 . the vertical dashed - dotted line indicates the switching times @xmath304 . in this example , the `` zooming - out '' stage finished at @xmath305 . we see the non - smoothness of @xmath306 and the increase of @xmath49 at the switching times @xmath304 because of switches and quantizer updates . not surprisingly , the adjustments of @xmath49 in and are conservative . we have proposed an update rule of dynamic quantizers to stabilize continuous - time switched systems with quantized output feedback . the average dwell - time property has been utilized for the state reconstruction in the `` zooming - out '' stage and for convergence to the origin in the `` zooming - in '' stage . the update rule not only periodically decreases the `` zoom '' parameter to drive the state to the origin , but also adjusts the parameter at each switching time to avoid quantizer saturation . future work involves designing the controller and the quantizer simultaneously , and addressing more general systems by incorporating disturbances and nonlinear dynamics .	in this paper , we study the problem of stabilizing continuous - time switched linear systems with quantized output feedback . we assume that the observer and the control gain are given for each mode . also , the plant mode is known to the controller and the quantizer . extending the result in the non - switched case , we develop an update rule of the quantizer to achieve asymptotic stability of the closed - loop system under the average dwell - time assumption . to avoid quantizer saturation , we adjust the quantizer at every switching time . switched systems , quantized control , output feedback stabilization .	7,265	141
quantum key distribution ( qkd ) is often said to be unconditionally secure @xcite . more precisely , qkd can be proven to be secure against any eavesdropping _ given _ that the users ( alice and bob ) devices satisfy some requirements , which often include mathematical characterization of users devices as well as the assumption that there is no side - channel . this means that no one can break mathematical model of qkd , however in practice , it is very difficult for practical devices to meet the requirements , leading to the breakage of the security of practical qkd systems . actually , some attacks on qkd have been proposed and demonstrated successfully against practical qkd systems @xcite . to combat the practical attacks , some counter - measures @xcite , including device independent security proof idea @xcite , have been proposed . the device independent security proof is very interesting from the theoretical viewpoint , however it can not apply to practical qkd systems where loopholes in testing bell s inequality @xcite can not be closed . as for the experimental counter - measures , battle - testing of the practical detection unit has attracted many researchers attention @xcite since the most successful practical attack so far is to exploit the imperfections of the detectors . recently , a very simple and very promising idea , which is called a measurement device independent qkd ( mdiqkd ) has been proposed by lo , curty , and qi @xcite . in this scheme , neither alice nor bob performs any measurement , but they only send out quantum signals to a measurement unit ( mu ) . mu is a willing participant of the protocol , and mu can be a network administrator or a relay . however , mu can be untrusted and completely under the control of the eavesdropper ( eve ) . after alice and bob send out signals , they wait for mu s announcement of whether she has obtained the successful detection , and proceed to the standard post - processing of their sifted data , such as error rate estimation , error correction , and privacy amplification . the basic idea of mdiqkd is based on a reversed epr - based qkd protocol @xcite , which is equivalent to epr - based qkd @xcite in the sense of the security , and mdiqkd is remarkable because it removes _ all _ the potential loopholes of the detectors without sacrificing the performance of standard qkd since alice and bob do not detect any quantum signals from eve . moreover , it is shown in @xcite that mdiqkd with infinite number of decoy states and polarization encoding can cover about twice the distance of standard decoyed qkd , which is comparable to epr - based qkd . the only assumption needed in mdiqkd is that the preparation of the quantum signal sources by alice and bob is ( almost ) perfect and carefully characterized . we remark that the characterization of the signal source should be easier than that of the detection unit since the characterization of the detection unit involves the estimation of the response of the devices to unknown input signals sent from eve . with mdiqkd in our hand , we do not need to worry about imperfections of mu any more , and we should focus our attention more to the imperfections of signal sources . one of the important imperfections of the sources is the basis - dependent flaw that stems from the discrepancy of the density matrices corresponding to the two bases in bb84 states . the security of standard bb84 with basis - dependent flaw has been analyzed in @xcite which show that the basis - dependent flaw decreases the achievable distance . thus , in order to investigate the practicality of mdiqkd , we need to generalize the above works to investigate the security of mdiqkd under the imperfection . another problem in mdiqkd is that the first proposal is based on polarization encoding @xcite , however , in some situations where birefringence effect in optical fiber is highly time - dependent , we need to consider mdiqkd with phase encoding rather than polarization encoding . in this paper , we study the above issues simultaneously . we first propose two schemes of the phase encoding mdiqkd , one employs phase locking of two separate laser sources and the other one uses the conversion of phase encoding into polarization encoding . then , we prove the unconditional security of these schemes with basis - dependent flaw by generalizing the quantum coin idea @xcite . based on the security proof , we simulate the key generation rate with realistic parameters , especially we employ a simple model to evaluate the basis - dependent flaw due to the imperfection of the phase modulators . our simulation results imply that the first scheme covers shorter distances and may require less accuracy of the state preparation , while the second scheme can cover much longer distances when we can prepare the state very precisely . we note that in this paper we consider the most general type of attacks allowed by quantum mechanics and establish unconditional security for our protocols . this paper is organized as follows . in sec . [ sec : protocol ] , we give a generic description of mdiqkd protocol , and we propose our schemes in sec . [ sec : phase encoding scheme i ] and sec . [ sec : phase encoding scheme ii ] . then , we prove the unconditional security of our schemes in sec . [ sec : proof ] , and we present some simulation results of the key generation rate based on realistic parameters in sec . [ sec : simulation ] . finally , we summarize this paper in sec . [ sec : summary ] . in this section , we introduce mdiqkd protocol whose description is generic for all the schemes that we will introduce in the following sections . the mdiqkd protocol runs as follows . step ( 1 ) : each of alice and bob prepares a signal pulse and a reference pulse , and each of alice and bob applies phase modulation to the signal pulse , which is randomly chosen from @xmath0 , @xmath1 , @xmath2 , and @xmath3 . here , @xmath4 ( @xmath5 ) defines @xmath6 ( @xmath7)-basis . alice and bob send both pulses through quantum channels to eve who possesses mu . step ( 2 ) : mu performs some measurement , and announces whether the measurement outcome is successful or not . it also broadcasts whether the successful event is the detection of type-0 or type-1 ( the two types of the successful outcomes correspond to two specific bell states @xcite ) . step ( 3 ) : if the measurement outcome is successful , then alice and bob keep their data . otherwise , they discard the data . when the outcome is successful , alice and bob broadcast their bases and they keep the data only when the bases match , which we call sifted key . depending on the type of the successful event and the basis that they used , bob may or may not perform bit - flip on his sifted key . step ( 4 ) : alice and bob repeat ( 1)-(3 ) many times until they have large enough number of the sifted key . step ( 5 ) : they sacrifice a portion of the data as the test bits to estimate the bit error rate and the phase error rate on the remaining data ( code bits ) . step ( 6 ) : if the estimated bit error and phase error rates are too high , then they abort the protocol , otherwise they proceed . step ( 7 ) : alice and bob agree over a public channel on an error correcting code and on a hash function depending on the bit and phase error rate on the code bits . after performing error correction and privacy amplification , they share the key . the role of the mu in eve is to establish a quantum correlation , i.e. , a bell state , between alice and bob to generate the key . if it can establish the strong correlation , then alice and bob can generate the key , and if it can not , then it only results in a high bit error rate to be detected by alice and bob and they abort the protocol . as we will see later , since alice and bob can judge whether they can generate a key or not by only checking the experimental data as well as information on the fidelity between the density matrices in @xmath6- basis and @xmath7-basis , it does not matter who performs the measurement nor what kind of measurement is actually done as long as mu broadcasts whether the measurement outcome was successful together with the information of whether the successful outcome is type-0 or type-1 . in the security proof , we assume that mu is totally under the control of eve . in practice , however , we should choose an appropriate measurement that establishes the strong correlation under the normal operation , i.e. , the situation without eve who induces the channel losses and noises . in the following sections , we will propose two phase encoding mdiqkd schemes . in this section , we propose an experimental setup for mdiqkd with phase encoding scheme , which is depicted in fig . [ setup1 ] . this scheme will be proven to be unconditionally secure , i.e. , secure against the most general type of attacks allowed by quantum mechanics . in this setup , we assume that the intensity of alice s signal ( reference ) pulse matches with that of bob s signal ( reference ) pulse when they enter mu . in order to lock the relative phase , we use strong pulses as the reference pulses . in pl unit in the figure , the relative phase between the two strong pulses is measured in two polarization modes separately . the measurement result is denoted by @xmath8 ( here , the arrow represents two entries that correspond to the two relative phases ) . depending on this information @xmath8 , appropriate phase modulations for two polarization modes are applied to incoming signal pulse from alice . then , alice s and bob s signal pulses are input into the 50/50 beam splitter which is followed by two single - photon threshold detectors . the successful event of type-0 ( type-1 ) in step ( 2 ) is defined as the event where only d0 ( d1 ) clicks . in the case of type-1 successful detection event , bob applies bit flip to his sifted key ( we define the phase relationship of bs in such a way that d1 never clicks when the phases of the two input signal coherent pulses are the same ) . . then , the phase shift of @xmath8 for each polarization mode is applied to one of the signal pulses , and they will be detected by d0 and d1 after the interference at the 50:50 beam splitter bs . [ setup1 ] ] roughly speaking , our scheme performs double bb84 @xcite , i.e. , each of alice and bob is sending signals in the bb84 states , without phase randomization @xcite . differences between our scheme and the polarization encoding mdiqkd scheme include that alice and bob do not need to share the reference frame for the polarization mode , since mu performs the feed - forward control of the polarization , and our scheme intrinsically possesses the basis - dependent flaw . to see how this particular setup establishes the quantum correlation under the normal operation , it is convenient to consider an entanglement distribution scheme @xcite , which is mathematically equivalent to the actual protocol . for the simplicity of the discussion , we assume the perfect phase locking for the moment and we only consider the case where both of alice and bob use @xmath6-basis . we skip the discussion for @xmath7-basis , however it holds in a similar manner @xcite . in this case , the actual protocol is equivalently described as follows . first , alice prepares two systems in the following state , which is a purification of the @xmath6-basis density matrix , @xmath9 and sends the second system to mu through the quantum channel . here , @xmath10 and @xmath11 represent coherent states that alice prepares in the actual protocol ( @xmath12 represents the mean photon number or inetensity ) , @xmath13 and @xmath14 are eigenstate of the computational basis ( @xmath6 basis ) , which is related with @xmath7-basis eigenstate through @xmath15 and @xmath16 . for the later convenience , we also define @xmath17-basis states as @xmath18 and @xmath19 . moreover , the subscript of @xmath20 in @xmath21 represents that alice is to measure her qubit along @xmath6-basis , the subscript of @xmath22 in @xmath12 refers to the party who prepares the system , and the superscript @xmath23 represents the relative phase of the superposition . similarly , bob also prepares two systems in a similar state @xmath24 , sends the second system to mu , and performs @xmath6-basis measurement . note that @xmath6-basis measurement by alice and bob can be delayed after eve s announcement of the successful event without losing any generalities in the security analysis , and we assume this delay in what follows . in order to see the joint state of the qubit pair after the announcement , note that the beam splitter converts the joint state @xmath25 into the following state @xmath26 @xmath27 here , for the simplicity of the discussion , we assume that there is no channel losses , we define @xmath28 , and @xmath29 represents the vacuum state . moreover , the subscripts @xmath30 and @xmath31 represent the output ports of the beam splitter . if detector d0 ( d1 ) detects photons and the other detector d1 ( d0 ) detects the vacuum state , i.e. , type-0 ( type-1 ) event , it is shown in the appendix a that the joint probability of having type-0 ( type-1 ) successful event and alice and bob share the maximally entangled state @xmath32 ( @xmath33 ) is @xmath34 . we note that since @xmath35 , alice and bob do not always share this state , and with a joint probability of @xmath36 , they have type-0 ( type-1 ) successful event and share the maximally entangled state with the phase error , i.e. , the bit error in @xmath7-basis , as @xmath37 ( @xmath38 ) . note that the bit - flip operation in type-1 successful detection can be equivalently performed by @xmath2 rotation around @xmath17-basis before bob performs @xmath6 basis measurement . in other words , @xmath2 rotation around @xmath17-basis before @xmath6-basis measurement does not change the statistics of the @xmath6-basis measurement followed by the bit - flip . thanks to this property , we can conclude that alice and bob share @xmath32 with probability of @xmath39 and @xmath37 with probability of @xmath40 after the rotation . this means that even if alice and bob are given the successful detection event , they can not be sure whether they share @xmath37 or @xmath32 , however , if they choose a small enough @xmath41 , then the phase error rate ( the rate of the state @xmath37 in the qubit pairs remaining after the successful events or equivalently , the rate of @xmath7-basis bit error among all the shared qubit pairs ) becomes small and they can generate a pure state @xmath32 by phase error correction , which is equivalently done by privacy amplification in the actual protocol @xcite . we note that the above discussion is valid only for the case without noises and losses , and we will prove the security against the most general attack in sec . [ sec : proof ] without relying on the argument given in this section . we remark that in the phase encoding scheme i , it is important that alice and bob know quite well about the four states that they prepare . this may be accomplished by using state tomography with homodyne measurement involving the use of the strong reference pulse @xcite . [ scheme ii ] in this section , we propose the second experimental setup for mdiqkd with phase encoding scheme . like scheme i , this scheme will also be proven to be unconditionally secure . in this scheme , the coherent pulses that alice and bob send out are exactly the same as those in the standard phase encoding bb84 , i.e. , @xmath42 where subscripts @xmath43 and @xmath44 respectively denote the signal pulse and the reference pulse , @xmath45 is a completely random phase , @xmath46 is randomly chosen from @xmath47 to encode the information . after entering the mu , each pulse pair is converted from a phase coding signal to a polarization coding signal by a phase - to - polarization converter ( see details below ) . we note that thanks to the phase randomization by @xmath45 , the joint state of the signal pulse and the reference pulse is a classical mixture of photon number states . in fig . [ setup2 ] , we show the schematics of the converter . this converter performs the phase - to - polarization conversion : @xmath48 to @xmath49 , where @xmath50 is a projector that projects the joint system of the signal and reference pulses to a two - dimensional single - photon subspace spanned by @xmath51 where @xmath0 and @xmath52 represent the photon number , and @xmath53 ( @xmath54 ) represents the horizontal ( vertical ) polarization state of a single - photon . to see how it works , let us follow the time evolution of the input state . at the polarization beam splitter ( pbs in fig . [ setup2 ] ) , the signal and reference pulses first split into two polarization modes , h and v , and we throw away the pulses being routed to v mode . then , in h mode , the signal pulse and the reference pulse are routed to different paths by using an optical switch , and we apply @xmath2-rotation only to one of the paths to convert h to v. at this point , we essentially have @xmath55 , where the subscripts of `` @xmath56 '' and `` @xmath57 '' respectively denote the upper path and the lower path . finally , these spatial modes @xmath56 and @xmath57 are combined together by using a polarization beam splitter so that we have @xmath49 in the output port depicted as `` out '' . in practice , since the birefringence of the quantum channel can be highly time dependent and the polarization state of the input pulses to mu may randomly change with time , i.e. , the input polarization state is a completely mixed state , we can not deterministically distill a pure polarization state , and thus the conversion efficiency can never be perfect . in other words , one may consider the same conversion of the v mode just after the first polarization beam splitter , however it is impossible to combine the resulting polarization pulses from v mode and the one from h mode into a single mode . we assume that mu has two converters , one is for the conversion of alice s pulse and the other one is for bob s pulse , and the two output ports `` out '' are connected to exactly the same bell measurement unit @xcite in the polarization encoding mdiqkd scheme in fig . [ bell m ] @xcite . this bell measurement unit consists of a 50:50 beam splitter , two polarization beam splitters , and four single - photon detectors , which only distinguishes perfectly two out of the four bell states of @xmath38 and @xmath33 . the polarization beam splitters discriminate between @xmath58 and @xmath59 ( note that we choose @xmath60 and @xmath61 modes rather than h and v modes since our computational basis is @xmath60 and @xmath61 ) . suppose that a single - photon enters both from alice and bob . in this case , the click of d0 + and d0- or d1 + and d1- means the detection of @xmath38 , and the click of d0 + and d1- or d0- and d1 + means the detection of @xmath33 ( see fig . [ bell m ] ) . in this scheme , since the use of coherent light induces non - zero bit error rate in @xmath7-basis ( @xmath62-basis ) , we consider to generate the key from @xmath63-basis and we use the data in @xmath7-basis only to estimate the bit error rate in this basis conditioned on that both of alice and bob emit a single - photon , which determines the amount of privacy amplification . by considering a single - photon polarization input both from alice and bob , one can see that bob should not apply the bit flip only when alice and bob use @xmath7-basis and @xmath64 is detected in mu , and bob should apply the bit flip in all the other successful events to share the same bit value . accordingly , the bit error in @xmath6-basis is given by the successful detection event conditioned on that alice and bob s polarization are identical . as for @xmath7-basis , the bit error is @xmath64 detection given the orthogonal polarizations or @xmath65 detection given the identical polarization . assuming completely random input polarization state , our converter successfully converts the single - photon pulse with a probability of @xmath66 . note in the normal experiment that the birefringence effect between alice and the converter and the one between bob and the converter are random and independent , however it only leads to fluctuating coincidence rate of alice s and bob s signals at the bell measurement , but does not affect the qber . moreover , the fluctuation increases the single - photon loss inserted into the bell measurement . especially , the events that the output of the converter for alice is the vacuum and the one for bob is a single - photon , and vice versa would increase compared to the case where we have no birefringence effect . however , this is not a problem since the bell measurement does not output the conclusive events in these cases unless the dark counting occurs . thus , the random and independent polarization fluctuation in the normal experiment is not a problem , and we will simply assume in our simulation in sec . [ simulationii ] that this fluctuation can be modeled just by @xmath66 loss . we emphasize that we do not rely on these assumptions at all when we prove the security , and our security proof applies to any channels and mus . for the better performance and also for the simplicity of analysis , we assume the use of infinite number of decoy states @xcite to estimate the fraction of the probability of successful event conditioned on that both of alice and bob emit a single - photon . one of the differences in our analysis from the work in @xcite is that we will take into account the imperfection of alice s and bob s source , i.e. , the decay of the fidelity between two density matrices in two bases . we also remark that since the h and v modes are defined locally in mu , alice and bob do not need to share the reference frame for the polarization mode , which is one of the qualitative differences from polarization encoding miqkd scheme @xcite . this section is devoted to the unconditional security proof , i.e. , the security proof against the most general attacks , of our schemes . since both of our schemes are based on bb84 and the basis - dependent flaw in both protocols can be treated in the same manner , we can prove the security in a unified manner . if the states sent by alice and bob were basis independent , i.e. , the density matrices of @xmath6-basis and @xmath7-basis were the same , then the security proof of the original bb84 @xcite could directly apply ( also see @xcite for a bit more detailed discussion of this proof ) , however they are basis dependent in our case . fortunately , security proof of standard bb84 with basis - dependent flaw has already been shown to be secure @xcite , and we generalize this idea to our case where we have basis - dependent flaw from both of alice and bob . in order to do so , we consider a virtual protocol @xcite that alice and bob get together and the basis choices by alice and bob are made via measurement processes on the so - called quantum coin . in this virtual protocol of the phase encoding scheme i , alice and bob prepare joint systems in the state @xcite @xmath67 since just replacing the state , for instance @xmath68 where @xmath52 and @xmath0 in the ket respectively represents the single - photon and the vacuum , is enough to apply the following proof to the phase encoding scheme ii , we discuss only the security of the phase encoding scheme i in what follows . in eq . ( [ coin - joint - state ] ) , the first system denoted by @xmath69 is given to eve just after the preparation , and it informs eve of whether the bases to be used by alice and bob match or not . the second system , denoted by @xmath70 , is a copy of the first system and this system is given to bob who measures this system with @xmath71 basis to know whether alice s and bob s bases match or not . if his measurement outcome is @xmath72 ( @xmath73 ) , then he uses the same ( the other ) basis to be used by alice ( note that no classical communication is needed in order for bob to know alice s basis since alice and bob get together ) . the third system , which is denoted by @xmath22 and we call `` quantum coin '' , is possessed and to be measured by alice along @xmath74 basis to determine her basis choice , and the measurement outcome will be sent to eve after eve broadcasts the measurement outcome at mu . moreover , all the second systems of @xmath75 , @xmath76 , @xmath24 , and @xmath77 are sent to eve . note in this formalism that the information , including classical information and quantum information , available to eve is the same as those in the actual protocol , and the generated key is also the same as the one of the actual protocol since the statistics of alice s and bob s raw data is exactly the same as the one of the actual protocol . thus , we are allowed to work on this virtual protocol for the security proof . the first system given to eve in eq . ( [ coin - joint - state ] ) allows her to know which coherent pulses contain data in the sifted key and she can post - select only the relevant pulses . thus , without the loss of any generalities of the security proof , we can concentrate only on the post - selected version of the state in eq . ( [ coin - joint - state ] ) as @xmath78 the most important quantity in the proof is the phase error rate in the code bits . the definition of the phase error rate is the rate of bit errors along @xmath7-basis in the sifted key if they had chosen @xmath7-basis as the measurement basis when both of them have sent pulses in @xmath6-basis . if alice and bob have a good estimation of this rate as well as the bit error rate in the sifted key ( the bit error rate in @xmath6-basis given alice and bob have chosen @xmath6-basis for the state preparation ) , they can perform hashing in @xmath7-basis and @xmath6-basis simultaneously @xcite to distill pairs of qubits in the state whose fidelity with respect to the product state of the maximally entangled state @xmath32 is close to @xmath52 . according to the discussion on the universal composability @xcite , the key distilled via @xmath6-basis measurement on such a state is composably secure and moreover exactly the same key can be generated only by classical means , i.e. , error correction and privacy amplification @xcite . thus , we are left only with the phase error estimation . for the simplicity of the discussion , we assume the large number of successful events @xmath79 so that we neglect all the statistical fluctuations and we are allowed to work on a probability rather than the relative frequency . the quantity we have to estimate is the bit error along @xmath7-basis , denoted by @xmath80 , given alice and bob send @xmath81 state , which is different from the experimentally available bit error rate along @xmath7-basis given alice and bob send @xmath82 state . intuitively , if the basis - dependent flaw is very small , @xmath80 and @xmath83 should be very close since the states are almost indistinguishable . to make this intuition rigorous , we briefly review the idea by @xcite which applies bloch sphere bound @xcite to the quantum coin . suppose that we randomly choose @xmath17-basis or @xmath6-basis as the measurement basis for each quantum coin . let @xmath84 and @xmath85 be fraction that those quantum coins result in @xmath52 in @xmath17-basis and @xmath6-basis measurement , respectively . what bloch sphere bound , i.e. , eq . ( 13 ) or eq . ( 14 ) in @xcite or eq . ( a1 ) in @xcite , tells us in our case is that no matter how the correlations among the quantum coins are and no matter what the state for the quantum coins is , thanks to the randomly chosen bases , the following inequality holds with probability exponentially close to @xmath52 in @xmath79 , @xmath86 by applying this bound separately to the quantum coins that are conditional on having phase errors and to those that are conditional on having no phase error , and furthermore by combining those inequalities using bayes s rule , we have @xmath87 here , @xmath88 is equivalent to the probability that the measurement outcome of the quantum coin along @xmath6-basis is @xmath14 given the successful event in mu . note that this probability can be enhanced by eve who chooses carefully the pulses , and eve could attribute all the loss events to the quantum coins being in the state @xmath13 . thus , we have an upper bound of @xmath88 in the worst case scenario as @xmath89 and @xmath90 where @xmath91 is the frequency of the successful event . note that we have not used the explicit form of @xmath92 and @xmath93 , where @xmath94 , in the derivation of eqs . ( [ phase error bound ] ) , ( [ fdelta ] ) , and ( [ delta ] ) , and the important point is that the state @xmath92 and @xmath93 are the purification of alice s and bob s density matrices for both bases . since there always exists purification states of @xmath95 and @xmath96 , which are respectively denoted by @xmath97 and @xmath98 , such that @xmath99 , @xmath100 can be rewritten by @xmath101/2\ , , \label{fiddelta}\end{aligned}\ ] ] where @xmath102 represents alice s density matrix of @xmath6 basis and all the other density matrices are defined by the same manner . our expression of @xmath100 has the product of two fidelities , while the standard bb84 with basis - dependent flaw in @xcite has only one fidelity ( the fidelity between alice s density matrices in @xmath6 and @xmath7 bases ) . the two products may lead to poor performance of our schemes compared to that of standard qkd in terms of the achievable distances , however our schemes have the huge advantage over the standard qkd that there is no side - channel in the detectors . finally , the key generation rate @xmath103 , given @xmath6-basis , in the asymptotic limit of large @xmath79 is given by @xmath104 where @xmath105 is the bit error rate in @xmath6-basis , @xmath106 is the inefficiency of the error correcting code , and @xmath107 . we can trivially obtain the key generation rate for @xmath7-basis just by interchanging @xmath6-basis in all the discussions above to @xmath7-basis . we remark in our security proof that we have assumed nothing about what kind of measurement mu conducts but that it announces whether it detects the successful event and the type of the event ( this announcement allows us to calculate @xmath108 and the error rates ) . thus , mu can be assumed to be totally under the control of eve . in the following subsections , we show some examples of the key generation rate of each of our schemes assuming typical experimental parameters taken from gobby - yuan - shields ( gys ) experiment @xcite unless otherwise stated . moreover , we assume that the imperfect phase modulation is the main source of the decay of the fidelity between the density matrices in two bases , and we evaluate the effect of this imperfection on the key generation rate . in the phase encoding scheme i , the important quantity for the security @xmath100 can be expressed as @xmath109\ , . \label{delta11}\end{aligned}\ ] ] note that this quantity is dependent on the intensity of alice s and bob s sources . as we have mentioned in sec . iii , this quantity may be estimated relatively easily via tomography involving homodyne measurement . ) of @xmath110 and @xmath111 . dashed line : ( a ) mu is at bob s side , i.e. , @xmath112 . solid line : ( b ) mu is just in the middle between alice and bob . the lines achieving the longer distances correspond to @xmath110 of @xmath113 . see also the main text for the explanation . [ fig : key33 ] ] ) that outputs fig . [ fig : key33 ] as a function of the distance between alice and bob . [ fig : intensity33 ] ] to simulate the resulting key generation rate , we assume that the bit error stems from the dark counting as well as alignment errors due to imperfect phase locking or imperfect optical components . the alignment error is assumed to be proportional to the probability of having a correct click caused only by the optical detection not by the dark counting . moreover , we make assumptions that all the detectors have the same characteristics for the simplicity of the analysis , and alice and bob choose the intensities of the signal lights in such a way that the intensities of the incoming pulses to mu are the same . finally , we assume the quantum inefficiency of the detectors to be part of the losses in the quantum channels . with all the assumptions , we may express the resulting experimental parameters as @xmath114(1-p_{\rm dark})\nonumber\\ & + & ( 1-p_{\rm dark})e^{-2\alpha_{\rm in}}p_{\rm dark}\nonumber\\ \gamma_{\rm suc}&=&\gamma_{\rm suc}^{(x)}+\gamma_{\rm suc}^{(y)}\nonumber\\ \delta_x&=&\delta_y=\big[e_{\rm ali}(1-p_{\rm dark})^2(1-e^{-2\alpha_{\rm in}})\nonumber\\ & + & ( 1-p_{\rm dark})e^{-2\alpha_{\rm in}}p_{\rm dark}\big]/\gamma_{\rm suc}^{(x)}\nonumber\\ \alpha_{\rm in}&\equiv&\alpha_{a}\eta_{a}=\alpha_{b}\eta_{b}\nonumber\\ \eta_{a}&=&\eta_{{\rm det } , a}10^{-\xi_{a } l_{a}/10}\nonumber\\ \eta_{b}&=&\eta_{{\rm det } , b}10^{-\xi_{b } l_{b}/10}\ , . \label{ex data i}\end{aligned}\ ] ] here , @xmath115 is the dark count rate of the detector , @xmath113 is the alignment error rate , @xmath116 is alice s ( bob s ) overall transmission rate , @xmath117 ( @xmath118 ) is the quantum efficiency of alice s ( bob s ) detector , @xmath119 is alice s ( bob s ) channel transmission rate , and @xmath120 ( @xmath121 ) is the distance between alice ( bob ) and mu . the first term and the second term in @xmath122 or @xmath123 respectively represent the alignment error , which is assumed to be proportional to the probability of having correct bit value due to the detection of the light , and errors due to dark counting ( one detector clicks due to the dark counting while the other one does not ) . we take the following parameters from gys experiment @xcite : @xmath124 , @xmath125 , @xmath126 ( db / km ) , @xmath127 , and @xmath128 , and we simulate the key generation rate as a function of the distance between alice and bob in fig . [ fig : key33 ] . in the figure , we consider two settings : ( a ) mu is at bob s side , i.e. , @xmath112 ( b ) mu is just in the middle between alice and bob . the reason why we consider these setting is that the basis - dependent flaw is dependent on intensities that alice and bob employ , and it is not trivial where we should place mu for the better performance . since mdiqkd polarization encoding scheme without basis - dependent flaw achieves almost twice the distance of bb84 @xcite , we may expect that the setting ( b ) could achieve almost twice the distance of bb84 without phase randomization that achieves about 13 ( km ) @xcite with the same experimental parameters . the simulation result , however , does not follow this intuition since we have the basis - dependent flaw not only from alice s side but also from bob s side . thus , the advantage that we obtain from putting mu between alice and bob is overwhelmed by the double basis - dependent flaw . in each setting , we have optimized the intensity of the coherent pulses @xmath12 for each distance ( see fig . [ fig : intensity33 ] ) . in order to explain why the optimal @xmath129 is so small , note that scheme i intrinsically suffers from the basis - dependent flaw due to eq . ( [ delta11 ] ) . this means that if we use relatively large @xmath129 , then we can not generate the key due to the flaw . actually , when we set @xmath130 , which is a typical order of the amplitude for decoy bb84 , one can see that the upper bound of the phase error rate is @xmath131 even in the zero distance , i.e. , @xmath132 , and we have no chance to generate the key with this amplitude . thus , alice and bob have to reduce the intensities in order to suppress the basis - dependent flaw . also , as the distance gets larger and the losses get increased , alice and bob have to use weaker pulses since larger losses can be exploited by eve to enhance the basis - dependent flaw according to eq . ( [ fdelta ] ) , and they can reduce the intensities until it reaches the cut - off value where the detection of the weak pulses is overwhelmed by the dark counts . in the above simulation , we have assumed that alice and bob can prepare states very accurately , however in reality , they can only prepare approximate states due to the imperfection of the sources . this imperfection gives more basis - dependent flaw , and in order to estimate the effect of this imperfection , we assume that the fidelity between the two actually prepared density matrices in two bases is approximated by the fidelity between the following density matrices ( see appendix b for the detail ) @xmath133 and @xmath134 where we assume an imperfect phase modulator whose degree of the phase modulation error is proportional to the target phase modulation value , and @xmath135 represents the imperfection of the phase modulation that is related with the extinction ratio @xmath136 as @xmath137 in this equation , we assume that the non - zero extinction ratio is only due to the imperfection of the phase modulators . since imperfect phase modulation results in the same effect as the alignment errors , i.e. , the pulses are routed to a wrong output port , we assume that the alignment error rate is increased with this imperfection . thus , in the simulation accommodating the imperfection of the phase modulation , we replace @xmath113 with @xmath138 . here , we have used a pessimistic assumption that the effect of the phase modulation becomes @xmath139-times higher than before since each of alice and bob has one phase modulator and mu has two phase modulators for the phase shift of two polarization modes ( note from eq . ( [ mimperfect phase modulator ] ) that @xmath136 is approximately proportional to @xmath140 , thus 4 times degradation in terms of the accuracy of the phase modulation results in @xmath139-times degradation in terms of the extinction ratio ) . we also remark that in practice , it is more likely that the phase encoding errors are independent , in which case a factor of 4 will suffice and the key rate will actually be higher than what is presented in our paper . on the other hand , we have to use the following @xmath100 when we consider the security : @xmath141/2\,.\end{aligned}\ ] ] in figs . [ fig : keyimpferfectpmi ] and [ fig : intensityimpferfectpmi ] , we plot the key generation rate and the corresponding optimal alice s mean photon numbers ( @xmath12 ) as a function of the distance between alice and bob . in the figures , we define @xmath142 that satisfies @xmath143 as @xmath144 , where @xmath145 is the typical order of @xmath136 in some experiments @xcite . we have confirmed that we can not generate the key when @xmath145 . however , we can see in the figures that if the accuracy of the phase modulation is increased three times or five times , i.e. , @xmath146 and @xmath147 , then we can generate the key . like the case in fig . [ fig : intensity33 ] , the small optimal mean photon number can be intuitively understood by the arguments that we have already made in this section . in order to investigate the feasibility of the phase encoding scheme i with the current technologies , we replace @xmath125 , @xmath127 , and @xmath128 with @xmath148 , @xmath149 @xcite , and @xmath150 @xcite . we see in fig . [ fig : keyimpferfectpminew ] that the key generation is possible over much longer distances with those parameters assuming the precise control of the intensities of the laser source . we also show the corresponding optimal mean photon number @xmath129 in fig . [ fig : intensityimpferfectpminew ] . we note that thanks to the higher quantum efficiency , the success probability becomes higher , following that alice and bob can use larger mean photon number @xmath129 compared to those in figs . [ fig : intensityimpferfectpmi ] and [ fig : intensityimpferfectpminew ] . ) of @xmath110 and imperfect phase modulators . @xmath151 represents the typical amount of the phase modulation error , and we plot the key rate for smaller imperfection of @xmath152 and @xmath153 . dashed line : mu is at bob s side , i.e. , @xmath112 . solid line : mu is just in the middle between alice and bob.[fig : keyimpferfectpmi ] ] ) that outputs fig . [ fig : keyimpferfectpmi ] as a function of the distance between alice and bob . [ fig : intensityimpferfectpmi ] ] with @xmath148 , @xmath149 @xcite , and @xmath151 . dashed line : mu is at bob s side , i.e. , @xmath112 . solid line : mu is just in the middle between alice and bob.[fig : keyimpferfectpminew ] ] ) that outputs fig . [ fig : keyimpferfectpminew ] as a function of the distance between alice and bob . [ fig : intensityimpferfectpminew ] ] in the phase encoding scheme ii , note that we can generate the key only from the successful detection event in mu given both of alice and bob send out a single - photon since if either or both of alice and bob emit more than one photon , then eve can employ the so - called photon number splitting attack @xcite . thus , the important quantities to estimate are @xmath154 , @xmath155 , @xmath105 , @xmath156 , which respectively represents gain in @xmath6-basis given both of alice and bob emit a single - photon , the phase error rate given alice and bob emit a single - photon , overall bit error rate in @xmath6-basis , and overall gain in @xmath6-basis . to estimate these quantities stemming from the simultaneous single - photon emission , we assume the use of infinite number of decoy states for the simplicity of analysis @xcite . another important quantity in our study is the fidelity @xmath157 ( @xmath158 ) between alice s ( bob s ) @xmath6-basis and @xmath7-basis density matrices of only single - photon component , _ not _ whole optical modes . if this fidelity is given , then we have @xmath159 for the simplicity of the discussion , we consider the case of @xmath160 in our simulation . the estimation of the fidelity only in the single - photon part is very important , however to the best of our knowledge we do not know any experiment directly measuring this quantity . this measurement may require photon number resolving detectors and very accurate interferometers . thus , we again assume that the degradation of the fidelity is only due to the imperfect phase modulation given by eq . ( [ mimperfect phase modulator ] ) , and we presume that the fidelity of the two density matrices between the two bases is approximated by the fidelity between the following density matrices ( see appendix b for the detail ) @xmath161\nonumber\\ \rho_{y}^{(1)}&=&\frac{1}{2}\big[{\hat p}\left(\frac{{\left\| 0_z \right\rangle}+i e^{i\|\delta\|/2}{\left\| 1_z \right\rangle}}{\sqrt{2}}\right)\nonumber\\ & + & { \hat p}\left(\frac{{\left\| 0_z \right\rangle}-i e^{-i\|\delta\|/2}{\left\| 1_z \right\rangle}}{\sqrt{2}}\right)\big]\,.\end{aligned}\ ] ] with these parameters , we can express the key generation rate given alice and bob use @xmath6-basis as @xcite @xmath162-f(\delta_{x})q_{x}h(\delta_{x})\,,\end{aligned}\ ] ] where @xmath163 is the @xmath164 version of @xmath80 in eq . ( [ key rate ] ) . to simulate the resulting key generation rate , the bit errors are assumed to stem from multi - photon component , the dark counting , and the misalignment that is assumed to be proportional to the probability of obtaining the correct bit values only due to the detection by optical pulses . like before , we also assume that all the detectors have the same characteristics , alice and bob choose the intensities of the signal lights in such a way that the intensities of the incoming pulses to mu are the same , and all the quantum inefficiencies of the detectors can be attributed to part of the losses in the quantum channel . finally , alice s and bob s coherent light sources are assumed to be phase randomized , and the imperfect phase modulation is represented by the increase of the alignment error rate . with these assumptions , we may have the following resulting experimental parameters @xmath165\nonumber\\ & + & w^{(2,1)}+w^{(2,0)}\nonumber\\ \delta_{x}^{(1,1)}&=&\big\{4\alpha_{a}\alpha_{b}\eta_{a}\eta_{b}e^{-2(\alpha_{a}+\alpha_{b})}p_{\rm dark}(1-p_{\rm dark})^2/2\nonumber\\ & + & 2(e_{\rm ali}+4\eta_{\rm ex})\alpha_{a}\alpha_{b}\eta_{a}\eta_{b}e^{-2(\alpha_{a}+\alpha_{b})}(1-p_{\rm dark})^2\nonumber\\ & + & ( w^{(2,1)}+w^{(2,0)})/2\big\}/q^{(1,1)}_{x}\nonumber\\ q^{(1,1)}_{y}&=&q^{(1,1)}_{x}\nonumber\\ \delta^{(1,1)}_{y}&=&\delta^{(1,1)}_{x}\nonumber\\ w^{(2,1)}&\equiv&8\alpha_{a}\alpha_{b}e^{-2(\alpha_{a}+\alpha_{b})}\big[\eta_{a}(1-\eta_{b})+(1-\eta_{a})\eta_{b}\big]\nonumber\\ & \times&p_{\rm dark}(1-p_{\rm dark})^2\nonumber\\ w^{(2,0)}&\equiv&16\alpha_{a}\alpha_{b}(1-\eta_{a})(1-\eta_{b})e^{-2(\alpha_{a}+\alpha_{b})}\nonumber\\ & \times&p_{\rm dark}^2(1-p_{\rm dark})^2\nonumber\\ q_{x}&=&2\left[1-(1-p_{\rm dark})e^{-\alpha_{\rm in}}\right]^2(1-p_{\rm dark})^2e^{-2\alpha_{\rm in}}+v \nonumber\\ \delta_{x}&=&v+(e_{\rm ali}+4\eta_{\rm ex})2\left(1-e^{-\alpha_{\rm in}}\right)^2 \nonumber\\ & \times&(1-p_{\rm dark})^2e^{-2\alpha_{\rm in}}\nonumber\\ v&\equiv&\frac{p_{\rm dark}(1-p_{\rm dark})}{2\pi}\nonumber\\ & \times&\int_{0}^{2\pi}d\theta\big[1-(1-p_{\rm dark})e^{-\alpha_{\rm in}\|1+e^{i\theta}\|^2}\big ] \nonumber\\ & \times&\big[(1-p_{\rm dark})e^{-\alpha_{\rm in}\|1-e^{i\theta}\|^2}\big]\nonumber\\ & + & \frac{p_{\rm dark}(1-p_{\rm dark})}{2\pi}\nonumber\\ & \times&\int_{0}^{2\pi}d\theta\big[1-(1-p_{\rm dark})e^{-\alpha_{\rm in}\|1-e^{i\theta}\|^2}\big ] \nonumber\\ & \times&\big[(1-p_{\rm dark})e^{-\alpha_{\rm in}\|1+e^{i\theta}\|^2}\big]\nonumber\\ \alpha_{\rm in}&\equiv&\alpha_{a}\eta_{a}=\alpha_{b}\eta_{b}\nonumber\\ \eta_{a}&=&\eta_{{\rm det } , a}10^{-\xi_{a } l_{a}/10}/2\nonumber\\ \eta_{b}&=&\eta_{{\rm det } , b}10^{-\xi_{b } l_{b}/10}/2\ , \label{exp - data - ii}\end{aligned}\ ] ] note that @xmath12 ( @xmath166 ) represents each of the intensity of alice s ( bob s ) signal light and the reference light , _ not _ the total intensity of them , and @xmath167 and @xmath168 are divided by @xmath169 since the conversion efficiency of our converter is @xmath66 . @xmath170 in @xmath171 again comes from the pessimistic assumption that each of alice s and bob s phase modulator is imperfect , and @xmath172 ( @xmath173 ) represents the probability of the event where both of alice and bob emit a single - photon and only one ( zero ) photon is detected but the successful detection event is obtained due to the dark counting . on the other hand , the quantity that quantifies the basis - dependent flaw @xmath88 in the present case is upper bounded by @xmath174\nonumber\\ q^{(1,1)}&\equiv&(q^{(1,1)}_{x}+q^{(1,1)}_{y})/2\end{aligned}\ ] ] where @xmath175 is the probability that mu receives a single - photon both from alice and bob simultaneously conditioned on that each of alice and bob sends out a single - photon . we remark that @xmath100 in this scheme is only dependent on the accuracy of the phase modulation . this is different from scheme i where the manipulation of the intensities of the pulses can affect the basis - dependent flaw . in the simulation , we again assume gys experimental parameters and we consider two settings : ( a ) mu is at bob s side and ( b ) mu is just in the middle between alice and bob . note that @xmath100 is independent of @xmath12 and @xmath166 in the phase encoding scheme ii case . in fig . [ f1 ] , we plot the key generation rates of ( a ) and ( b ) for @xmath176 , @xmath177 , @xmath178 , @xmath179 ( recall from eq . ( [ mimperfect phase modulator ] ) that @xmath180 that corresponds to the typical extinction ratio of @xmath181 ) , which respectively correspond to @xmath182 , @xmath183 , @xmath184 , and @xmath185 , and the achievable distances of ( a ) and ( b ) increase with the improvement of the accuracy , i.e. , with the decrease of @xmath135 . we have confirmed that no key can be distilled in ( a ) and ( b ) when @xmath186 . the figure shows that the achievable distance drops significantly with the degradation of the accuracy of the phase modulator , and the main reason of this fast decay is that @xmath88 is approximated by @xmath187 and this dominator decreases exponentially with the increase of the distance . we also plot the corresponding optimal @xmath129 in fig . notice that the mean photon number increases in some regime in some cases of ( a ) , and recall that this increase does not change @xmath100 . if we increased the intensity in scheme i with the distance , then we would have more basis - dependent flaw , resulting in shortening of the achievable distance . this may be an intuitive reason why we see no such increase in figs . [ fig : intensity33 ] , [ fig : intensityimpferfectpmi ] , and 9 . like in the phase encoding scheme i , we investigate the feasibility of the phase encoding scheme ii with the current technologies by replacing @xmath125 , @xmath127 , and @xmath128 with @xmath148 , @xmath149 @xcite , and @xmath150 @xcite . with this upgrade , we have confirmed the impossibility of the key generation , however if we double the quantum efficiency of the detector or equivalently , if we assume the polarization encoding so that the factor of @xmath131 , which is introduced by the phase - to - polarization converter , is removed both from @xmath167 and @xmath168 in eq . ( [ exp - data - ii ] ) , then we can generate the key , which is shown in fig . [ f1new ] ( also see fig . finally , we note that our simulation is essentially the same as the polarization coding since the fact that we use phase encoding is only reflected by the dominator of 2 in @xmath188 and @xmath189 in eq . ( [ exp - data - ii ] ) . thus , the behavior of the key generation rate against the degradation of the state preparation is the same also in polarization based mdiqkd . also note that even in the standard bb84 , @xmath88 decays exponentially with increasing distance . thus , we conclude that very precise state preparation is very crucial in the security of not only mdiqkd but also in standard qkd . we also note that our estimation of the fidelity might be too pessimistic since we have assumed that the degradation of the extinction ratio is only due to imperfect phase modulation . in reality , the imperfection of mach - zehnder interferometer and other imperfections should contribute to the degradation , and the fidelity should be closer to @xmath52 than the one based on our model . . solid line : ( b ) mu is just in the middle between alice and bob . we plot the key generation rates of each case when @xmath176 , @xmath177 , @xmath178 , @xmath179 where @xmath135 is proportional to the amount of the phase modulation error , and for each case of ( a ) and ( b ) the key generation rates monotonously increase with the decrease of @xmath135 . , i.e. , with the improvement of the phase modulation . the key rates of ( a ) and ( b ) when @xmath190 are almost superposed . see also the main text for the explanation . [ f1 ] ] ) that outputs fig . the bold lines correspond to ( a ) . see also the main text for the explanation . [ f2 ] ] , @xmath191 , and @xmath151 . note that we double @xmath192 compared to the one of @xcite , or we effectively consider the polarization encoding @xcite . dashed line : ( a ) mu is at bob s side , i.e. , @xmath112 . solid line : ( b ) mu is just in the middle between alice and bob . the key rates are almost superposed . see also the main text for the explanation . [ f1new ] ] ) that outputs fig . [ f1new ] . the bold lines correspond to ( a ) . see also the main text for the explanation . [ f2new ] ] in summary , we have proposed two phase encoding mdiqkd schemes . the first scheme is based on the phase locking technique and the other one is based on the conversion of the pulses in the standard phase encoding bb84 to polarization modes . we proved the security of the first scheme , which intrinsically possesses basis - dependent flaw , as well as the second scheme with the assumption of the basis - dependent flaw in the single - photon part of the pulses . based on the security proof , we also evaluate the effect of imperfect state preparation , and especially we focus our attention to the imperfect phase modulation . while the first scheme can cover relatively short distances of the key generation , this scheme has an advantage that the basis - dependent flaw can be controlled by the intensities of the pulses . thanks to this property , we have confirmed based on a simple model that 3 or 5 times of the improvement in the accuracy of the phase modulation is enough to generate the key . moreover , we have confirmed that the key generation is possible even without these improvements if we implement this scheme by using the up - to - date technologies and the control of intensities of the laser source is precise . on the other hand , it is not so clear to us how accurate we can lock the phase of two spatially separated laser sources , which is important for the performance of scheme i. our result still implies that scheme i can tolerate up to some extent of the imperfect phase locking errors , which should be basically the same as the misalignment errors , but further analysis of the accuracy from the experimental viewpoint is necessary . we leave this problem for the future studies . -basis when @xmath176 , @xmath193 , @xmath194 where @xmath135 is the amount of the phase modulation error . [ sf1 ] ] ) that outputs fig . [ sf2 ] ] the second scheme can cover much longer distances when the fidelity of the _ single - photon components _ of @xmath7-basis and @xmath6-basis density matrices is perfect or extremely close to perfect . when we consider the slight degradations of the fidelity , however , we found that the achievable distances drop significantly . this suggests that we need a photon source with a very high fidelity , and very accurate estimation of the fidelity of the single - photon subspace is also indispensable . in our estimation of the imperfection of the phase modulation , we simply assume that the degradation of the extinction ratio is only due to imperfect phase modulation , which might be too pessimistic , and the imperfection of mach - zehnder interferometer and other imperfections contribute to the degradation . thus , the actual fidelity between the density matrices of the single - photon part in two bases might be very close to 1 , which should be experimentally confirmed for the secure communication . we note that the use of the passive device to prepare the state @xcite may be a promising way for the very accurate state preparation . we remark that the accurate preparation of the state is very important not only in mdiqkd but also in standard qkd where eve can enhance the imbalance of the quantum coin exponentially with the increase of the distance . to see this point , we respectively plot in fig . [ sf1 ] and fig . [ sf2 ] the key generation rate of standard bb84 with infinite decoy states in @xmath6-basis and its optimal mean photon number assuming @xmath148 , @xmath195 @xcite , @xmath150 , @xmath124 , and @xmath126 . again , @xmath196 is the typical value of the phase modulation error , and we see in the figure that the degradation of the phase modulator in terms of the accuracy significantly decreases the achievable distance of secure key generation . one notices that standard decoy bb84 is more robust against the degradation since the probability that the measurement outcome of the quantum coin along @xmath6-basis is @xmath14 given the successful detection of the signal by bob is written as @xmath197 rather than @xmath198 . on the other hand , one has to remember that we trust the operation of bob s detectors in this simulation , which may not hold in practice . finally , we neglect the effect of the fluctuation of the intensity and the center frequency of the laser light in our study , which we will analyze in the future works . in summary , our work highlights the importance of very accurate preparation of the states to avoid basis - dependent flaws . we thank x. ma , m. curty , k. azuma , t. yamamoto , r. namiki , t. honjo , h. takesue , y. tokunaga , and especially g. kato for enlightening discussions . part of this research was conducted when k. t and c - h . f. f visited the university of toronto , and they express their sincere gratitude for all the supports and hospitalities that they received during their visit . this research is in part supported by the project `` secure photonic network technology '' as part of `` the project uqcc '' by the national institute of information and communications technology ( nict ) of japan , in part by the japan society for the promotion of science ( jsps ) through its funding program for world - leading innovative r@xmath199d on science and technology ( first program ) " , in part by rgc grant no . 700709p of the hksar government , and also in part by nserc , canada research chair program , canadian institute for advanced research ( cifar ) and quantumworks . in this appendix , we give a detailed calculation about how scheme i works when there is no channel losses and noises . in order to calculate the joint probability that alice and bob obtain type-0 successful event , where only the detector d0 clicks , and they share the maximally entangled state @xmath32 , we introduce a projector @xmath200 that corresponds to type-0 successful event . here , @xmath201 represents the non - vacuum state . the state after alice and bob have the type-0 successful event @xmath202 ( see eq . ( [ normal schi ] ) for the definition of @xmath26 ) can be expressed by @xmath203 here , @xmath204 is an identity operator on @xmath205 and @xmath206 , @xmath207 and @xmath208 are complex numbers , and @xmath209 and @xmath210 are orthonormal bases , which are related with each other through @xmath211 by a direct calculation , one can show that latexmath:[\ ] ] here , @xmath240 , @xmath241 , and @xmath242 , and @xmath243 is defined by @xmath244 where @xmath245 is a purification of @xmath246 , which is the state that alice actually prepares for the bit value @xmath247 in basis @xmath248 , and @xmath249 is alice s qubit system . one can choose any purification for @xmath245 , and in particular it should be chosen in such a way that it maximizes the inner product in eq . ( c2 ) or ( c3 ) . one can similarly define @xmath250 , and @xmath46 is introduced via considering a joint state involving the quantum coin as @xmath251 due to this change , the figures for the key generation rate have to be revised . as the examples of revised figures , we show the revised version of figs . 8 , 9 , 12 , and 13 , which are the most important figures for our main conclusions to hold . notice that there are only minor changes in figs . 8 and 9 and the changes in figs . 12 and 13 are relatively big . however , the big changes do not affect the validity of the main conclusions in our paper , which is the importance of the state preparation in mdiqkd and the fact that our schemes can generate the key with the practical channel mode that we have assumed . for the derivation of eq . ( [ 0 ] ) , we invoke koashi s proof @xcite . to apply koashi s proof , it is important to ensure that i ) one of the two parties holds a virtual * qubit * ( rather than a higher dimensional system ) and ii ) the fictitious measurements performed on the virtual qubit have to form * conjugate * observables . therefore , it is not valid to consider fidelity alone ( which allows arbitrary purifications that may not satisfy the conjugate observables requirement ) . fortunately , it turns out to be easy to modify our equation to satisfy the above two requirements . since the difference between eq . ( c2 ) and eq . ( c3 ) comes from whether we consider alice s virtual qubit or bob s virtual qubit , we focus only on eq . ( c2 ) and the same argument holds for eq . ( c3 ) . in koashi s proof , the security is guaranteed via two alternative tasks , ( i ) agreement on x ( key distillation basis ) and ( ii ) alice s or bob s preparation of an eigenstate of y , the conjugate basis of x , with use of an extra communication channel . the problem with the original ( i.e. uncorrected ) version of eq . ( 9 ) is the following . if we use the uncorrected version of eq . ( 9 ) in our paper , then the use of the fidelity means that the real part in eq . ( c2 ) is equivalent to @xmath252 with the maximization over _ all possible _ local unitary operators @xmath253 . in this case , if alice performs a measurement along x basis , then it violates the correspondence between her sending state @xmath246 and her qubit state @xmath254 in general , and thus , in the uncorrected version of eq . ( 9 ) in our paper , the argument based on the fidelity does not guarantee the security of the protocol . in contrast , with the corrected version of eq . ( 9 ) in our paper , since the maximization over @xmath46 and @xmath255 in eq . ( c2 ) preserves the relationship between alice s sending state and her qubit state as well as the conjugate relationship between x and y , we can apply koashi s proof for the security argument of the protocol . b. qi , c .- h . f. fung , h .- k . lo and x. ma , quant . 73 - 82 ( 2007 ) , y. zhao , c .- h . f. fung , b. qi , c. chen and h .- k . lo , phys . a 78 , 042333 ( 2008 ) , c .- h . f. fung , b. qi , k. tamaki and h .- k . lo , phys . a 75 , 032314 ( 2007 ) , f. xu , b. qi and h .- k . lo , new j. phys . 12 , 113026 ( 2010 ) . l. lydersen , c. wiechers , c. wittmann , d. elser , j. skaar and v. makarov , nature photonics 4 , pp . 686 - 689 ( 2010 ) , z. l. yuan , j. f. dynes and a. j. shields , nature photonics 4 , pp . 800 - 801 ( 2010 ) , l. lydersen , c. wiechers , c. wittmann , d. elser , j. skaar and v. makarov , nature photonics 4 , 801 ( 2010 ) , i. gerhardt , q. liu , a. lamas - linares , j. skaar , c. kurtsiefer and v. makarov , nature comm . 2 , 349 ( 2011 ) , l. lydersen , m. k. akhlaghi , a. h. majedi , j. skaar and v. makarov , arxiv : 1106.2396 . f. fung , k. tamaki , b. qi , h .- k . lo and x. ma , quant . inf . comp . 9 , 131 ( 2009 ) , l. lydersen , j. skaar , quant . inf . comp . * 10 * , 0060 ( 2010 ) , . mary , l. lydersen , j. skaar , phys . a * 82 * , 032337 ( 2010 ) . d. mayers and a. c .- c . yao , in proceedings of the 39th annual symposium on foundations of computer science ( focs98 ) , ( ieee computer society , washington , dc , 1998 ) , p. 503 , a. acin , n. brunner , n. gisin , s. massar , s. pironio , and v. scarani , phys . lett . * 98 * , 230501 ( 2007 ) . the definition of the four bell state is as follows . @xmath257=\frac{1}{\sqrt{2}}[{\left\| 0_x \right\rangle}_{a1}{\left\| 0_x \right\rangle}_{b1}-{\left\| 1_x \right\rangle}_{a1}{\left\| 1_x \right\rangle}_{b1}]$ ] , @xmath258=\frac{1}{\sqrt{2}}[{\left\| 0_x \right\rangle}_{a1}{\left\| 1_x \right\rangle}_{b1}+{\left\| 1_x \right\rangle}_{a1}{\left\| 0_x \right\rangle}_{b1}]$ ] , @xmath259=\frac{1}{\sqrt{2}}[{\left\| 0_x \right\rangle}_{a1}{\left\| 0_x \right\rangle}_{b1}+{\left\| 1_x \right\rangle}_{a1}{\left\| 1_x \right\rangle}_{b1}]$ ] , and @xmath260 . one of the most simplest proofs is shor - preskill s proof @xcite . the intuition of this proof is as follows . note that if alice and bob share some pairs of @xmath32 , ( i.e. , alice has one half of each pair and bob has the other half ) , then they can generate a secure key by performing @xmath6-basis measurement . the reason of the security is that this state is a pure state , which means that this state has no correlations with the third system including eve s system . due to the intervention by eve , alice and bob do not share this pure state in general , but instead they share noisy pairs . the basic idea of the proof is to consider the distillation of @xmath32 from the noisy pairs . for the distillation , note that @xmath32 is only one qubit pair state that has no bit errors in @xmath6-basis ( we call this error as the bit error ) and has no bit errors in @xmath7-basis ( we call this error as the phase error ) . it is known that if alice and bob employ the so - called css code ( calderbank - shor - steane code ) @xcite , then the noisy pairs are projected to a classical mixture of the four bell states , i.e. , @xmath32 , @xmath37 ( @xmath32 with the phase error ) , @xmath33 ( @xmath32 with the bit error ) , and @xmath38 ( @xmath32 with both the phase and bit errors ) . moreover , if alice and bob choose a correct css code , which can be achieved by random sampling procedure , then css code can detect the position of the erroneous pair with high probability . thus , by performing bit and phase flip operation depending on the detected error positions , alice and bob can distill some qubit pairs that are very close in fidelity to the product state of @xmath32 . in general , implementation of the above scheme requires a quantum computer . fortunately , shor - preskill showed that the bit error detection and bit flip operation can be done classically , and the phase error detection and phase flip operation need not be done , but exactly the same key can be obtained by the privacy amplification , so that we do not need to possess a quantum computer for the key distillation . r. renner , and r. koenig , proc . of tcc 2005 , lncs , springer , vol . * 3378 * ( 2005 ) , m. ben - or , and dominic mayers , arxiv : quant - ph/0409062 , m. ben - or , michal horodecki , d. w. leung , d. mayers , j. oppenheim , theory of cryptography : second theory of cryptography conference , tcc 2005 , j.kilian ( ed . ) springer verlag 2005 , vol . * 3378 * of lecture notes in computer science , pp . 386 - 406 .	in this paper , we study the unconditional security of the so - called measurement device independent quantum key distribution ( mdiqkd ) with the basis - dependent flaw in the context of phase encoding schemes . we propose two schemes for the phase encoding , the first one employs a phase locking technique with the use of non - phase - randomized coherent pulses , and the second one uses conversion of standard bb84 phase encoding pulses into polarization modes . we prove the unconditional security of these schemes and we also simulate the key generation rate based on simple device models that accommodate imperfections . our simulation results show the feasibility of these schemes with current technologies and highlight the importance of the state preparation with good fidelity between the density matrices in the two bases . since the basis - dependent flaw is a problem not only for mdiqkd but also for standard qkd , our work highlights the importance of an accurate signal source in practical qkd systems . + * note : we include the erratum of this paper in appendix c. the correction does not affect the validity of the main conclusions reported in the paper , which is the importance of the state preparation in mdiqkd and the fact that our schemes can generate the key with the practical channel mode that we have assumed . *	20,399	308
in this letter , we will focus on holographic qcd from d4 branes . we will discuss some problems with the usual correspondence between the confinement / deconfinement transition in qcd and the scherk - schwarz transition between a solitonic d4 brane and a black d4 brane in the gravity dual . some of these problems were first discussed in @xcite . we will specifically show that the black d4 brane can not be identified with the ( strong coupling continuation of the ) deconfinement phase in qcd in four dimensions . as a resolution of these problems , we will propose an alternative scenario in which the confinement / deconfinement transition corresponds to a gregory - laflamme transition @xcite between a uniformly distributed soliton and a localized soliton in the iib frame . the scenario we propose suggests that we need to reconsider several previous results in holographic qcd including phenomena related to the sakai - sugimoto model @xcite . in this section , we will review the construction of four dimensional @xmath0 pure yang - mills theory ( ym4 ) from @xmath1 d4 branes @xcite . let us first consider a 10 dimensional euclidean spacetime with an @xmath2 and consider d4 branes wrapping on the @xmath2 . we define the coordinate along this @xmath2 as @xmath3 and its periodicity as @xmath4 . the effective theory on this brane is a 5 dimensional supersymmetric yang - mills theory ( sym5 ) on the @xmath5 . for the fermions on the brane , the boundary condition along the circle can be ap ( antiperiodic ) or p ( periodic ) ; to specify the theory , we must pick one of these two boundary conditions . let us take the ap boundary condition . this gives rise to fermion masses proportional to the kaluza - klein scale @xmath6 , leading to supersymmetry breaking ( this is called the ss scherk - schwarz mechanism ) . this , in turn , induces masses for the adjoint scalars and for @xmath7 , which are proportional to @xmath8 at one - loop . therefore , if @xmath9 is sufficiently small and the dynamical scale @xmath10 and temperature are much less than both the above mass scales , then the fermions , adjoint scalars and kk modes are decoupled and the 5 dimensional supersymmetric yang - mills theory is reduced to a four dimensional pure yang - mills theory . the precise conditions to obtain ym4 from sym5 are @xmath11 by taking the large @xmath1 ( maldacena ) limit of this system @xcite at low temperatures , we obtain the dual gravity description of the compactified 5 dimensional sym theory @xcite , which consists , at low temperatures , of a solitonic d@xmath12 brane solution wrapping the @xmath5 . the explicit metric is given by @xmath13 , \nonumber \\ & f_4(u)=1-\left ( \frac{u_0}{u}\right)^3 , \quad e^{\phi}=\frac{\lambda_5}{(2\pi)^2n } \left(\frac{u^{3/2}}{\sqrt{d_5 \lambda_5 } } \right)^{1/2 } , \quad u_0=\frac{\pi \lambda_4}{9 l_4 } , \label{metric - sd4}\end{aligned}\ ] ] where @xmath14 and @xmath15 . this gravity solution is not always valid . e.g. in order that the stringy modes can be ignored , we should ensure that the curvature in string units must be small @xcite . this condition turns out to be equivalent to @xmath16 this is opposite to the condition ( [ gauge - cond ] ) . thus , the gravity solution can describe sym5 but can not directly describe ym4 . this is a common problem in the construction of holographic duals of non - supersymmetric gauge theories . however , in principle , leading order effects of stringy modes can be computed by perturbatively evaluating @xmath17 corrections in the gravity theory and , as in case of the standard strong coupling expansion in gauge theory ( see , e.g. @xcite ) , we may obtain appropriate weak coupling results , _ provided no phase transition occurs between the weak and strong coupling regimes_. although the gravity solution ( [ metric - sd4 ] ) is just the leading term of such an expansion , we should be able to infer qualitative properties of the gauge theory by extrapolating . many interesting results , including the qualitative predictions in @xcite , have been obtained using this prescription . we will proceed to describe the thermodynamics of yang - mills theory from gravity in this spirit . in this section we investigate the thermodynamic phase structure of the gravity solutions and compare the result with the phase structure of pure yang - mills theory . to discuss holographic qcd at finite temperatures , we begin by compactifying euclidean time in the boundary theory on a circle with periodicity @xmath18 . in order to determine the gravitational theory , we need to fix the periodicity of fermions in the gauge theory along the time cycle . let us recall that the gauge theory of interest here is pure yang mills theory in four dimensions , which does not have fermions . fermions reappear when the validity condition of the gravity description ( [ gravity - cond - sd4 ] ) is enforced and @xmath10 goes above the kk scale @xmath6 . in this sense , fermions are an artifact of the holographic method and in the region of validity of the pure ym theory , the periodicity of the fermion should not affect the gauge theory results . indeed we can obtain the thermal partition function of ym4 from sym5 on @xmath19 with either the ( ap , ap ) or ( p , ap ) boundary condition here and elsewhere the boundary conditions will always refer to those of the boundary theory along @xmath20 , respectively . ] : @xmath21 where the limit ensures that the fermions and adjoint scalars are decoupled because of large mass . since we recover pure ym4 in both cases , it is pertinent to study gravity solutions corresponding to both these boundary conditions . the gravity solutions appearing in the ( ap , ap ) case @xcite are summarized as @xmath22 here the metric of the black d4 solution is given by @xmath23 , ~~ u_0=\frac{\pi \lambda_4 l_4}{9\beta^2 } \label{metric - bd4}\end{aligned}\ ] ] note that this metric is related to the solitonic d4 brane ( [ metric - sd4 ] ) through the @xmath24 symmetry : @xmath25 . thus , the free energies of these solutions are coincident at the self - dual point @xmath26 and a phase transition called scherk - schwarz transition happens there . in ( [ table - apap ] ) , @xmath27 and @xmath28 are the polyakov loop operators : @xmath29 vacuum expectation values of these operators characterize the phases of the gauge theory . these also act as order parameters for the @xmath30 ` centre symmetry ' ( see @xcite for details ) and their vanishing is related to large-@xmath1 volume independence @xcite . the gravity solutions appearing in the ( p , ap ) case are @xmath31 note that we need to take a t - dual along the temporal @xmath32 circle and go to the iib frame to see the high temperature region , since the masses of the winding modes of the iia string wrapping along the temporal circle become light when @xmath33 . this t - dual maps the solitonic d4 solution in the iia to solitonic d3 branes uniformly smeared along the dual temporal circle in the iib , with metric given by @xmath34 . \label{metric - ssd3 } \end{aligned}\ ] ] here @xmath35 is the dual of @xmath36 , hence @xmath37 has a periodicity @xmath38 . this uniform configuration becomes meta - stable above a critical temperature @xmath39 , and the solitonic d3 branes get localized on the dual temporal circle . this phase transition is a gregory - laflamme ( gl ) type transition @xcite . the metric of the localized solitonic d3 brane is approximately described in @xcite . the phases in the 4 dimensional pure yang - mills theory are @xmath40 here @xmath28 makes an appearance because ym4 is viewed as a limit of sym5 . that @xmath41 follows from the fact that kk reduction from sym5 to ym4 only works in this phase ; in the phase with vanishing @xmath28 the thermodynamics is independent of @xmath4 ( through the large @xmath1 volume independence mentioned above ) and can not describe the kk reduction . note , further , that the above table works for either p or ap boundary condition for the sym5 fermions along the temporal circle , as we argued in ( [ 4d - limit ] ) . let us now compare this phase structure in yang - mills theory with the one found above for gravity . in the low temperature regime , the properties of the polyakov loop operators in the solitonic d4 brane agree with those of the confinement phase . thus these two phases may be identified . in the high temperature regime , it is the localized d3 soliton which matches the deconfinement phase in terms of properties of @xmath27 and @xmath28 whereas the black d4 brane solution does not . this , in particular , means that the @xmath30 symmetry is realized in the same was in the localized d3 soliton phase as in the deconfinement phase , whereas it is realized differently in the black d4 solution . thus the localized d3 solution may correspond to the deconfinement phase but the black d4 brane does not . indeed the expected phase structures of sym5 with the ( ap , ap ) and ( p , ap ) boundary condition are shown in figure [ fig - d4-phase ] . in the ( ap , ap ) case , at least one phase transition has to occur between the black d4 brane solution and the deconfinement phase . ( this is also expected through the @xmath24 symmetry : @xmath42 @xcite . see also @xcite . ) therefore the previous conjecture that the black d4 brane corresponds to the deconfinement phase of the yang - mills theory is not correct , since these two phases are not smoothly connected ( see the remarks below in _ italics _ ) . by contrast , in the ( p , ap ) case , the gravity solutions in the strong coupling regime may smoothly continue to the phases of the weakly coupled 4 dimensional yang mills theory . thus we propose that we should look at these gravity solutions in the ( p , ap ) case to investigate the dynamics of the yang - mills theory . especially the gl transition between the smeared d3 soliton and the localized d3 soliton would correspond to the confinement / deconfinement transition in the yang - mills theory where d4 branes are replaced by d1 branes . this provides support for the proposal presented in this paper . ] . with the ( ap , ap ) and ( p , ap ) boundary condition . the gravity analysis is valid in the strong coupling region ( the blue region ) . the 4 dimensional ym description is valid in the upper green region , see ( [ gauge - cond ] ) . the lower green region in the ( ap , ap ) case is the mirror of the upper one via the @xmath24 symmetry @xmath25 . the solid black lines correspond to a minimal extrapolation of the phase boundaries through the intermediate region . the dotted line denotes another possible phase transition which is allowed by the @xmath24 symmetry . ] our proposal opens up the interesting possibility of a relation between the gl transition and the hagedorn transition . it is known that the gl instability is an instability of the kk modes of the graviton along the compact circle @xcite ( such modes develop an imaginary frequency and grow exponentially ) . in our case , the kk modes along the dual temporal circle , which cause the gl instability in the iib description , would be mapped to winding modes around the temporal circle through the t - duality @xcite . this indicates that the gl transition is associated with the tachyonic instability of the winding modes of the iia string . this is similar to the hagedorn transition in string theory @xcite , where the temporal winding modes cause the instability . thus the gl transition in the iib description might correspond to the hagedorn transition in the iia description . note that on the gauge theory side also , the confinement / deconfinement transition has been shown to be related to the hagedorn transition @xcite . this makes it plausible that the hagedorn transition in the yang - mills theory continues to the hagedorn transition in the iia string , which , as we argued above , is possibly the dual of the gl transition in the iib supergravity . in this study , we showed that the identification between the deconfinement phase and the black d4 brane solution in the ( ap , ap ) boundary condition is not correct and proposed a resolution by using the ( p , ap ) boundary condition . this result suggests that we need to reconsider several previous results , in which the black d4 brane was employed in high temperature holographic qcd including the sakai - sugimoto model @xcite . one important ingredient in the sakai - sugimoto model is the mechanism of chiral symmetry restoration at high temperatures @xcite . in @xcite , we proposed a new mechanism for chiral symmetry restoration in our framework . another important issue is to explore what geometry corresponds to the deconfinement phase in the real time formalism ; this is important , e.g. , for the investigation of the viscosity ratio . problems similar to the above had also been encountered in the study of two dimensional bosonic gauge theory in @xcite . this indicates that the issues addressed in this paper are rather general in the discussion of holography for non - supersymmetric gauge theories at finite temperatures . we would like to thank ofer aharony , avinash dhar , saumen datta , koji hashimoto , yoshimasa hidaka , elias kiritsis , matthew lippert , shiraz minwalla , rene meyer , rob myers , vasilis niarchos , tadakatsu sakai , shigeki sugimoto and tadashi takayanagi for valuable discussions and comments . is partially supported by regional potential program of the e.u . fp7-regpot-2008 - 1 : cretehepcosmo-228644 and by marie curie contract pirg06-ga-2009 - 256487 . 999 e. witten , `` anti - de sitter space , thermal phase transition , and confinement in gauge theories , '' adv . theor . math . * 2 * ( 1998 ) 505 [ arxiv : hep - th/9803131 ] . o. aharony , s. s. gubser , j. m. maldacena , h. ooguri and y. oz , `` large n field theories , string theory and gravity , '' phys . rept . * 323 * , 183 ( 2000 ) [ arxiv : hep - th/9905111 ] . d. j. gross and h. ooguri , `` aspects of large n gauge theory dynamics as seen by string theory , '' phys . d * 58 * ( 1998 ) 106002 [ arxiv : hep - th/9805129 ] . m. kruczenski , d. mateos , r. c. myers and d. j. winters , `` towards a holographic dual of large - n(c ) qcd , '' jhep * 0405 * ( 2004 ) 041 [ arxiv : hep - th/0311270 ] . t. sakai and s. sugimoto , `` low energy hadron physics in holographic qcd , '' prog . theor . * 113 * ( 2005 ) 843 [ arxiv : hep - th/0412141 ] . o. aharony , j. sonnenschein and s. yankielowicz , `` a holographic model of deconfinement and chiral symmetry restoration , '' annals phys . * 322 * ( 2007 ) 1420 [ arxiv : hep - th/0604161 ] . r. gregory and r. laflamme , `` the instability of charged black strings and p - branes , '' nucl . b * 428 * ( 1994 ) 399 [ arxiv : hep - th/9404071 ] . t. harmark and n. a. obers , `` black holes on cylinders , '' jhep * 0205 * ( 2002 ) 032 [ hep - th/0204047 ] . o. aharony , j. marsano , s. minwalla and t. wiseman , `` black hole - black string phase transitions in thermal 1 + 1-dimensional supersymmetric yang - mills theory on a circle , '' class . * 21 * , 5169 ( 2004 ) [ arxiv : hep - th/0406210 ] . t. harmark and n. a. obers , `` new phases of near - extremal branes on a circle , '' jhep * 0409 * ( 2004 ) 022 [ arxiv : hep - th/0407094 ] . h. kudoh and t. wiseman , `` connecting black holes and black strings , '' phys . lett . * 94 * ( 2005 ) 161102 [ hep - th/0409111 ] . b. kol , `` the phase transition between caged black holes and black strings : a review , '' phys . * 422 * ( 2006 ) 119 [ hep - th/0411240 ] . n. itzhaki , j. m. maldacena , j. sonnenschein and s. yankielowicz , `` supergravity and the large n limit of theories with sixteen supercharges , '' phys . d * 58 * ( 1998 ) 046004 [ arxiv : hep - th/9802042 ] . j. m. drouffe , c. itzykson , `` lattice gauge fields , '' phys . rept . * 38 * , 133 - 175 ( 1978 ) . t. eguchi and h. kawai , `` reduction of dynamical degrees of freedom in the large n gauge theory , '' phys . lett . * 48 * , 1063 ( 1982 ) . a. gocksch and f. neri , `` on large n qcd at finite temperature , '' phys . * 50 * ( 1983 ) 1099 . o. aharony , j. marsano , s. minwalla , k. papadodimas , m. van raamsdonk and t. wiseman , `` the phase structure of low dimensional large n gauge theories on tori , '' jhep * 0601 * , 140 ( 2006 ) [ arxiv : hep - th/0508077 ] . s. catterall , a. joseph , t. wiseman , `` thermal phases of d1-branes on a circle from lattice super yang - mills , '' jhep * 1012 * ( 2010 ) 022 . [ arxiv:1008.4964 [ hep - th ] ] . t. harmark , v. niarchos and n. a. obers , `` instabilities of black strings and branes , '' class . * 24 * ( 2007 ) r1 [ arxiv : hep - th/0701022 ] . s. f. ross , t. wiseman , `` smeared d0 charge and the gubser - mitra conjecture , '' class . * 22 * ( 2005 ) 2933 - 2946 . [ hep - th/0503152 ] . b. sundborg , `` the hagedorn transition , deconfinement and n=4 sym theory , '' nucl . b * 573 * , 349 ( 2000 ) [ arxiv : hep - th/9908001 ] . o. aharony , j. marsano , s. minwalla , k. papadodimas and m. van raamsdonk , `` the hagedorn - deconfinement phase transition in weakly coupled large n gauge theories , '' adv . * 8 * ( 2004 ) 603 [ arxiv : hep - th/0310285 ] . g. mandal , t. morita , `` phases of a two dimensional large n gauge theory on a torus , '' phys . * d84 * ( 2011 ) 085007 . [ arxiv:1103.1558 [ hep - th ] ] .	we study the gravity dual of four dimensional pure yang - mills theory through d4 branes , as proposed by witten ( holographic qcd ) . in this holographic qcd , it has been widely believed that the confinement phase in the pure yang - mills theory corresponds to the solitonic d4 brane in gravity and the deconfinement phase corresponds to the black d4 brane . we inspect this conjecture carefully and show that the correspondence between the black d4 brane and the deconfinement phase is not correct . instead , by using a slightly different set up , we find an alternative gravity solution called `` localized soliton '' , which would be properly related to the deconfinement phase . in this case , the confinement / deconfinement transition is realized as a gregory - laflamme type transition . we find that our proposal naturally explains several known properties of qcd . tifr / th/11 - 49 + cctp-2011 - 39 # 1([#1 ] )	5,985	275
in the past 30 years , the development of linear algebra libraries has been tremendously successful , resulting in a variety of reliable and efficient computational kernels . unfortunately these kernels are limited by a rigid interface that does not allow users to pass knowledge specific to the target problem . if available , such knowledge may lead to domain - specific algorithms that attain higher performance than any traditional library @xcite . the difficulty does not lay so much in creating flexible interfaces , but in developing algorithms capable of taking advantage of the extra information . in this paper , we present preliminary work on a linear algebra compiler , written in mathematica , that automatically exploits application - specific knowledge to generate high - performance algorithms . the compiler takes as input a target equation and information on the structure and properties of the operands , and returns as output algorithms that exploit the given information . in the same way that a traditional compiler breaks the program into assembly instructions directly supported by the processor , attempting different types of optimization , our linear algebra compiler breaks a target operation down to library - supported kernels , and generates not one but a family of viable algorithms . the decomposition process undergone by our compiler closely replicates the thinking process of a human expert . we show the potential of the compiler by means of a challenging operation arising in computational biology : the _ genome - wide association study _ ( gwas ) , an ubiquitous tool in the fields of genomics and medical genetics @xcite . as part of gwas , one has to solve the following equation @xmath0 where @xmath1 , @xmath2 , and @xmath3 are known quantities , and @xmath4 is sought after . the size and properties of the operands are as follows : @xmath5 , @xmath6 is full rank , @xmath7 is symmetric positive definite ( spd ) , @xmath8 , @xmath9 , and @xmath10 ; @xmath11 , @xmath12 , @xmath13 , and @xmath14 is either @xmath15 or of the order of @xmath16 . at the core of gwas lays a linear regression analysis with non - independent outcomes , carried out through the solution of a two - dimensional sequence of the generalized least - squares problem ( gls ) @xmath17 while gls may be directly solved , for instance , by matlab , or may be reduced to a form accepted by lapack @xcite , none of these solutions can exploit the specific structure pertaining to gwas . the nature of the problem , a sequence of correlated glss , allows multiple ways to reuse computation . also , different sizes of the input operands demand different algorithms to attain high performance in all possible scenarios . the application of our compiler to gwas , eq . [ eq : probdef ] , results in the automatic generation of dozens of algorithms , many of which outperform the current state of the art by a factor of four or more . the paper is organized as follows . related work is briefly described in section [ sec : related ] . sections [ sec : principles ] and [ sec : system - overview ] uncover the principles and mechanisms upon which the compiler is built . in section [ sec : generation - algs ] we carefully detail the automatic generation of multiple algorithms , and outline the code generation process . in section [ sec : performance ] we report on the performance of the generated algorithms through numerical experiments . we draw conclusions in section [ sec : conclusions ] . a number of research projects concentrate their efforts on domain - specific languages and compilers . among them , the spiral project @xcite and the tensor contraction engine ( tce ) @xcite , focused on signal processing transforms and tensor contractions , respectively . as described throughout this paper , the main difference between our approach and spiral is the inference of properties . centered on general dense linear algebra operations , one of the goals of the flame project is the systematic generation of algorithms . the flame methodology , based on the partitioning of the operands and the automatic identification of loop - invariants @xcite , has been successfully applied to a number of operations , originating hundreds of high - performance algorithms . the approach described in this paper is orthogonal to flame . no partitioning of the operands takes place . instead , the main idea is the mapping of operations onto high - performance kernels from available libraries , such as blas @xcite and lapack . in this section we expose the human thinking process behind the generation of algorithms for a broad range of linear algebra equations . as an example , we derive an algorithm for the solution of the gls problem , eq . [ eq : fgls ] , as it would be done by an expert . together with the derivation , we describe the rationale for every step of the algorithm . the exposed rationale highlights the key ideas on top of which we founded the design of our compiler . given eq . [ eq : fgls ] , the * first concern is the inverse operator * applied to the expression @xmath18 . since @xmath19 is not square , the inverse can not be distributed over the product and the expression needs to be processed first . the attention falls then on @xmath20 . the inversion of a matrix is costly and not recommended for numerical reasons ; therefore , since @xmath21 is a general matrix , we * factor * it . given the structure of @xmath21 ( spd ) , we choose a cholesky factorization , resulting in @xmath22 where @xmath23 is square and lower triangular . as @xmath23 is square , the inverse may now be distributed over the product @xmath24 , yielding @xmath25 . next , we process @xmath26 ; we observe that the quantity @xmath27 * appears multiple times * , and may be computed and reused to * save computation * : @xmath28 at this point , since @xmath29 is not square and the inverse can not be distributed , there are two * alternatives * : 1 ) multiply out @xmath30 ; or 2 ) factor @xmath29 , for instance through a qr factorization . in this example , we choose the former : @xmath31 one can prove that @xmath32 is spd , suggesting yet another factorization . we choose a cholesky factorization and distribute the inverse over the product : @xmath33 now that all the remaining inverses are applied to triangular matrices , we are left with a series of products to compute the final result . since all operands are matrices except the vector @xmath34 , we compute eq . [ eq : exalg1step4 ] from right to left to * minimize the number of flops. the final algorithm is shown in alg . [ alg : exalg - chol ] , together with the names of the corresponding blas and lapack building blocks . .... $ l l^t = m$ ( ! \sc potrf ! ) $ w : = l^{-1 } x$ ( ! \sc trsm ! ) $ s : = w^t w$ ( ! \sc syrk ! ) $ g g^t = s$ ( ! \sc potrf ! ) $ y : = l^{-1 } y$ ( ! \sc trsv ! ) $ b : = w^t y$ ( ! \sc gemv ! ) $ b : = g^{-1 } b$ ( ! \sc trsv ! ) $ b : = g^{-t } b$ ( ! \sc trsv ! ) .... three ideas stand out as the guiding principles for the thinking process : the first concern is to deal , whenever it is not applied to diagonal or triangular matrices , with the inverse operator . two scenarios may arise : a ) it is applied to a single operand , @xmath35 . in this case the operand is factored with a suitable factorization according to its structure ; b ) the inverse is applied to an expression . this case is handled by either computing the expression and reducing it to the first case , or factoring one of the matrices and analyzing the resulting scenario . * when decomposing the equation , we give priority to a ) common segments , i.e. , common subexpressions , and b ) segments that minimize the number of flops ; this way we reduce the amount of computation performed . * if multiple alternatives leading to viable algorithms arise , we explore all of them . our compiler follows the above guiding principles to closely replicate the thinking process of a human expert . to support the application of these principles , the compiler incorporates a number of modules ranging from basic matrix algebra support to analysis of dependencies , including the identification of building blocks offered by available libraries . in the following , we describe the core modules . matrix algebra : : the compiler is written using mathematica from scratch . we implement our own operators : addition ( plus ) , negation ( minus ) , multiplication ( times ) , inversion ( inv ) , and transposition ( trans ) . together with the operators , we define their precedence and properties , as commutativity , to support matrices as well as vectors and scalars . we also define a set of rewrite rules according to matrix algebra properties to freely manipulate expressions and simplify them , allowing the compiler to work on multiple equivalent representations . inference of properties : : in this module we define the set of supported matrix properties . as of now : identity , diagonal , triangular , symmetric , symmetric positive definite , and orthogonal . on top of these properties , we build an inference engine that , given the properties of the operands , is able to infer properties of complex expressions . this module is extensible and facilitates incorporating additional properties . building blocks interface : : this module contains an extensive list of patterns associated with the desired building blocks onto which the algorithms will be mapped . it also contains the corresponding cost functions to be used to construct the cost analysis of the generated algorithms . as with the properties module , if a new library is to be used , the list of accepted building blocks can be easily extended . analysis of dependencies : : when considering a sequence of problems , as in gwas , this module analyzes the dependencies among operations and between operations and the dimensions of the sequence . through this analysis , the compiler rearranges the operations in the algorithm , reducing redundant computations . code generation : : in addition to the automatic generation of algorithms , the compiler includes a module to translate such algorithms into code . so far , we support the generation of matlab code for one instance as well as sequences of problems . to complete the overview of our compiler , we provide a high - level description of the compiler s _ reasoning_. the main idea is to build a tree in which the root node contains the initial target equation ; each edge is labeled with a building block ; and each node contains intermediate equations yet to be mapped . the compiler progresses in a breadth - first fashion until all leaf nodes contain an expression directly mapped onto a building block . while processing a node s equation , the search space is constrained according to the following criteria : 1 . if the expression contains an inverse applied to a single ( non - diagonal , non - triangular ) matrix , for instance @xmath20 , then the compiler identifies a set of viable factorizations for @xmath21 based on its properties and structure ; 2 . if the expression contains an inverse applied to a sub - expression , for instance @xmath36 , then the compiler identifies both viable factorizations for the operands in the sub - expression ( e.g. , @xmath37 ) , and segments of the sub - expression that are directly mapped onto a building block ( e.g. , @xmath38 ) ; 3 . if the expression contains no inverse to process ( as in @xmath39 , with @xmath40 and @xmath23 triangular ) , then the compiler identifies segments with a mapping onto a building block . when inspecting expressions for segments , the compiler gives priority to common segments and segments that minimize the number of flops . all three cases may yield multiple building blocks . for each building block either a factorization or a segment both a new edge and a new children node are created . the edge is labeled with the corresponding building block , and the node contains the new resulting expression . for instance , the analysis of eq . [ eq : exalg1step2 ] creates the following sub - tree : in addition , thanks to the _ inference of properties _ module , for each building block , properties of the output operands are inferred from those of the input operands . each path from the root node to a leaf represents one algorithm to solve the target equation . by assembling the building blocks attached to each edge in the path , the compiler returns a collection of algorithms , one per leaf . our compiler has been successfully applied to equations such as pseudo - inverses , least - squares - like problems , and the automatic differentiation of blas and lapack operations . of special interest are the scenarios in which sequences of such problems arise ; for instance , the study case presented in this paper , genome - wide association studies , which consist of a two - dimensional sequence of correlated gls problems . the compiler is still in its early stages and the code is not yet available for a general release . however , we include along the paper details on the input and output of the system , as well as screenshots of the actual working prototype . we detail now the application to gwas of the process described above . box [ box : input ] includes the input to the compiler : the target equation along with domain - specific knowledge arising from gwas , e.g , operands shape and properties . as a result , dozens of algorithms are automatically generated ; we report on three selected ones . .... equation = { equal[b , times [ inv[times [ trans[x ] , inv[plus [ times[h , phi ] , times[plus[1 , minus[h ] ] , i d ] ] ] , x ] ] , trans[x ] , inv[plus [ times[h , phi ] , times[plus[1 , minus[h ] ] , i d ] ] ] , y ] ] } ; operandproperties = { { x , { `` input '' , `` matrix '' , `` fullrank '' } } , { y , { `` input '' , `` vector '' } } , { phi , { `` input '' , `` matrix '' , `` symmetric '' } } , { h , { `` input '' , `` scalar '' } } , { b , { `` output '' , `` vector '' } } } ; expressionproperties = { inv[plus [ times[h , phi ] , times[plus[1 , minus[h ] ] , i d ] ] ] , `` spd '' } ; sizeassumptions = { rows[x ] > cols[x ] } ; .... to ease the reader , we describe the process towards the generation of an algorithm similar to alg . [ alg : exalg - chol ] . the starting point is eq . [ eq : probdef ] . since @xmath19 is not square , the inverse operator applied to @xmath41 can not be distributed over the product ; thus , the inner - most inverse is @xmath42 . the inverse is applied to an expression , which is inspected for viable factorizations and segments . among the identified alternatives are a ) the factorization of the operand @xmath43 according to its properties , and b ) the computation of the expression @xmath44 . here we concentrate on the second case . the segment @xmath44 is matched as the scal - add building block ( scaling and addition of matrices ) ; the operation is made explicit and replaced : @xmath45 now , the inner - most inverse is applied to a single operand , @xmath21 , and the compiler decides to factor it using multiple alternatives : cholesky ( @xmath46 ) , qr ( @xmath47 ) , eigendecomposition ( @xmath48 ) , and svd ( @xmath49 ) . all the alternatives are explored ; we focus now on the cholesky factorization ( potrf routine from lapack ) : @xmath50 after @xmath21 is factored and replaced by @xmath24 , the inference engine propagates a number of properties to @xmath23 based on the properties of @xmath21 and the factorization applied . concretely , @xmath23 is square , triangular and full - rank . next , since @xmath23 is triangular , the inner - most inverse to be processed in eq . [ eq : alg1step1 ] is @xmath51 . in this case two routes are explored : either factor @xmath19 ( @xmath23 is triangular and does not need further factorization ) , or map a segment of the expression onto a building block . we consider this second alternative . the compiler identifies the solution of a triangular system ( trsm routine from blas ) as a common segment appearing three times in eq . [ eq : alg1step1 ] , makes it explicit , and replaces it : @xmath52 since @xmath23 is square and full - rank , and x is also full - rank , @xmath29 inherits the shape of @xmath19 and is labelled as full - rank . as @xmath29 is not square , the inverse can not be distributed over the product yet . therefore , the compiler faces again two alternatives : either factoring @xmath29 or multiplying @xmath30 . we proceed describing the latter scenario while the former is analyzed in sec . [ subsec : alg - two ] . @xmath30 is identified as a building block ( syrk routine of blas ) , and made explicit : @xmath53 the inference engine plays an important role deducing properties of @xmath32 . during the previous steps , the engine has inferred that @xmath29 is full - rank and rows[w ] > cols[w ] ; therefore the following rule states that @xmath29 is spd . is spd if @xmath54 is full rank and has more rows than columns . ] .... isspdq [ times [ trans [ a_?isfullrankq ] , a _ ] / ; rows[a ] > cols[a ] : = true ; .... this knowledge is now used to determine possible factorizations for @xmath32 . we concentrate on the cholesky factorization : @xmath55 in eq . [ eq : alg1step4 ] , all inverses are applied to triangular matrices ; therefore , no more treatment of inverses is needed . the compiler proceeds with the final decomposition of the remaining series of products . since at every step the inference engine keeps track of the properties of the operands in the original equation as well as the intermediate temporary quantities , it knows that every operand in eq . [ eq : alg1step4 ] are matrices except for the vector @xmath34 . this knowledge is used to give matrix - vector products priority over matrix - matrix products , and eq . [ eq : alg1step4 ] is decomposed accordingly . in case the compiler can not find applicable heuristics to lead the decomposition , it explores the multiple viable mappings onto building blocks . the resulting algorithm , and the corresponding output from mathematica , are assembled in alg . [ alg : alg - chol ] , chol - gwas . .... $ m : = h\phi + ( 1-h)i$ ( ! \sc scal - add ! ) $ l l^t = m$ ( ! \sc potrf ! ) $ w : = l^{-1 } x$ ( ! \sc trsm ! ) $ s : = w^t w$ ( ! \sc syrk ! ) $ g g^t = s$ ( ! \sc potrf ! ) $ y : = l^{-1 } y$ ( ! \sc trsv ! ) $ b : = w^t y$ ( ! \sc gemv ! ) $ b : = g^{-1 } b$ ( ! \sc trsv ! ) $ b : = g^{-t } b$ ( ! \sc trsv ! ) .... in this subsection we display the capability of the compiler to analyze alternative paths , leading to multiple viable algorithms . at the same time , we expose more examples of algebraic manipulation carried out by the compiler . the presented algorithm results from the alternative path arising in eq . [ eq : alg1step3 ] , the factorization of @xmath29 . since @xmath29 is a full - rank column panel , the compiler analyzes the scenario where @xmath29 is factored using a qr factorization ( geqrf routine in lapack ) : @xmath56 at this point , the compiler exploits the capabilities of the _ matrix algebra _ module to perform a series of simplifications : @xmath57 first , it distributes the transpose operator over the product . then , it applies the rule .... times [ trans [ q_?isorthonormalq , q _ ] - > i d , .... included as part of the knowledge - base of the module . the rule states that the product @xmath58 , when @xmath59 is orthogonal with normalized columns , may be rewritten ( - > ) as the identity matrix . next , since @xmath60 is square , the inverse is distributed over the product . more mathematical knowledge allows the compiler to rewrite the product @xmath61 as the identity . in eq . [ eq : alg2step4 ] , the compiler does not need to process any more inverses ; hence , the last step is to decompose the remaining computation into a sequence of products . once more , @xmath34 is the only non - matrix operand . accordingly , the compiler decomposes the equation from right to left . the final algorithm is put together in alg . [ alg : alg - qr ] , qr - gwas . .... $ m : = h\phi + ( 1-h)i$ ( ! \sc scal - add ! ) $ l l^t$ = $ m$ ( ! { \sc potrf } ! ) $ w$ : = $ l^{-1 } x$ ( ! { \sc trsm } ! ) $ q r = w$ ( ! { \sc geqrf } ! ) $ y$ : = $ l^{-1 } y$ ( ! { \sc trsv } ! ) $ b$ : = $ q^t y$ ( ! { \sc gemv } ! ) $ b$ : = $ r^{-1 } b$ ( ! { \sc trsv } ! ) .... this third algorithm exploits further knowledge from gwas , concretely the structure of @xmath21 , in a manner that may be overlooked even by human experts . again , the starting point is eq . [ eq : probdef ] . the inner - most inverse is @xmath62 . instead of multiplying out the expression within the inverse operator , we now describe the alternative path also explored by the compiler : factoring one of the matrices in the expression . we concentrate in the case where an eigendecomposition of @xmath43 ( syevd or syevr from lapack ) is chosen : @xmath63 where @xmath64 is a square , orthogonal matrix with normalized columns , and @xmath29 is a square , diagonal matrix . in this scenario , the _ matrix algebra _ module is essential ; it allows the compiler to work with alternative representations of eq . [ eq : alg3step1 ] . we already illustrated an example where the product @xmath58 , @xmath59 orthonormal , is replaced with the identity matrix . the freedom gained when defining its own operators , allows the compiler to perform also the opposite transformation : .... i d - > times [ q , trans [ q ] ] ; i d - > times [ trans [ q ] , q ] ; .... to apply these rules , the compiler inspects the expression @xmath65 for orthonormal matrices : @xmath64 is found to be orthonormal and used instead of @xmath59 in the right - hand side of the previous rules . the resulting expression is @xmath66 the algebraic manipulation capabilities of the compiler lead to the derivation of further multiple equivalent representations of eq . [ eq : alg3step2 ] . we recall that , although we focus on a concrete branch of the derivation , the compiler analyzes the many alternatives . in the branch under study , the quantities @xmath64 and @xmath67 are grouped on the left- and right - hand sides of the inverse , respectively : @xmath68 then , since both @xmath64 and @xmath69 are square , the inverse is distributed : @xmath70 finally , by means of the rules : .... inv [ q_?isorthonormalq ] - > trans [ q ] ; inv [ trans [ q_?isorthonormalq ] ] - > q ; .... which state that the inverse of an orthonormal matrix is its transpose , the expression becomes : @xmath71 the resulting equation is @xmath72 the inner - most inverse in eq . [ eq : alg3step3 ] is applied to a diagonal object ( @xmath29 is diagonal and @xmath73 a scalar ) . no more factorizations are needed , @xmath69 is identified as a scal - add building block , and exposed : @xmath74 @xmath75 is a diagonal matrix ; hence only the inverse applied to @xmath76 remains to be processed . among the alternative steps , we consider the mapping of the common segment @xmath77 , that appears three times , onto the gemm building block ( matrix - matrix product ) : @xmath78 from this point on , the compiler proceeds as shown for the previous examples , and obtains , among others , alg . [ alg : alg - eigen ] , eig - gwas . .... $ z w z^t$ = $ \phi$ ( ! { \sc syevx } ! ) $ d : = h w + ( 1 - h ) i$ ( ! { \sc add - scal } ! ) $ k : = x^t z$ ( ! { \sc gemm } ! ) $ v : = k d^{-1}$ ( ! { \sc scal } ! ) $ s$ : = $ v k^t$ ( ! { \sc gemm } ! ) $ q r$ = $ s$ ( ! { \sc geqrf } ! ) $ y : = z^t y$ ( ! { \sc gemv } ! ) $ b : = v y$ ( ! { \sc gemv } ! ) $ b : = q^t b$ ( ! { \sc gemv } ! ) $ b : = r^{-1 } b$ ( ! { \sc trsv } ! ) .... at first sight , alg . [ alg : alg - eigen ] might seem to be a suboptimal approach . however , as we show in sec . [ sec : performance ] , it is representative of a family of algorithms that play a crucial role when solving a certain sequence of gls problems within gwas . we have illustrated how our compiler , closely replicating the reasoning of a human expert , automatically generates algorithms for the solution of a single gls problem . as shown in eq . [ eq : probdef ] , in practice one has to solve one - dimensional ( @xmath79 ) or two - dimensional ( @xmath80 ) sequences of such problems . in this context we have developed a module that performs a loop dependence analysis to identify loop - independent operations and reduce redundant computations . for space reasons , we do not further describe the module , and limit to the automatically generated cost analysis . the list of patterns for the identification of building blocks included in the _ building blocks interface _ module also incorporates the corresponding computational cost associated to the operations . given a generated algorithm , the compiler composes the cost of the algorithm by combining the number of floating point operations performed by the individual building blocks , taking into account the loops over the problem dimensions . table [ tab : cost ] includes the cost of the three presented algorithms , which attained the lowest complexities for one- and two - dimensional sequences . while qr - gwas and chol - gwas share the same cost for both types of sequences , suggesting a very similar behavior in practice , the cost of eig - gwas differs in both cases . for the one - dimensional sequence the cost of eig - gwas is not only greater in theory , the practical constants associated to its terms increase the gap . on the contrary , for the two - dimensional sequence , the cost of eig - gwas is lower than the cost of the other two . this analysis suggests that qr - gwas and chol - gwas are better suited for the one - dimensional case , while eig - gwas is better suited for the two - dimensional one . in sec . [ sec : performance ] we confirm these predictions through experimental results . [ cols="<,^,^,^",options="header " , ] the translation from algorithms to code is not a straightforward task ; in fact , when manually performed , it is tedious and error prone . to overcome this difficulty , we incorporate in our compiler a module for the automatic generation of code . as of now , we support matlab ; an extension to fortran , a much more challenging target language , is planned . we provide here a short overview of this module . given an algorithm as derived by the compiler , the code generator builds an _ abstract syntax tree _ ( ast ) mirroring the structure of the algorithm . then , for each node in the ast , the module generates the corresponding code statements . specifically , for the nodes corresponding to _ for _ loops , the module not only generates a for statement but also the specific statements to extract subparts of the operands according to their dimensionality ; as for the nodes representing the building blocks , the generator must map the operation to the specific matlab routine or matrix expression . as an example of automatically generated code , the matlab routine corresponding to the aforementioned eig - gwas algorithm for a two - dimensional sequence is illustrated in fig . [ fig : eig - code ] . we turn now the attention to numerical results . in the experiments , we compare the algorithms automatically generated by our compiler with lapack and genabel @xcite , a widely used package for gwas - like problems . for details on genabel s algorithm for gwas , gwfgls , we refer the reader to @xcite . we present results for the two most representative scenarios in gwas : one - dimensional ( @xmath81 ) , and two - dimensional ( @xmath82 ) sequences of gls problems . the experiments were performed on an 12-core intel xeon x5675 processor running at 3.06 ghz , with 96 gb of memory . the algorithms were implemented in c , and linked to the multi - threaded gotoblas and the reference lapack libraries . the experiments were executed using 12 threads . we first study the scenario @xmath81 . we compare the performance of qr - gwas and chol - gwas , with genabel s gwfgls , and gels - gwas , based on lapack s gels routine . the results are displayed in fig . [ fig : oney ] . as expected , qr - gwas and chol - gwas attain the same performance and overlap . most interestingly , our algorithms clearly outperform gels - gwas and gwfgls , obtaining speedups of 4 and 8 , respectively . , @xmath83 , @xmath81 . the improvement in the performance of our algorithms is due to a careful exploitation of both the properties of the operands and the sequence of gls problems . ] next , we present an even more interesting result . the current approach of all state - of - the - art libraries to the case @xmath82 is to repeat the experiment @xmath14 times with the same algorithm used for @xmath79 . on the contrary , our compiler generates the algorithm eig - gwas , which particularly suits such scenario . as fig . [ fig : manyy ] illustrates , eig - gwas outperforms the best algorithm for the case @xmath79 , chol - gwas , by a factor of 4 , and therefore outperforms gels - gwas and gwfgls by a factor of 16 and 32 respectively . , @xmath83 , @xmath84 . chol - gwas is best suited for the scenario @xmath79 , while eig - gwas is best suited for the scenario @xmath85 . ] the results remark two significant facts : 1 ) the exploitation of domain - specific knowledge may lead to improvements in state - of - the - art algorithms ; and 2 ) the library user may benefit from the existence of multiple algorithms , each matching a given scenario better than the others . in the case of gwas our compiler achieves both , enabling computational biologists to target larger experiments while reducing the execution time . we presented a linear algebra compiler that automatically exploits domain - specific knowledge to generate high - performance algorithms . our linear algebra compiler mimics the reasoning of a human expert to , similar to a traditional compiler , decompose a target equation into a sequence of library - supported building blocks . the compiler builds on a number of modules to support the replication of human reasoning . among them , the _ matrix algebra _ module , which enables the compiler to freely manipulate and simplify algebraic expressions , and the _ properties inference _ module , which is able to infer properties of complex expressions from the properties of the operands . the potential of the compiler is shown by means of its application to the challenging _ genome - wide association study _ equation . several of the dozens of algorithms produced by our compiler , when compared to state - of - the - art ones , obtain n - fold speedups . as future work we plan an extension to the _ code generation _ module to support fortran . also , the asymptotic operation count is only a preliminary approach to estimate the performance of the generated algorithms . there is the need for a more robust metric to suggest a `` best '' algorithm for a given scenario . the authors gratefully acknowledge the support received from the deutsche forschungsgemeinschaft ( german research association ) through grant gsc 111 . bientinesi , p. , eijkhout , v. , kim , k. , kurtz , j. , van de geijn , r. : sparse direct factorizations through unassembled hyper - matrices . computer methods in applied mechanics and engineering * 199 * ( 2010 ) 430438 anderson , e. , bai , z. , bischof , c. , blackford , s. , demmel , j. , dongarra , j. , du croz , j. , greenbaum , a. , hammarling , s. , mckenney , a. , sorensen , d. : users guide . third edn . society for industrial and applied mathematics , philadelphia , pa ( 1999 ) pschel , m. , moura , j.m.f . , johnson , j. , padua , d. , veloso , m. , singer , b. , xiong , j. , franchetti , f. , gacic , a. , voronenko , y. , chen , k. , johnson , r.w . , rizzolo , n. : : code generation for dsp transforms . proceedings of the ieee , special issue on `` program generation , optimization , and adaptation '' * 93*(2 ) ( 2005 ) 232 275 baumgartner , g. , auer , a. , bernholdt , d.e . , bibireata , a. , choppella , v. , cociorva , d. , gao , x. , harrison , r.j . , hirata , s. , krishnamoorthy , s. , krishnan , s. , chung lam , c. , lu , q. , nooijen , m. , pitzer , r.m . , ramanujam , j. , sadayappan , p. , sibiryakov , a. , bernholdt , d.e . , bibireata , a. , cociorva , d. , gao , x. , krishnamoorthy , s. , krishnan , s. : synthesis of high - performance parallel programs for a class of ab initio quantum chemistry models . in : proceedings of the ieee . ( 2005 ) 2005 fabregat - traver , d. , bientinesi , p. : knowledge - based automatic generation of partitioned matrix expressions . in gerdt , v. , koepf , w. , mayr , e. , vorozhtsov , e. , eds . : computer algebra in scientific computing . volume 6885 of lecture notes in computer science . , springer berlin / heidelberg ( 2011 ) 144157 fabregat - traver , d. , bientinesi , p. : automatic generation of loop - invariants for matrix operations . in : computational science and its applications , international conference , los alamitos , ca , usa , ieee computer society ( 2011 ) 8292 fabregat - traver , d. , aulchenko , y.s . , bientinesi , p. : fast and scalable algorithms for genome studies . technical report , aachen institute for advanced study in computational engineering science ( 2012 ) available at http://www.aices.rwth-aachen.de:8080/aices/preprint/documents/aices-2012-05-01.pdf .	we present a prototypical linear algebra compiler that automatically exploits domain - specific knowledge to generate high - performance algorithms . the input to the compiler is a target equation together with knowledge of both the structure of the problem and the properties of the operands . the output is a variety of high - performance algorithms , and the corresponding source code , to solve the target equation . our approach consists in the decomposition of the input equation into a sequence of library - supported kernels . since in general such a decomposition is not unique , our compiler returns not one but a number of algorithms . the potential of the compiler is shown by means of its application to a challenging equation arising within the _ genome - wide association study_. as a result , the compiler produces multiple `` best '' algorithms that outperform the best existing libraries .	10,585	198
supersymmetry is a well - motivated way to extend the standard model ( sm ) . most impressively , supersymmetry can stabilize the large disparity between the size of the electroweak scale and the planck scale @xcite . in addition , the minimal supersymmetric extension of the sm @xcite , the mssm , leads to an excellent unification of the @xmath1 , @xmath2 , and @xmath3 gauge couplings @xcite near @xmath4 gev , a scale that is large enough that grand - unified theory ( gut ) induced nucleon decay is not a fatal problem . the mssm also contains a new stable particle if @xmath5-parity is an exact symmetry . this new stable particle can potentially make up the dark matter . the main obstacles facing supersymmetric extensions of the sm come from the requirement that supersymmetry be ( softly ) broken . to preserve the natural supersymmetric hierarchy between @xmath6 and @xmath7 , every mssm operator that breaks supersymmetry should be accompanied by a dimensionful coupling of size less than about a tev . however , for a generic set of soft terms of this size , consistent with all the symmetries of the theory , the amount of flavor mixing and cp violation predicted by the model is much greater than has been observed . instead , the experimental constraints require that the soft terms be nearly flavor - diagonal in the super - ckm basis @xcite , and that nearly all the independent cp violating phases be very small @xcite , or finely - tuned to cancel @xcite . from the low - energy perspective , it is not clear why this should be so . a number of approaches to the supersymmetric flavor and cp problems have been put forward , such as adding new flavor symmetries to the mssm @xcite , or mediating supersymmetry breaking through gauge interactions @xcite or the superconformal anomaly @xcite . these models also face new difficulties . new flavor symmetries typically require additional matter fields and hence the complications that go with them . gauge mediation generates flavor - universal soft masses and trivially small @xmath8 terms , but does not fully solve the cp problem , and makes it difficult to generate both the @xmath9 and @xmath10 terms with the correct size . anomaly mediation in its most simple form suffers from tachyonic slepton soft masses @xcite . a more radical approach to the flavor and cp problems is to push the scale of the soft supersymmetry breaking scalar couplings to be much larger than the electroweak scale @xcite . if this is done while keeping the gauginos relatively light , it is possible to preserve gauge unification and a good dark matter candidate @xcite . of course , supersymmetry would no longer directly solve the gauge hierarchy problem if the scalar superpartners are very heavy . another way to address the supersymmetric flavor problem , and the one we consider in the present work , is to have the soft scalar masses and the @xmath8 terms vanish simultaneously at a scale @xmath0 @xcite . if this scale is much larger than the electroweak scale , and if the gaugino masses do not vanish at @xmath0 , non - zero values for the scalar soft terms will be generated by radiative effects as the theory is evolved to lower energies . since the scalar soft terms thus induced are family - universal , the resulting soft spectrum does not have a flavor problem . the supersymmetric cp problem is also improved but not solved by this approach . besides the ckm phases , the only remaining phases are those of the gaugino soft masses , and the @xmath9 and @xmath10 terms . if the gaugino soft mass phases are universal , there are three new phases of which two can be removed by making field redefinitions @xcite . the remaining phase can be eliminated as well within particular models @xcite . near - vanishing soft scalar terms can arise in a number of ways . the canonical examples are the no - scale models of gravity mediated supersymmetry breaking . in these models , the absence of scalar soft terms is related to the flatness of the hidden sector potential that allows the gravitino mass to be determined by loop - corrections due to light fields @xcite . a more recent construction that leads to near - vanishing soft scalar operators is gaugino - mediated supersymmetry breaking @xcite . here , the mssm chiral multiplets are separated from the source of supersymmetry breaking by an extra - dimensional bulk , while the gauge multiplets propagate in the bulk . locality in the extra dimension(s ) leads to gaugino masses that are much larger than the scalar soft terms . small scalar soft terms can also be obtained from strong conformal dynamics in either the visible or the hidden sectors . in these constructions , the conformal running suppresses the scalar soft terms exponentially relative to the gaugino soft masses @xcite . the main difficulty with very small input scalar soft masses is that the lightest sm superpartner particle is usually a mostly right - handed slepton , which can be problematic for cosmology . this is nearly always the case if gaugino universality is assumed to hold above @xmath11 , and @xmath12 . on the other hand , if @xmath0 is an order of magnitude or more above @xmath11 ( with gaugino universality ) , the lightest superpartner becomes a mostly bino neutralino . a viable low - energy spectrum can be obtained in this way @xcite . for @xmath13 , a neutralino lsp can be obtained by relaxing the requirement that all soft scalar terms vanish at @xmath0 . one such generalization that does not re - introduce a flavor problem is to allow the higgs soft masses @xmath14 and @xmath15 to be non - zero at @xmath0 . these soft masses contribute to the running of the slepton masses through a hypercharge fayet - iliopoulos @xmath16-term , and can push the slepton masses above that of the lightest neutralino @xcite . in the present work , we study the phenomenology of the mssm subject to vanishing scalar soft terms . we generalize our study by including non - vanishing higgs boson soft masses , as well as a higgs boson bilinear @xmath17 term . this does not reintroduce a flavor problem . inspired by grand - unified theories , we take our input scale to be @xmath11 , and demand universal gaugino masses at this scale . after imposing consistent electroweak symmetry breaking , the independent parameters of this theory , which we call higgs exempt no - scale ( hens ) supersymmetry , are m_1/2 , , m_h_u^2 , m_h_d^2 , sgn ( ) , where @xmath18 is the universal gaugino mass at @xmath11 , @xmath19 is the ratio of the higgs boson expectation values , @xmath14 and @xmath15 are the soft higgs masses at @xmath11 , and @xmath20 is the sign of the supersymmetric higgs bilinear @xmath9 term . the outline of this paper is as follows . in section [ theo ] we motivate this scenario , and show how it can emerge in a number of different ways . section [ mass ] discusses the low - energy mass spectrum of the model , the acceptable regions of parameter space , and the most important constraints on these regions . in section [ dark ] we investigate the prospects for dark matter in the model , and discuss some of the potential signatures this might induce . section [ coll ] contains an investigation of some of the potential collider signatures of the model at the tevatron and the lhc , as well as a discussion of the discovery prospects at these machines . finally , section [ conc ] is reserved for our conclusions . while this work was in preparation , we became aware of refs . @xcite , which investigate the mass spectrum and dark matter prospects of gaugino mediation with unsuppressed higgs soft masses . there is some overlap between their work and the material in sections [ mass ] and [ dark ] of this paper . however , unlike refs @xcite , we allow negative higgs soft squared masses at the input scale @xcite , and have a more extensive discussion of the phenomenological constraints . on the other hand , they consider specific details of gaugino mediated models , and discuss the possibility of gravitino dark matter in more detail than we do . our primary motivation for considering models with small scalar soft terms is data driven : they provide a simple and elegant reason for small flavor violations induced by supersymmetry . even so , it is comforting that this framework can arise from a number of theoretical constructions . in this section we describe some of these models , and discuss how they can be modified to allow for non - vanishing soft masses for the higgs fields . vanishing scalar soft terms have traditionally been associated with no - scale models @xcite . these models are attractive because the gravitino mass , and therefore the scale of supersymmetry breaking , is determined dynamically . the basic assumption underlying no - scale constructions is that the effective superspace khler density and superpotential have the form @xcite & = & -3m_pl^2 + f(x ) + f^(x^ ) + g(,^),[g - noscale ] + & = & w(),where @xmath21 is a hidden sector field , @xmath22 represents a visible sector field , and @xmath23 is a holomorphic cubic function . with this form of the khler density and superpotential , the tree - level potential along the direction of the hidden sector field @xmath21 is flat , the gravitino mass , @xmath24 is undetermined , and no soft terms are generated for the visible sector scalars . supersymmetry breaking is communicated to the visible sector by non - trivial @xmath21-dependent gauge kinetic functions , which generate gaugino soft masses on the order of @xmath24 . the one - loop corrections from the gauginos lift the potential in the @xmath21 direction and fix the value of @xmath24 to lie close to the electroweak scale , which is determined dynamically by the large top yukawa coupling @xcite . it is difficult to maintain @xmath25 in no - scale models if there are other larger scales in the theory because of the radiative corrections these contribute to the effective potential for @xmath24 @xcite . if such large scales exist , such as in a gut , the heavy sector must be completely sequestered from the supersymmetry breaking , since in the supersymmetric limit , they do not alter the effective potential . within a gut where the higgs superfields are components of complete multiplets that also contain heavy fields , it is therefore essential to prevent these gut multiplets from obtaining a supersymmetry breaking mass . having separated the higgs in this way , it is natural to sequester the other chiral multiplets as well . this is the origin of the vanishing scalar soft terms in no - scale models . gaugino masses can be induced by a non - minimal kinetic function for those and only those components of the gut vector multiplet that remain light . thus if the higgs multiplets are components of a larger gut multiplet , of which some components develop gut scale masses , it is not possible to generate soft masses for the higgs fields without destabilizing @xmath24 . on the other hand , soft higgs masses might be possible in more general unification scenarios in which the higgs fields do not belong to complete gut multiplets @xcite . the form of the no - scale khler density and superpotential , eq . ( [ g - noscale ] ) , is an input to these models . such a form does arise to lowest order in several string- and m - theory constructions , but is typically corrected at higher orders @xcite . more generally , a superspace khler density in which the visible and hidden sectors appear as disjoint terms , as in eq . ( [ g - noscale ] ) , is said to be _ sequestered _ @xcite . a sequestered khler density and superpotential guarantees that no direct soft terms are generated . this is a necessary ingredient for anomaly mediation @xcite . complete sequestering can be obtained geometrically by confining the visible and hidden sectors to branes separated by an extra - dimensional bulk @xcite . a partial sequestration can also be realized if the mssm gauge multiplets are allowed to propagate in the bulk , as in gaugino mediated supersymmetry breaking @xcite . if so , the gauginos will develop soft masses through their local couplings to the hidden sector , while the soft terms of the chiral multiplets will only be generated by loops passing across the bulk . the resulting supersymmetric spectrum at the compactification scale of the extra dimension is therefore close to the one we are interested in : non - zero gaugino masses , and much smaller soft scalar terms . by allowing the higgs multiplets to propagate in the bulk , non - zero higgs soft terms can be generated as well . this is perhaps the simplest way to realize the hens models we shall consider . sequestering can also be realized in four dimensions through strongly - coupled conformal dynamics in the hidden sector @xcite . even without strong conformal running , the contributions to the gaugino masses and the trilinear @xmath8 are naturally suppressed relative to the gravitino mass if there are no singlets in the hidden sector . this is not true for the scalar soft masses , which are generated by terms like d^4^_i_j x^x _ i^_j , [ softmass ] where @xmath26 is the messenger scale , @xmath21 is a hidden sector field , and @xmath27 is a visible field . operators of the form of eq . ( [ softmass ] ) can be suppressed relative to the gravitino mass by strong conformal dynamics in the hidden sector that couples at the renormalizable level to @xmath21 @xcite . if all such soft mass contributions are sufficiently suppressed , and if there are no hidden sector singlets , the visible sector is sequestered and the leading contribution to the soft terms comes from anomaly mediation . by allowing singlets in the hidden sector , both gaugino masses and @xmath8 terms can be generated that are much larger than the anomaly - mediated terms @xcite . this is close to , but not quite , the spectrum we are interested in , because it is unclear how to avoid suppressing the higgs soft masses while squashing the rest . a partial sequestration of soft terms , as well as an explanation for the yukawa hierarchy , can also be obtained from strong conformal dynamics in the visible sector @xcite . in these constructions , there is a new gauge group @xmath28 that approaches a strongly - coupled fixed point in the ir . the mssm fields are not charged under @xmath28 , but they do couple to fields that are through cubic operators in the superpotential . as the theory flows towards the fixed point , the mssm fields develop large anomalous dimensions which suppress their corresponding ( physical ) yukawa couplings . since different ( linear combinations of ) fields develop distinct anomalous dimensions , related to their effective superconformal @xmath5 charges , a yukawa hierarchy can be generated in this way @xcite . the conformal running also produces a general suppression of the soft scalar masses , as well as a hierarchy of trilinear @xmath8 terms that mirrors the yukawa couplings @xcite . conversely , the gaugino masses are largely unaffected because they do not couple directly to the strongly - coupled sector . the third generation multiplets and the higgs multiplets must also be shielded from the conformal running effects to avoid suppressing the top quark yukawa coupling . as a result , the third generation and the higgs soft masses do not get suppressed . thus , the spectrum from visible sector conformal running is similar to one we shall consider , but augmented by third generation soft masses and @xmath8 terms . we expect the phenomenology of both scenarios to be similar over much of the allowed parameter space . from the discussion above , we see that a hens soft mass spectrum can arise from gaugino mediation with the higgs multiplets in the bulk , or from conformal running in the hidden sector up to additional contributions to the third generation states . before proceeding , however , let us comment on our choice of @xmath11 as the input scale for the soft spectrum . in gaugino mediation , the input scale is on the order of the compactification scale , @xmath29 . , is less than the cutoff of the theory . ] for visible - sector conformal running , the input scale for the soft spectrum is the scale at which the conformal running ceases , which we will also call @xmath29 . our motivation to set @xmath30 is partly conventional , but is also motivated by gauge unification and our wish to strongly suppress the scalar soft masses . in both cases , gauge unification can be preserved with @xmath31 , but the process will be more complicated than in the standard picture . in gaugino mediation , kaluza - klein states appear above @xmath29 and can lead to an accelerated power - law running @xcite . with conformal dynamics in the visible sector , the sm gauge coupling beta functions will be modified by the large anomalous dimensions of the mssm fields . gauge unification will still occur , albeit at a lower scale , provided the conformal dynamics respects a global symmetry into which the sm gauge group can be embedded @xcite . thus , in each case having @xmath29 below @xmath11 can induce an effective unification of the sm gauge couplings below the apparent unification scale @xmath32 . this is problematic for many gut completions of the mssm , which predict baryon and lepton number violation . typically , some additional structure is needed if @xmath29 is much smaller than @xmath11 . this motivates us to consider @xmath33 . it is clear that gauge unification can also be maintained with @xmath34 . if @xmath29 is larger than @xmath11 , the renormalization group running from @xmath29 down to @xmath11 will induce non - vanishing ( flavor - universal ) soft masses at @xmath11 . the size of these corrections from running above @xmath11 depends on the precise @xmath35 completion of the theory , but even for minimal @xmath35 models they can be significant , on the order of @xcite a & & c_a ( ) m_1/2,[abovemg ] + m^2 & & c_m^2 ( ) m_1/2 ^ 2,where @xmath36 is the gut coupling , and @xmath37 and @xmath38 are dimensionless constants on the order of or slightly larger than unity . these contributions can be large enough for an acceptable low - energy spectrum to be obtained @xcite . on the other hand , in both gaugino mediation and conformal sequestering , @xmath29 can not be more than about an order of magnitude above @xmath11 because the suppression of soft terms ( and yukawa couplings ) requires a separation of scales . let @xmath39 be the scale at which conformal running begins in the case of conformal dynamics , or the @xmath40 cutoff of the extra - dimensional gauge theory in gaugino mediation . presumably @xmath41 . the amount of suppression of the soft scalar terms from conformal running is expected to be an order - one power of @xmath42 , whereas the required suppression is typically on the order of @xmath43 @xcite . an even stronger upper bound on @xmath0 can be obtained if the conformal dynamics are responsible for the small electron yukawa coupling as in ref . the condition for this is y_e ( ) ( ) ^(_l+_e)/2 , where @xmath44 denote the anomalous dimensions of @xmath45 and @xmath46 , which are generally smaller than 2 . if we take this bound seriously , @xmath29 can be at most only slightly larger than @xmath11 . in gaugino mediation , flavor - mixing contact interactions between the mssm chiral multiplets and hidden sector operators , arising from bulk states with masses above the uv cutoff scale , are suppressed by a factor of @xmath47 @xcite . again this factor must be less than about @xmath43 to avoid various experimental flavor constraints , which translates into @xmath0 being within an order of magnitude larger than @xmath11 ( for @xmath48 ) . given the above considerations , we will set @xmath30 throughout this paper , and not concern ourselves with the precise mechanism by which the scalar soft terms are suppressed . while beyond the scope of the present work , it also interesting to speculate that the breaking of the gut symmetry is related to the geometry of the extra dimension , or the escape from conformal running . such a construction would further justify our choice of @xmath30 . finally , let us also note that within particular models there is typically some residual flavor violation due to an incomplete suppression of the scalar terms at @xmath29 . the amount of flavor suppression can be close to the level probed by current experiments . however , without specifying a particular model , it is not possible to perform an analysis of the constraints due to flavor physics . thus , we assume as our starting point that at the scale @xmath49 , all scalar masses except those of the higgs bosons are precisely zero and that corrections to that assumption are inconsequential to the phenomenology discussed below . the essential features of the hens mass spectrum are well illustrated by a simple one - loop analysis . at this order , the ratio @xmath50 , @xmath51 , is scale invariant for all three gaugino masses . if the gauge couplings unify and the gaugino masses are universal at @xmath11 , it follows that at lower scales q , @xmath52 ^ 2 m_{1/2}$ ] . for @xmath53 , this gives m_1 ( 0.43)m_1/2 , m_2 ( 0.83)m_1/2 , m_3 ( 2.6)m_1/2.[2:m12 ] the one - loop running of the scalar soft masses is given by @xcite ( 4)^2 x_i -8_ac^a_ig_a^2\|m_a\|^2 + g_1 ^ 2y_is , [ 2:msoftrun ] where @xmath54 depends on the soft masses and @xmath8 terms and is usually proportional to yukawa couplings , @xmath55 is the quadratic casimir for the representation @xmath56 under gauge group @xmath57 , and s = ( m_h_u^2-m_h_d^2 ) + tr_f(m_q^2 - 2m_u^2+m_e^2+m_d^2-m_l^2 ) , [ 2:sterm ] with the trace above running over flavors . at one - loop order , the rg equation for the @xmath58 term is particularly simple , ( 4)^2 = g_1 ^ 2s . because of this simple form , the effect of the @xmath58-term on the low - scale soft masses is to simply shift the value they would have with @xmath59 by the amount m_i^2 = -s_gut -(0.052)y_is_gut , where @xmath60 evaluated at @xmath11 . neglecting yukawa effects , the low - scale slepton soft masses at @xmath61 are m_l^2 & & ^2 + ( 0.052)s_gut , [ 2:mslepton ] + m_e^2 & & ^2 - ( 0.052)s_gut . if mixing effects are small , the physical slepton masses will be close to @xmath62 and @xmath63 , up to the @xmath3 @xmath16-term contributions . the mass of the lightest neutralino is usually close to @xmath64 ( under the assumption of gaugino universality ) unless @xmath9 is relatively small . comparing eq . with eq . , we see that for @xmath65 , @xmath66 is less than @xmath64 and the lightest superpartner tends to be a mostly right - handed slepton . on the other hand , if @xmath67 , the right - handed slepton soft mass is pushed up relative to @xmath64 , allowing for a mostly bino neutralino lsp . for @xmath68 very large and negative , the lsp can be a mostly left - handed slepton . relative to the sleptons and the electroweak gauginos , the squarks and gluino are very heavy because the @xmath1 gauge coupling grows large in the infrared . to confirm the simple analysis given above , we have performed a scan over the hens parameter space using suspect 2.34 @xcite . this code performs the renormalization group running at two - loop order with one - loop threshold effects , and includes radiative and mixing corrections to the physical particle masses . we take @xmath69 @xcite and @xmath70 gev @xcite in our analysis . for each model parameter point we require consistent electroweak symmetry breaking , and superpartner masses above the lep ii and tevatron bounds ( @xmath71 gev , @xmath72 , @xmath73 ) . we also impose the lower - energy constraints & & [ -8,24]10 ^ -4 @xcite + br(bs ) & & [ 3.0,4.0]10 ^ -4 @xcite + a _ & & [ -5,50]10 ^ -10 @xcitethese ranges correspond approximately to the @xmath74 allowed values , although we have allowed a slightly larger range for @xmath75 . the constraint from the muon magnetic moment is particularly interesting in hens scenarios , and we shall discuss it more extensively below . in the immediate analysis we do not include the lep ii bound on the lightest higgs boson mass . we will discuss this constraint below as well . and the differently colored regions in the figure indicate the identity of the lightest superpartner . the quantities @xmath14 and @xmath15 are evaluated at the input scale @xmath11.,scaledwidth=70.0% ] and @xmath77 . the differently colored regions in the figure indicate the identity of the lightest superpartner . the quantities @xmath14 and @xmath15 are evaluated at the input scale @xmath11.,scaledwidth=70.0% ] and @xmath77 . the differently colored regions in the figure indicate the identity of the lightest superpartner . the quantities @xmath14 and @xmath15 are evaluated at the input scale @xmath11.,scaledwidth=70.0% ] figures [ lsp-10 - 300 ] , [ lsp-10 - 500 ] , and [ lsp-30 - 500 ] show the allowed regions of @xmath78 and @xmath79 , for @xmath80 equal to @xmath81 , @xmath82 , and @xmath83 , subject to the constraints described above . the soft higgs masses in these plots are re - expressed in terms of the more convenient combinations @xmath84 , and @xmath85 , where @xmath86 denotes the signed square root ( @xmath87 ) . also shown in these plots is the identity of the lightest superpartner at each allowed parameter point . these figures confirm our previous approximate analysis . when @xmath68 is positive or zero , the lsp is a mostly right - handed stau or selectron . as @xmath68 becomes more negative , a neutralino becomes the lsp , while for very large and negative values of @xmath68 the lsp is a sneutrino . for extremely large positive or negative values of @xmath68 , one of the slepton soft masses becomes tachyonic . the allowed parameter region is cut off at larger positive values of @xmath88 because @xmath89 only has a negative solution , implying that electroweak symmetry breaking is not possible . reaches zero by the @xmath90 and the chargino mass constraints . ] note that in fig . [ lsp-10 - 500 ] , there is a thin strip along the upper border of the allowed region in which the lsp is a neutralino . in this strip , the @xmath9 term is smaller than @xmath64 and the neutralino lsp is mostly higgsino . for larger negative values of @xmath88 , @xmath91 and the parameter space gets cut off by the bound from @xmath90 . as @xmath88 becomes even smaller , electroweak symmetry breaking ceases to occur . the effects of the @xmath92 yukawa coupling and left - right mixing can be seen by comparing figs . [ lsp-10 - 500 ] and [ lsp-30 - 500 ] . in the models we are considering , the value of the yukawa - dependent term in eq . for the right - handed stau soft mass is x_e_3 2\|y_\|^2m_h_d^2 . the left - right mixing is also proportional to the @xmath92 yukawa . as @xmath19 increases , so too does the @xmath92 yukawa , and therefore also the yukawa effect on the running and the mixing . left - right mixing tends to push the lighter stau mass lower , and for this reason it is more difficult to obtain a neutralino lsp at larger values of @xmath19 . however , there is also a competing effect from the influence of the @xmath92 yukawa on the running of @xmath93 . when @xmath15 is large and negative , the @xmath94 term increases the value of @xmath93 at low energies . thus , a selectron or an electron sneutrino is the lsp in some parts of the parameter space . in parts of the parameter space shown in figs . [ lsp-10 - 300]-[lsp-30 - 500 ] the value of @xmath14 is large and negative , while the slepton soft masses are considerably smaller in magnitude ( but positive ) . in these regions , it is likely that the standard mssm minimum is only metastable , and that a deeper charge - breaking minimum exists at large field values @xcite . the precise constraints on the existence of such non - standard global minima depend on the details of the thermal history of the universe @xcite , and we do not investigate them in the present work . however , a necessary condition is that the lifetime of the metastable mssm vacuum at @xmath95 should be greater than the age of the universe . while beyond the scope of the present work , it is possible that some regions of the parameter space shown in figs . [ lsp-10 - 300]-[lsp-30 - 500 ] , especially the lower left region and at larger values of @xmath19 , may not be populated after a more detailed analysis . since the sleptons in hens models are relatively light , the corrections to the anomalous magnetic moment of the muon , @xmath97 can be significant @xcite . currently , the measured value of @xmath98 exceeds the sm prediction by about two standard deviations @xcite , a _ = a_^exp - a_^sm = ( 2210)10 ^ -10 . this result is suggestive of new physics . in the mssm , there are additional contributions to @xmath99 from loops involving a virtual chargino and muon sneutrino , and loops with a virtual neutralino and smuon . for the hens scenarios we are studying , in which all masses scale predominantly with @xmath18 and the gaugino masses are universal ( and assumed real and positive ) , the leading supersymmetry contribution to @xmath98 is proportional to @xmath19 , scales roughly as @xmath100 , and has a sign equal to the sign of the @xmath9 term , @xmath20 @xcite . given the tension between the measured value of @xmath75 and the sm prediction , @xmath101 is strongly favored . indeed , we find that negative @xmath20 is only possible for very large values of @xmath18 . conversely , if @xmath20 is positive the new supersymmetric contribution can help to explain this possible discrepancy between the sm prediction and experiment . as a function of @xmath18 for several ranges of @xmath19 , and @xmath101 . the spread of points come from scanning over the acceptable input values of @xmath14 and @xmath15 . the red points indicate @xmath102 , the green points @xmath103 , the blue points @xmath104 , and the magenta points @xmath105 . , scaledwidth=70.0% ] the value of @xmath106 is shown as a function of @xmath18 in fig . [ g2-mu ] . in generating this figure , we have taken @xmath101 , and have scanned over input values of @xmath14 and @xmath15 at @xmath11 . the distribution for @xmath107 looks the same , except the sign of @xmath108 is opposite . with @xmath109 , the new physics contribution is frequently too large , and from this we obtain a lower bound on @xmath18 as a function of @xmath19 . for @xmath110 , this bound is @xmath111 , while for @xmath112 , it increases to @xmath113 . plane of solutions that respect the bounds of @xmath114 and @xmath115 . due to uncertainty in the top quark mass , and the theoretical uncertainty in the computation of @xmath116 , a more conservative constraint on this theoretically computed value of @xmath116 is @xmath117 , which is also shown in the figure.,scaledwidth=70.0% ] a further constraint on hens models , and one we have not yet imposed , is that the sm higgs boson mass should exceed the lep ii bound @xcite , m_h > 114.4 . this bound also applies to the lightest cp - even higgs boson in much of the parameter space of the mssm . however at tree - level in the mssm , the lightest cp - even higgs boson has a mass below @xmath118 . it is only because of large loop corrections to the mass , predominantly due to the scalar tops , that this higgs state can be raised above the lep ii bound . with vanishing input scalar soft masses , the stop masses scale with @xmath18 . the higgs boson mass bound therefore imposes a further lower bound on the universal input gaugino mass . the combined bounds on @xmath18 as a function of @xmath19 from the conditions @xmath119 and @xmath120 gev are shown in fig . [ tanb - m12 ] . we impose a slightly weaker @xmath117 lower bound on the higgs boson mass than the @xmath121 lep ii bound to account for various uncertainties associated with the theoretical computation of @xmath116 . we have taken @xmath122 in our analysis . at smaller @xmath19 , less than about 15 , the higgs mass bound imposes the stronger constraint , while the upper bound on @xmath75 is more significant for values of @xmath123 . for any value of @xmath19 , @xmath18 must be greater than about @xmath124 gev if we impose the weaker higgs mass bound ( @xmath125 gev ) , and larger than about @xmath126 gev to satisfy the stronger bound ( @xmath127 gev ) . note that as @xmath18 grows , the phenomenological constraints on the model tend to weaken , but usually at the cost of increased fine - tuning in the higgs sector @xcite . in the previous section , we found that in a large region of the hens parameter space a slepton or a sneutrino is the lsp . such an lsp can be problematic for cosmology . if the lsp is a charged slepton , the very strong constraints that exist for charged stable particles imply that such a scenario is all but ruled out @xcite . these bounds do not apply to a sneutrino lsp , but in this case the direct detection rate is much too high ( for a mass of @xmath128 ) @xcite . without invoking some additional and interesting mechanisms that allow the sneutrino as a more viable lsp @xcite , we will focus on the case of a neutralino lsp . also , a slepton as the lightest superpartner particle , charged or not , may still provide a consistent picture of dark matter if the gravitino is the lsp @xcite . we will comment briefly on that possibility later . we investigate the relic density of neutralino lsps at various points in the allowed parameter space with the aid of darksusy 4.1 @xcite . this computer program performs a fully relativistic computation of the relic density , and includes all relevant coannhilation channels . figures [ 10 - 300 ] and [ 10 - 500 ] show the neutralino lsp relic density for @xmath110 , @xmath129 gev and @xmath126 gev , and @xmath130 for the full range of allowed input values of @xmath14 and @xmath15 at @xmath49 . in both figures the black plus signs indicate the regions in which the lightest neutralino is not the lsp , but that are otherwise allowed . the red triangles correspond to parameter points where the neutralino relic density is acceptably small , @xmath131 . this is to be compared with the observed dark matter density @xcite , h^2 = 0.1045^+0.0072_-0.0095 ( ) . in the blue , green , and magenta regions , the neutralino relic density exceeds @xmath132 . these regions can still be consistent with the wmap measurements if there is a late - time injection of entropy into the universe after the neutralinos have frozen out @xcite . , @xmath129 gev , and @xmath133 . the region in which the lightest neutralino is not the lsp is denoted by the black plus signs . the red triangles indicate parameter points where the neutralino lsp relic density is less than @xmath131 . in the blue and green regions , the neutralino lsp relic density exceeds this value.,scaledwidth=70.0% ] , @xmath134 gev , and @xmath133 . the region in which the lightest neutralino is not the lsp is denoted by the black plus signs . the red triangles indicate parameter points where the neutralino lsp relic density is less than @xmath131 . in the blue , green , and magenta regions , the neutralino lsp relic density exceeds this value.,scaledwidth=70.0% ] the shape of the neutralino relic density contours in figs . [ 10 - 300]-[10 - 500 ] can be understood in terms of the mass spectrum of the model . in most of the parameter space , the lightest neutralino is predominantly bino . the main annihilation channels in this bulk region are @xmath135-channel slepton exchanges . this is not efficient enough to reduce the relic density to an acceptable level for the values of @xmath18 that are relevant , although it does come close for @xmath129 gev . along the left and right edges of the neutralino region , the mass of the neutralino lsp approaches that of the lightest slepton . this near degeneracy allows for coannihilation between the neutralino lsp and the lightest slepton to become effective , pushing the neutralino relic density to a value well below the wmap value . moving away from these edges towards the bulk region , the coannihilation efficiency falls off quickly , roughly as @xmath136 $ ] , and the neutralino density goes up . in this transitional region , where slepton coannihilation is only moderately efficient , the correct dark matter density is obtained . the strip of low relic density along the top of the neutralino lsp region arises because the @xmath9 parameter becomes small . in this strip , the neutralino lsp develops a significant higgsino component allowing it to annihilate effectively through gauge bosons , and by coannihilation with the lightest chargino . , @xmath134 gev , and @xmath133 . the region in which the lightest neutralino is not the lsp is denoted by the black plus signs . the red triangles indicate parameter points where the neutralino lsp relic density is less than @xmath131 . in the blue , green , and magenta regions , the neutralino lsp relic density exceeds this value.,scaledwidth=70.0% ] in the regions where the lightest neutralino is not the lightest sm superpartner ( denoted by black plus signs in figs . [ 10 - 300]-[10 - 500 ] ) the lightest superpartner is always a slepton . on the left - hand side of the neutralino region , this particle is either a tau or an electron sneutrino , as illustrated in figs . [ lsp-10 - 300]-[lsp-30 - 500 ] . on the right , the lightest superpartner is a mostly right - handed stau or selectron . the annihilation of sleptons is very efficient in the early universe . if such a particle were stable , the relic density in the regions discussed above would be on the order of @xmath137 . a relic density of heavy charged particles of this size is firmly ruled out by direct searches @xcite . even for a stable sneutrino , a relic density of this size is ruled out by dark matter direct detection searches @xcite . a charged or neutral slepton lighter than the lightest neutralino may still be acceptable provided the gravitino is the true lsp . since the gravitino couples very weakly to the mssm states , the slepton nlsp would freeze out as if it were the lsp , and decay into gravitinos at a much later time . the final gravitino density produced by these decays is determined by the quasi - stable nlsp density , @xmath138 , through the relation @xcite _ 3/2^decay h^2 = _ h^2 , where @xmath24 is the gravitino mass . in the parameter regions we are considering , this density is too small to account for all the dark matter if nlsp decays are the only source of relic gravitinos . however , there are other possible sources of relic gravitinos , such as thermal production after inflationary reheating @xcite and non - thermal production through heavy particle decays @xcite that can bring the total gravitino relic density up to the value needed to explain all the dark matter . let us also note that this scenario is constrained by the requirement that the late - time decays not overly disrupt the predictions of big - bang nucleosynthesis or the black - body spectrum of the cosmic microwave background radiation . these constraints are relatively weak for a sneutrino nlsp , but they are quite severe for a charged slepton ( or neutralino ) nlsp , and require @xmath139 mev for most of the range of nlsp masses we are considering @xcite . we have also examined the effect of varying @xmath19 on the predictions for dark matter within the model . smaller values of this ratio , @xmath140 , do not change the qualitative features of the picture described above . for @xmath141 , there are a couple of important changes . most importantly , larger values of @xmath19 induce more mixing between the left- and right - handed staus , which has the effect of lowering the mass of the lightest stau . because of this , for @xmath112 and @xmath134 gev there is no longer an acceptable region of parameter space in which the lightest neutralino is both the lightest superpartner , and mostly higgsino . in fig . [ 30 - 500 ] we plot the neutralino lsp relic density for @xmath142 , @xmath143 , and @xmath133 . the thin slice of acceptable @xmath144 arises due to the coannihilation of the lsp with a light slepton . the prospects for direct and indirect detection of neutralino dark matter were investigated using darksusy 4.1 @xcite . for the most part , the direct and indirect detection signals within our scenario are very similar to those of a generic msugra model with a mostly bino lsp @xcite . in general , these potential signals are very weak . however , much stronger signals can arise when the neutralino lsp has a significant higgsino component . and @xmath134 gev . the black plus signs indicate the parameter region that is consistent with various phenomenological constraints , but need not have a neutralino lsp . the red , green , blue , magenta points indicate parameter points for which a neutralino is the lsp , for several ranges of @xmath145.,scaledwidth=70.0% ] dark matter in the local halo can be detected directly by its elastic scattering with heavy nuclei . the most strongly constrained neutralino - nucleus cross - sections are the spin - independent ones . it is conventional to express the experimental limits on these cross - sections in terms of an effective neutralino - proton cross - section . the values of the effective spin - independent cross - sections in the present model , for @xmath110 and @xmath18 = 500 gev , are shown in fig . [ dd-10 - 500 ] .. ] except in a thin strip at the top of the allowed region , where the neutralino lsp is predominantly higgsino , these cross - sections are much smaller than the current experimental bound ( for a standard set of assumptions about the local halo density ) of @xmath146 pb @xcite for the range of neutralino masses we consider here . the scattering cross - sections in the higgsino region are within an order of magnitude this bound . and @xmath129 , 500 , and 700 gev , as well as for @xmath112 and @xmath147 , as a function of the ratio @xmath148 . the current experimental limit for standard assumptions about the local halo density is @xmath146 pb for the range of neutralino masses we consider here @xcite . , scaledwidth=70.0% ] spin - independent scattering between a neutralino and a nucleon is mediated predominantly by the exchange of cp - even higgs bosons and squarks . in the allowed parameter regions discussed above , the squarks tend to be much heavier than the neutralino lsp , thereby suppressing their contribution . thus , the leading contribution to the spin - independent neutralino - nucleon cross - section usually comes from cp - even higgs exchange . since the neutralino higgs vertices are proportional to the higgsino component of the neutralino , the effective cross - section depends sensitively on the ratio @xmath148 . this is illustrated in fig . [ dd - mu - m1 ] for @xmath110 and @xmath149 gev , as well as for @xmath112 and @xmath147 . the dependence on other parameters such as @xmath150 and @xmath151 is much weaker . and @xmath134 gev . a detector threshold of 1gev is assumed . the best current bound is @xmath152 coming from super - kamiokande @xcite . future sensitivities from icecube and antares are @xmath153 . , scaledwidth=70.0% ] relic neutralinos can also be searched for indirectly by looking for their annihilation products in regions where they tend to clump , and have a local density much larger than the average value . in these particularly dense patches , the rate of neutralino annihilation can become large enough that their products are potentially observable . the most stringent of these indirect detection constraints on neutralino dark matter usually comes from their annihilation in the core of the sun and the earth .- channel cp - odd higgs resonance . ] the neutrinos produced by this process can lead to a signal , in the form of a muon flux , in neutrino telescopes . the best current bound on such a muon flux comes from the super - kamiokande experiment @xcite , and is on the order of @xmath152 . this sensitivity or bound is expected to be tightened to @xmath154 in the next few years by icecube @xcite and antares @xcite . the values of @xmath155 due to annihilation in both the core of the earth and the sun in the present model with @xmath110 and @xmath134 gev are shown in fig . [ muflux ] as a function of the ratio @xmath148 , assuming a detector threshold of @xmath156 . as for the direct detection signal , the largest indirect detection signal is obtained when @xmath148 is on the order of unity and the neutralino lsp has a significant higgsino component . in fact , for @xmath157 , the signal as at or slightly above the super - kamiokande bound . for larger values of @xmath148 the signal falls off very quickly to values that are well below the reach of upcoming experiments . in hens scenarios , the sleptons and the electroweak gauginos are generally very light relative to the squarks and the gluino . if the lightest neutralino is the lsp , which we assume throughout this section , the distinguishing feature of these scenarios at colliders are multi - lepton events with missing @xmath158 . in this section we discuss the prospects for discovery and identification of hens models at the tevatron and the lhc . the most promising search channel at the tevatron is the trilepton signal with missing @xmath158 @xcite . this can be induced , for example , by the electroweak production of @xmath159 , with subsequent cascades of the form @xmath160 and @xmath161 . for @xmath162 , the significant source of susy events comes from the electroweak production of gauginos , making this channel a copious and clean one . in hens scenarios , the @xmath163 and @xmath164 states tend to be mostly wino and have two - body decays into left - handed sleptons . because of this feature , the branching fractions of the abovementioned decay cascades can be significant , leading to a sizeable trilepton cross - section . indeed , the mass spectrum derived from hens models is close to being optimal for trilepton production . to estimate the effective tevatron trilepton cross - sections , we have simulated susy production from @xmath165 collisions at @xmath166 tev using isajet 7.74 @xcite . following the treatment in ref . @xcite , we use the isajet subroutines calsim and calini ( in the isaplt package ) as a simple detector model with coverage in the range @xmath167 , and calorimeter cells of size @xmath168 . to simulate energy resolution uncertainties , the electromagnetic calorimeter cells are smeared by an amount @xmath169 , while the hadronic calorimeter cells are smeared by an amount @xmath170 . we define jets as hadronic clusters with @xmath171 gev within a cone of size @xmath172 , and use the getjet subroutine to perform the clustering . isolated leptons are defined to be @xmath173 s or @xmath9 s having @xmath174 gev , with net visible hadronic activity @xmath175 gev within a cone of size @xmath176 about the lepton direction . we focus on a particular set of cuts , corresponding to the hc2 set in ref . @xcite , that is well - suited to the hens mass spectrum @xcite . in each event , we require three isolated leptons with @xmath177 , and @xmath178 . in addition to this , the total missing @xmath158 must exceed @xmath179 gev , the invariant mass of same - flavor opposite - sign dileptons must lie in the range @xmath180 , and the transverse invariant mass between each lepton and the missing @xmath158 vector must lie outside the range @xmath181 . the dilepton invariant mass veto is designed to remove background events from off - shell @xmath182 and @xmath183 decays , while the @xmath184 veto removes leptons from @xmath23 decays . with these cuts , the sm background is estimated to be @xmath185 @xcite , and is due mostly to the remaining @xmath186 and @xmath187 events in which both off - shell gauge bosons decay leptonically . and @xmath110 . the estimated background is @xmath188 , scaledwidth=70.0% ] in fig . [ tevscan ] we show the trilepton cross - section subject to the cuts for @xmath129 gev and @xmath110 . this value of @xmath18 is about as small as possible within the model given the lower bound on the light higgs boson mass , and represents a best - case scenario at the tevatron . note that for considerably larger values of @xmath19 , the constraint from the anomalous magnetic moment of the muon requires larger values of @xmath18 as well , as can be seen in fig . [ tanb - m12 ] . like in the previous plots , the values of @xmath14 and @xmath15 in fig . [ tevscan ] are those at the input scale , @xmath30 . the dependence of the trilepton cross - section on @xmath14 and @xmath15 can be understood in terms of the mass spectrum . except in the upper - right portion of the allowed parameter space , the effective cross - section increases smoothly from bottom to top as the value of @xmath9 decreases . in most of the parameter space , @xmath9 is larger than @xmath189 and the @xmath163 and @xmath190 states are mostly wino . as @xmath9 approaches @xmath189 , these states develop a larger higgsino fraction and their masses are reduced by the mixing . the heavier chargino and neutralino states become lighter as well . on account of these effects , the total gaugino cross section is increased leading to more trilepton events . this pattern is broken in the upper right corner of the parameter space because the mass of the @xmath163 state approaches the left - handed slepton masses from above , again due to higgsino mixing . when this mass difference becomes small , the branching fraction for @xmath191 goes down . the leptons produced by the cascades become relatively soft as well . since this decay mode plays a prominent role in the trilepton signal subject to the hc2 cuts , the effective cross - section falls off rapidly when the decay fraction is suppressed . the effective cross - section in this region can be increased by using slightly weaker lepton @xmath192 cuts , such as the sc2 set discussed in ref . @xcite , but at the expense of an increase in background . the effective trilepton cross - sections shown in fig . [ tevscan ] fall within the range of @xmath193 . given the estimated background of @xmath194 , the signal significance level is marginal . for example , the poisson probability @xmath195 for a total of ten events , corresponding to the maximal expected signal and background with @xmath196 , is about @xmath197 . while this is unfortunately not enough for a discovery , an excess of clean trilepton events at the tevatron would provide a tantalizing hint of a light hens scenario . we also note that other event signatures involving leptons can be searched for in these scenarios . of particular noteworthiness is the same - sign dilepton signature , which has small standard model background . if nature is supersymmetric and has a hens spectrum , the prospects for discovery at the lhc with @xmath198 of data are excellent provided @xmath18 is less than about 700 gev . to quantify this , we focus on six inclusive lhc susy search channels , classified by the number of isolated leptons in the event : @xmath199 ; @xmath200 ; @xmath201 ; @xmath202 ; @xmath203 ; @xmath204 @xcite . ( here , @xmath205 and @xmath206 refer to opposite - sign and same - sign dileptons , respectively . ) besides an excess of events in these channels , which is expected in many susy scenarios , the relative numbers of events within different channels can point towards small input scalar soft masses . supersymmetric events at the lhc were simulated using isajet 7.74 @xcite . we use the isajet subroutines calsim and calini ( in the isaplt package ) as a simple detector model with coverage in the range @xmath207 , and calorimeter cells of size @xmath208 . a gaussian smearing of the calorimeter cells is included to simulate energy resolution uncertainties . the electromagnetic calorimeter cells are smeared by an amount @xmath209 , where @xmath210 denotes addition in quadrature . hadronic calorimeter cells are smeared by an amount @xmath211 for @xmath212 , and @xmath213 for @xmath214 . we define jets as clusters with @xmath215 gev and @xmath212 within a cone of size @xmath172 , and use the getjet subroutine to perform the clustering . isolated leptons are defined to be @xmath173 s or @xmath9 s having @xmath216 gev and @xmath217 , with total visible activity @xmath218 gev within a cone of size @xmath219 about the lepton direction . for all channels studied , we choose a cut energy @xmath220 and demand that each event have at least two hard jets , @xmath221 , with @xmath222 , as well as missing transverse energy @xmath223 . this cut substantially reduces the sm backgrounds relative to the susy signals . would be preferable to reduce the large sm backgrounds in the @xmath224 channel . however , the @xmath225 susy signal is still easily distinguishable from the large background for the parameter points we consider here . ] we also require that the transverse sphericity of the event satisfies @xmath226 to reduce the dijet background @xcite . in zero - lepton events , we demand that the transverse angle between the missing momentum vector and the nearest jet must lie in the range @xmath227 . in the one - lepton channel , we require a single isolated lepton with @xmath228 , as well as @xmath229 to reduce the leptonic @xmath23 background . for events with two or more isolated leptons , we demand that @xmath230 for the two hardest leptons . and @xmath110 for the five sample points described in the text . for comparison , the sm backgrounds are estimated to be about @xmath231 , @xmath232 , @xmath233 , @xmath234 , @xmath235 , and @xmath236 for the @xmath225 , @xmath237 , @xmath238 , @xmath239 , @xmath240 and @xmath241 channels respectively @xcite.,scaledwidth=70.0% ] the cross sections after cuts for @xmath242 and @xmath110 are given in fig . [ lhcbar ] for five sample points . for comparison , the sm backgrounds are estimated to be about @xmath243 , @xmath244 , @xmath245 , @xmath246 , @xmath247 , and @xmath248 for the @xmath225 , @xmath237 , @xmath238 , @xmath239 , @xmath240 , and @xmath241 channels respectively @xcite . the locations of the sample points , @xmath249 and @xmath250 , in the @xmath251 plane are listed in the appendix , and are also indicated in figs . [ 10 - 500-scan-3l ] and [ 10 - 500-scan-1l ] . among the five sample points , @xmath8 and @xmath252 have a neutralino relic density within the wmap allowed range , while the other points lead to relic densities that are too large ( but could be acceptable with a non - standard cosmology ) . at point e , the @xmath9 term is on the same order as @xmath253 , but it is greater than @xmath254 at the other sample points . thus , except for point e , the lsp is a mostly bino neutralino , while the lightest chargino and the next - to - lightest neutralino are predominantly wino . at point e where @xmath255 , all the chargino and neutralino states are fairly light and have significant higgsino components , which , as we shall discuss below , is the reason for the increase in the @xmath240 and @xmath241 rates . cross - sections after cuts at the lhc for @xmath256 and @xmath110 . the estimated background is @xmath247.,scaledwidth=70.0% ] in each of the six search channels , the susy signal is easily distinguishable from the background with @xmath257 of luminosity at the lhc . a more challenging task beyond an initial discovery is to distinguish this class of models from other ( susy ) scenarios and to deduce the model parameters . the number of leptonic events relative to the number of @xmath224 events is useful in this regard . compared to a generic msugra input spectrum with @xmath258 , the ratio of @xmath259 events to @xmath224 events is much larger for a given value of @xmath18 . for example , in msugra with @xmath260 , the ratio of @xmath261 to @xmath259 events is greater than four , whereas this ratio is close to unity for all five sample points considered . the ratio of the number of @xmath224 events to the number of @xmath262 events is also much larger in a generic msugra framework than it is here . this is the consequence of having left - handed sleptons lighter than the @xmath163 and @xmath164 states , which are in turn light enough to be generated by squark decays . for example , cascade chains such as @xmath263 have a significant branching probability , and are a rich source of leptons . the dependence of the effective @xmath240 cross - section on the input higgs soft mass parameters is shown in fig . [ 10 - 500-scan-3l ] . for the most part , this dependence is fairly mild except in the upper right portion of the allowed region . here , the @xmath9 term approaches the bino mass @xmath64 , and in the thin tail extending to the right , @xmath9 even falls below @xmath64 . in this region , all the neutralinos and charginos are significantly lighter than the squarks and gluinos . as a result , the decay cascades initiated by the strong superpartners are frequently very long , involving several chargino and neutralino states . at each step in the cascade chain there is a chance of producing a lepton , and thus the total fraction of events containing multiple leptons is very high . for example , the decay chain @xmath264 is kinematically allowed when @xmath9 is small , and has a significant branching fraction . the preponderance of leptons can be so high that the number of @xmath225 and @xmath237 events ( and even to some extent some @xmath265 events ) are significantly suppressed . this can be seen in fig . [ 10 - 500-scan-1l ] , which shows the @xmath237 effective cross section for @xmath147 and @xmath110 . note that the small @xmath9 region is strongly constrained by direct and indirect searches for dark matter , and will be probed by upcoming experiments , as was discussed above . cross - sections after cuts at the lhc for @xmath256 and @xmath110 . the estimated background is @xmath244.,scaledwidth=70.0% ] in the leftmost portion of the allowed region of fig . [ 10 - 500-scan-3l ] , there is also a net decrease in the cross section , which occurs in the other leptonic channels as well . within this region , leptons typically originate from decays of the mostly wino @xmath163 and @xmath164 states into left - handed sleptons and sneutrinos , which subsequently decay into the neutralino lsp . however , these left - handed states are only slightly heavier than the lsp , so the lepton emitted from the slepton decays tends to be soft , making it less likely to pass the lepton @xmath192 cuts . a particularly distinctive signature of the hens models are inclusive @xmath241 events . we find effective cross - sections above @xmath266 for @xmath110 and @xmath147 , which is sufficient for a @xmath196 lhc discovery given the sm background of about @xmath236 @xcite . there is a sharp increase in the @xmath241 cross - section in the small @xmath9 region at the upper right of the parameter space . the dominant sources of this increase are cascades initiated by right - handed squarks of the type described previously . furthermore , because the left - handed sleptons are lighter than @xmath267 but heavier than @xmath268 and @xmath269 in this region , superpartner cascades such as @xmath270 accompanied by many leptons have a non - trivial branching fraction and can produce three leptons from the single squark parent . is very small in this part of the parameter space . however , the other two leptons in the cascade tend to be quite hard . ] as a result , @xmath241 rates greater than @xmath271 can occur . we have also investigated the exclusive clean trilepton channel . it does not appear to be as promising as the inclusive channels . and @xmath112 for the five sample points described in the text . for comparison , the sm backgrounds are estimated to be about @xmath231 , @xmath232 , @xmath233 , @xmath234 , @xmath235 , and @xmath236 for the @xmath225 , @xmath237 , @xmath238 , @xmath239 , @xmath240 and @xmath241 channels respectively @xcite.,scaledwidth=70.0% ] varying @xmath19 does not qualitatively affect our findings . the cross sections after cuts for @xmath242 and @xmath112 are given in fig . [ lhcbar30 ] for five sample points , @xmath272 . details of these sample points are given in the appendix . the cross sections in all six channels are similar to those in fig . [ lhcbar ] with @xmath110 . in particular , the ratio of @xmath225 to @xmath237 events is still close to unity , and the @xmath240 and @xmath241 rates are observably large . the main difference that occurs at larger values of @xmath19 is that there is no small @xmath9 region . since the entire mass spectrum in hens models scales with @xmath18 , so too do the event rates . we have checked that for @xmath18 as large @xmath273 , corresponding to a gluino mass of @xmath274 , the inclusive event rates in all channels other than the @xmath225 and @xmath241 are large enough that a discovery with @xmath275 of lhc data is feasible . on the other hand , the event rates become even larger for smaller values of @xmath18 , making discovery even easier . supersymmetry has been recognized as a viable theory of physics beyond the standard model for many years now . it was quickly realized that it was not only viable , but also potentially useful in the quest to understand stability of the electroweak potential , radiative electroweak symmetry breaking , grand unification , dark matter , and baryogenesis . in subsequent years , there has been much effort devoted to understanding how supersymmetry breaking is to be achieved without creating additional phenomenological problems , such as too large fcnc . there are two particularly simple alternatives that keep the good features of supersymmetry while ( mostly ) dismissing the bad features . one approach is to raise the scalar masses significantly higher than the supersymmetric fermion masses . this is the idea of split supersymmetry discussed in the introduction . one drawback of this scenario is the apparent finetuning in the electroweak sector . the approach we pursue in this article is in some sense the opposite of split supersymmetry . here , rather than introduce a huge hierarchy of scalar masses over fermion masses , we wish to zero out the superpartner scalar masses at some scale ( i.e. , splat supersymmetry " ) . the simplest model of all scalar masses having a zero boundary condition does not work . however , applying a small alteration to the most minimal idea , namely that the higgs bosons are exempt from zero boundary condition requirement , preserves the good features of these theories , while satisfying the phenomenological requirements described in the text . this idea of higgs exempt no - scale ( hens ) supersymmetry has many phenomenological implications worthy of consideration at current and future experimental facilities . for example , we have found that the scenario can accommodate the tantalizing ( but small ) deviation of @xmath276 of the muon compared to the sm prediction . it also suggests a near maximal leptonic signal for the tevatron , and thus provides an excellent benchmark theory for the tevatron to either discover this form of supersymmetry or rule out large regions of parameter space in a clean way . furthermore , the lhc signatures are of many multi - lepton events . perhaps the most distinctive of them is the inclusive @xmath241 channel which can effectively rule out hens models up to gaugino mass scales that are uncomfortably large from the normal finetuning point of view . for the most part , this form of supersymmetry is rather straightforward for the lhc to find . an important exception is the lightest higgs boson , whose mass is pressured in this scenario to be low , and thus perhaps close to the current bound of @xmath121 . given the difficulties of finding a higgs boson less than @xmath277 @xcite , discovering the higgs boson might be one of the more challenging steps in confirming the complete structure of this theory . acknowledgements * we would like to thank p. kumar for discussions , and for contributions at the early stages of this work . we also wish to thank g. kane , s. martin , a. pierce and s. thomas for helpful discussions . this work is supported in part by the department of energy and the michigan center for theoretical physics . in this appendix , we list the relevant properties of the sample points @xmath278 chosen for @xmath110 and @xmath242 . the locations of these points in the @xmath279 plane are shown in figs . [ 10 - 500-scan-3l ] and [ 10 - 500-scan-1l ] . we also list properties of the points @xmath280 corresponding to @xmath112 and @xmath147 . .model parameters and particle masses for sample points @xmath278 , all with @xmath110 and @xmath147 . all dimensionful quantities in the table are listed in gev units . the @xmath144 values are valid computations for the assumption of standard thermal cosmological evolution and stable lightest neutralino . viability of points @xmath17 and @xmath16 require alterations to the standard assumptions . 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k. t. matchev and d. m. pierce , phys . b * 467 * , 225 ( 1999 ) [ hep - ph/9907505 ] . h. baer , m. drees , f. paige , p. quintana and x. tata , phys . d * 61 * , 095007 ( 2000 ) [ hep - ph/9906233 ] . f. e. paige , s. d. protopopescu , h. baer and x. tata , hep - ph/0312045 . h. baer , c. h. chen , f. paige and x. tata , phys . d * 52 * , 2746 ( 1995 ) [ hep - ph/9503271 ] . h. baer , c. h. chen , f. paige and x. tata , phys . d * 53 * , 6241 ( 1996 ) [ hep - ph/9512383 ] . atlas detector and physics performance . technical design report . 2 , cern - lhcc-99 - 15 , atlas - tdr-15 , may 1999 .	one of the most straightforward ways to address the flavor problem of low - energy supersymmetry is to arrange for the scalar soft terms to vanish simultaneously at a scale @xmath0 much larger than the electroweak scale . this occurs naturally in a number of scenarios , such as no - scale models , gaugino mediation , and several models with strong conformal dynamics . unfortunately , the most basic version of this approach that incorporates gaugino mass unification and zero scalar masses at the grand unification scale is not compatible with collider and dark matter constraints . however , experimental constraints can be satisfied if we exempt the higgs bosons from flowing to zero mass value at the high scale . we survey the theoretical constructions that allow this , and investigate the collider and dark matter consequences . a generic feature is that the sleptons are relatively light . because of this , these models frequently give a significant contribution to the anomalous magnetic moment of the muon , and neutralino - slepton coannihilation can play an important role in obtaining an acceptable dark matter relic density . furthermore , the light sleptons give rise to a large multiplicity of lepton events at colliders , including a potentially suggestive clean trilepton signal at the tevatron , and a substantial four lepton signature at the lhc . mctp-06 - 32 + higgs boson exempt no - scale supersymmetry + and its collider and cosmology implications + jason l. evans , david e. morrissey , james d. wells + michigan center for theoretical physics ( mctp ) + physics department , university of michigan , ann arbor , mi 48109	25,613	432
single electron tunneling ( set ) devices@xcite are promising candidates for a wide variety of nanoelectronics applications , such as sensitive electrometers@xcite , thermometers@xcite , electron pumps and turnstiles for current standards@xcite , and quantum bits for quantum information processing@xcite . in recent years , silicon has drawn a lot of attention as a candidate for practical set devices for several reasons . these advantages include compatibility with complementary metal oxide semiconductor ( cmos ) processing , good electrostatic control of the tunnel barriers@xcite , greater device stability as demonstrated by a lack of charge offset drift@xcite , and a relative lack of nuclear spins , an important source of decoherence in spin - based quantum information applications@xcite . however , to become truly viable in any of these applications , devices must be fabricated which overcome the device to device variations and low yield associated with the single device processing typical of small scale research programs . although , at the single device level , the gate voltage variation from one device to another may not be an important parameter , uniform device operation becomes crucial when trying to operate several set devices simultaneously , e.g. , in the large scale integration of set devices . the choice of device architecture can also impact the integrability of devices . for example , gate to gate variations in an architecture where more than one gate@xcite controls a single tunnel barrier can make finding the desired operating point a laborious iterative process . in this paper we demonstrate robust behavior and good unformity of easily - tuned , fully cmos single electron devices , which contain only silicon , thermally grown silicon dioxide ( sio@xmath3 ) and phorphorous doped polycrystalline silicon ( poly si ) in the active device region . the motivation for a fully cmos approach to fabrication is twofold : 1 ) to minimize the number of impurities and defects near the active device region and 2 ) to avoid the instabilities associated with metallic oxides and , in particular , aluminum oxide . in this way , we avail ourselves of the best opportunity to fabricate uniform , robust devices . below , we will discuss and demonstrate the robustness of our devices with respect to basic metal oxide semiconductor field effect transistor ( mosfet ) characteristics and set device operation . in particular , we show that these devices exhibit only small variations of the threshold voltage from device to device , dielectrics which are robust against breakdown , and charge offset stability of the order of 0.01 e over a period of several days . ( color online ) ( a ) a schematic view of a sample . device operation is described in the text . ( b ) left : an optical micrograph of a sample . right : an sem micrograph ( before upper gate deposition ) of the active device area , and schematics of an electrical measurement circuit ( does not show v@xmath4 ) . lower gates lgs , lgc and lgd are poly si and the conducting channel ( s / d ) is single crystal si . channel and lower gates sit on top of the buried silicon oxide ( box ) . the white arrow next to the finger gate indicates the dimension we call gate length . ( c ) a cross sectional sem image of a device along the dashed white line in ( b ) . the darker areas are si , the gray areas are sio@xmath5 and the bright layer on top is a protective layer of pt deposited prior to the fib cut . ] flow chart of the condensed fabrication process described in the text . ] our devices each contain a lightly boron doped ( p type ) mesa etched single crystal si nanowire , n@xmath6type source and drain , and two layers of gates ; see fig . [ sample ] ( a ) . the topmost gate layer , which we call the upper gate ( ug ) , covers the entire device between the heavily doped source and drain . applying a positive voltage to the upper gate inverts the underlying si nanowire and provides conduction . the second gate layer , which we call the lower gates ( lg ) , consists of three finger gates which wrap around the si nanowire . these are denoted as lgs ( closest to the source ) , lgc ( center gate ) and lgd ( closest to the drain ) ; see fig . [ sample ] ( b ) . the lower gate fingers are primarily used to locally deplete the electron gas and therefore to create electrostatically controlled tunnel barriers ( lgs and lgd ) , or to modulate the electrostatic potential of a quantum dot ( lgc ) . the devices are fabricated on a 6 inch silicon on insulator ( soi ) wafer , with doping density of about 10@xmath7 @xmath8 , an initial soi thickness of 100 nm , and a buried oxide ( box ) thickness of 200 nm . to minimize the interface trap density at the gate oxide interface of the nanowire@xcite , we fabricate the soi nanowires at a 45@xmath9 angle with respect to the flat ( @xmath10110@xmath11 ) of the wafer in order to obtain a @xmath10100@xmath11 crystallographic equivalent orientation on each facet of the nanowire . as previously mentioned , we fabricate these devices with a fully cmos process flow developed at the center for nanoscale science and technology ( cnst ) nanofabrication user facility at nist . the fabrication process is presented in fig . [ fabflowchart ] . the nanowire , lower gate and upper gate lithography and etching are performed with negative tone electron beam lithography ( ebl ) using hydrogen silsesquioxane ( hsq ) as a resist and dry etching in cl@xmath3 chemistry . source and drain areas located about 10 @xmath12 m away from the active device area are implanted with phosphorous at 30 kev with a dose of 10@xmath7 @xmath13 . we grow sacrificial oxide on both the nanowire and the lower gate layer in order to remove possible etch damage produced during the dry etch . both sacrificial oxide and the gate oxide on the nanowire are grown in a tube furnace at 850 @xmath9c and 950 @xmath9c , respectively . the sacrificial oxide is removed with a short 100:1 hf dip . the lower and upper gate layers are 75 nm thick _ in situ _ phosphorous doped poly si deposited by low pressure chemical vapor deposition ( lpcvd ) at 625 @xmath9c . both gate layers are degenerately doped to ensure electrical conduction at low temperatures with a typical resistivity of 1030 m@xmath14 cm ( determined from two terminal measurement at 2.2 k ) . the sacrificial oxide on the lower gate and the isolation oxide between the lower gate and the upper gate are grown with rapid thermal oxidation ( rto ) at 1000 @xmath9c . the final steps of the process are metallization of ohmic contacts with sputter deposited al1%si and a forming gas anneal at 425 @xmath9c for 30 min . to date , we have fabricated devices on two 6 inch wafers which we call a and b@xcite ; see fig . [ fabflowchart ] . the main differences between the wafers are the nominal gate oxide thickness and the finger gate lengths . the soi nanowire width and length are 70 nm and 800 nm respectively for both wafers . each wafer contained 48 dies : 36 with two devices as in fig . [ sample ] on each and 12 diagnostics dies located on the diagonals of the wafer . the diagnostics dies contained conventional field effect transistors ( fet ) with a 70 nm wide soi nanowire as a channel , and test structures to measure the resistance of ohmic contacts and the resistivity of the poly si . a cross - sectional sem image produced by a focused ion beam ( fib ) cut along the lgc finger ( white dashed line in fig . [ sample](b ) ) of a finished device is shown in fig [ sample](c ) . the darker areas in the micrograph are si and the gray areas are sio@xmath5 . the cross sectional image shows that both poly si films of upper gate and lower gate layers conformally coat the layers underneath as is expected from lpcvd growth , and that the oxides are continuous , as needed for electrical isolation . we characterized many devices and fets from randomly chosen dies across both wafers at room temperature and at 2.2 k. in addition , one of the devices was cooled down and measured in a dilution refrigerator to 30 mk . the summary of the electrical characterization is presented in table [ samplesummary ] . figure [ 3gchar ] ( a ) shows turn on characteristics of different devices ( as in fig . [ sample ] ) on wafer a ( 40 nm gate oxide ) . the solid lines and dashed lines correspond to data taken at 300 k and 2.2 k , respectively . at room temperature the threshold voltages , as obtained by linearly extrapolating the current to zero@xcite , were @xmath15 v with the standard deviation of 0.1 v and there was a threshold shift of about 0.6 v when devices were cooled down . a simple estimate of the threshold voltage@xcite which ignores the presence of any fixed oxide charge and which treats the devices as planar fets yields @xmath16 v at room temperature and @xmath17 v at 2.2 k , which is in reasonable agreement with our measured values . wafer b ( 25 nm gate oxide ) showed uniform turn on characteristics with @xmath18 v at room temperature and a nearly equal shift in the threshold when cooled down . the diagnostic transistors on each wafer also showed similar turn on behavior . a typical subthreshold slope for these devices was 80 mv / decade at room temperature ( the ideal subthreshold slope is 60 mv / decade@xcite ) with an on off ratio of 10@xmath19 , see inset in fig . [ 3gchar ] ( a ) . typical turn off characteristics for each of the finger gates ( lgs , lgc and lgd ) measured at both room temperature and 2.2 k for wafer b are shown in fig . [ 3gchar ] ( b ) . the room temperature data was taken with an upper gate voltage @xmath20 v , while the low temperature data was taken with @xmath21 v. the range for turn off voltages , i.e. the lower gate voltage @xmath22 at 100 pa of drain current @xmath23 , was from -1.5 v to -1 v at room temperature and -1 v to -0.5 v at 2.2 k for all measured lower gates for all devices . ( color online ) turn on characteristics , i.e. drain current as a function of gate voltage of a mosfet transistor with 25 nm gate oxide thickness . data was taken with the drain voltage @xmath24 of 1 mv . solid ( black ) line is initial turn on curve , dashed ( red , gray ) line is turn on curve after @xmath2510 v ( 4 mv / cm ) excursion in the gate voltage . the indentical curves indicate immunity to dielectric breakdown up to @xmath254 mv / cm . ] we also tested the robustness of the gate oxide and the isolation oxide on wafer b. in these tests , the gate voltage was swept in steps up to @xmath2510 v while the source - drain and leakage currents were simultaneously measured to the other gates and to the channel . all leakage resistances between the channel and either layer of gates or between gates were @xmath1110 g@xmath1 up to gate voltages of @xmath2510 v. after each gate voltage excursion , the turn on characteristics were remeasured in order to determine if there was a change in the threshold voltage or slope . diagnostic fets were immune to electric fields up to 4 mv / cm ( @xmath26 v ) , showing no change in @xmath27 nor generation of a leakage path ( fig . [ transistor ] ) . similar robustness measurements for set devices showed no threshold shift up to 2.8 mv / cm ( @xmath257 v ) and only a small ( 0.05 v and 0.2 v ) threshold shift after a gate excursion of 4 mv / cm ( @xmath2510 v ) in two out of four devices . no observable leakage path developed during the sweep . a typical literature value of the breakdown field of metal oxide semiconductor capacitor ( moscap ) is about 10 mv / cm , before generating a leakage path@xcite . we also performed robustness measurements of the isolation oxide between lg and ug on about 50 different lower gate fingers on different devices across the wafer . only three fingers developed a breakdown path during the @xmath2510 v sweep . [ cols="^,^,^,^,^",options="header " , ] ( color online ) set oscillations of a device at 2.2 k , vertical dashed lines ( red , gray ) are separated by a period of 12.9 mv . the data show good uniformity of the gate capacitance over 90 periods . ] ( color online ) coulomb diamonds measured at 30 mk for an set device , n is the electron number . charging energy @xmath281.2 mev , drain capacitance @xmath29=7 af and total capacitance of the island @xmath30=70 af , extracted from the diamond data . ] above , we have discussed the robustness and uniformity of devices in terms of mosfet performance , and in the following we present device characteristics when operated in a single electron device mode . first , we discuss ease of tuning . the right hand side of fig . [ sample ] ( b ) shows a schematic of a typical measurement circuit for a device . tuning the device to display set oscillations took very little time , on the order of minutes , because there was very little cross capacitance between gates and each barrier was controlled by a single gate voltage . to tune a device into set mode , we first applied a small bias voltage to the drain ( of order 1 mv ) and set the upper gate to a voltage ( obtained from a short upper gate sweep , typically about 2 v ) which gave about 1 na of current . next , a two dimensional sweep of @xmath31 and @xmath32 ( with @xmath33 well above the turn off voltage ) was performed to find the voltages where each of these gates began to turn off conduction . typically , the barrier voltages were about -0.6 v. we note that barrier resistances responded symmetrically to @xmath31 and @xmath32 . after fine tuning @xmath31 and @xmath32 , we measured set oscillations by sweeping @xmath33 with the other gate voltages fixed . coulomb oscillations of a device taken at 2.2 k are presented in fig . [ setosc ] . the oscillations were very regular with period of 12.9 mv over an lgc gate voltage range of 1.2 v , corresponding to about 90 periods . the capacitance of lgc to the dot , extracted from the set oscillation period , was about 12 af . coulomb diamond data recorded at 30 mk for the same device is shown in fig . [ diamonds ] . the charging energy and lever arm @xmath34 , which converts the gate voltage to the electrostatic potential of the island @xmath35 , extracted from the diamond data were 1.2 mev and 0.09 , respectively . the capacitance between the dot and each of the other two lower gates ( lgs and lgd ) was measured relative to the lgc capacitance by following the position of a current peak when sweeping lgs ( or lgd ) and lgc ( data not shown ) . the capacitance values for both lgs and lgd were about 5 af , indicating the dot was located in the center of the device . this , together with the agreement between the measured capacitance to lgc ( 12 af ) and that calculated from the geometry ( 14 af ) gives us confidence that the dot being modulated was an intentional dot formed through electrostatic control of lgs and lgd rather than through barriers formed by defects . as a more strict test of the quality of our fabrication we performed charge offset drift measurements on several devices . this measurement consisted of repeatedly measured coulomb blockade oscillations at a fixed time interval over several days . figure [ cbdata ] ( a ) shows a typical collection of set oscillations taken at 2.2 k and spanning the total duration of the charge offset drift measurement . to obtain charge offset drift values for each curve , a sinusoidal function of the form @xmath36 $ ] was fit to the measured data . here @xmath37 is the amplitude of the oscillations , @xmath38 is the gate voltage and @xmath39 the oscillation period . the phase of the sinusoidal fit function , @xmath40 , is a charge offset value for each curve . the result of this procedure is shown in fig . [ cbdata ] ( b ) . the devices exhibited very stable behavior with a drift in @xmath41 e over 8 days of measurement . moreover , these results rival those of similar si devices fabricated in other foundries@xcite . in addition , after eight days of measurement , we thermally cycled the device to room temperature . figure [ cbdata ] shows this data as well . while the thermal cycle resulted in a shift in the charge offset value of about ( 0.1@xmath42 ) e , the level of drift observed was identical both before and after the thermal cycle . finally , a measurement of @xmath43 vs. @xmath33 voltage with the same lgs , lgd , and ug voltages after the thermal cycle not only reproduced the set oscillations at the same value @xmath33 with charge offset of 0.1 e , but also reproduced the aperiodic features which become prominent when the gate begins to turn off conduction ( inset of fig . [ cbdata ] ) . while the previous results indicate that the cleanliness of our cmos fabrication is quite good , many devices in this first device run failed electrically by either not turning on or through an inability to turn off conduction with the lower gate fingers . this drove our yield of fully functioning ( in which we were able to measure intentional coulomb blockade ) devices down to 4/34 devices measured . we have been able to identify the gross fabrication failures , by cross - sectioning devices with fib in conjunction with the electrical results . in brief , the failure modes are the result of over oxidation of the soi nanowire and the lg fingers , as well as the overall amount of oxide removed in the processing . we believe further development of our process flow will ameliorate these failures . ( color online ) ( a ) an example of single electron oscillations taken at different times with interval of about 3 hours ; curves are offset vertically for clarity . ( b ) charge offset drift derived from coulomb oscillations , red and blue ( gray ) dashed horizontal lines are a guide for the eye . the gray area indicates the time interval of the thermal cycle . after the thermal cycle the charge offset value @xmath40 changed by 0.1 e , but remained as stable as before . inset : drain current as a function of lgc voltage @xmath44 before ( black line ) and after ( red , gray line ) the thermal cycle . the data in the inset before and after the thermal cycle are taken with the same gate voltages . all features in the data are reproduced after the thermal cycle . ] in summary , we have demonstrated devices which show good uniformity in electrical characteristics from device to device within a wafer and between wafers . moreover , the devices are quite robust against dielectric breakdown up to electric fields of 4 mv / cm . finally , and most importantly , when operated as a single electron device , these devices show very stable behavior . taken together , these characteristics indicate a relatively clean and stable electrostatic environment throughout the fabrication process . we attribute these successes to the minimization of impurities and defects which result from our cmos processing and material restrictions . to further improve the usefulness of these devices as current standards and quantum information devices , our next tasks include i ) substantially increasing the yield , and ii ) making shorter finger gates so that we can use those gates to both generate barriers and as plunger gates . while these results indicate that a fully cmos process pays dividends in device performance , it also complicates the fabrication . we believe that our results in terms of reliability , ease of tuning , and clean set behavior all justify the cost of the increased complexity of fabrication . we thank ted thorbeck , john bonevich , jerry bowser , and vincent luciani for fruitful discussions . we also thank jerry bowser and vincent luciani for guidance with the fabrication . research was performed in part at the nist center for nanoscale science and technology ( cnst ) . 20ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ _ , ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop _ _ ( , , ) @noop * * , ( )	we present electrical data of silicon single electron devices fabricated with cmos techniques and protocols . the easily tuned devices show clean coulomb diamonds at @xmath0 mk and charge offset drift of 0.01 e over eight days . in addition , the devices exhibit robust transistor characteristics including uniformity within about 0.5 v in the threshold voltage , gate resistances greater than 10 g@xmath1 , and immunity to dielectric breakdown in electric fields as high as @xmath2 mv / cm . these results highlight the benefits in device performance of a fully cmos process for single electron device fabrication .	5,692	143
recently , the generalizations of the logarithmic and exponential functions have attracted the attention of researchers . one - parameter logarithmic and exponential functions have been proposed in the context of non - extensive statistical mechanics @xcite , relativistic statistical mechanics @xcite and quantum group theory @xcite . two and three - parameter generalization of these functions have also been proposed @xcite . these generalizations are in current use in a wide range of disciplines since they permit the generalization of special functions : hyperbolic and trigonometric @xcite , gaussian / cauchy probability distribution function @xcite etc . also , they permit the description of several complex systems @xcite , for instance in generalizing the stretched exponential function @xcite . as mentioned above , the one - parameter generalizations of the logarithm and exponential functions are not univoquous . the @xmath1-logarithm function @xmath2 is defined as the value of the area underneath the non - symmetric hyperbole , @xmath3 , in the interval @xmath4 $ ] @xcite : @xmath5 this function is _ not _ the ordinary logarithmic function in the basis @xmath1 , namely @xmath6 $ ] , but a generalization of the natural logarithmic function definition , which is recovered for @xmath7 . the area is negative for @xmath8 , it vanishes for @xmath9 and it is positive for @xmath10 , independently of the @xmath1 values . given the area @xmath11 underneath the curve @xmath12 , for @xmath13 $ ] , the upper limit @xmath14 is the generalized @xmath1-exponential function : @xmath15 . this is the inverse function of the @xmath1-logarithmic @xmath16=x=\ln_{\tilde q}[e_{\tilde q}(x)]$ ] and it is given by : @xmath17 this is a non - negative function @xmath18 , with @xmath19 , for any @xmath1 . for @xmath20 , one has that @xmath21 , for @xmath22 and @xmath23 , for @xmath24 . notice that letting @xmath9 one has generalized the euler s number : @xmath25 instead of using the standard entropic index @xmath26 in eqs . ( [ eq : gen_log ] ) and ( [ eq : eqtilde ] ) , we have adopted the notation @xmath27 . the latter notation permits us to write simple relations as : @xmath28 or @xmath29 , bringing the inversion point around @xmath30 . these relations lead to simpler expressions in population dynamics problems @xcite and the generalized stretched exponential function @xcite contexts . also , they simplify the generalized sum and product operators @xcite , where a link to the aritmethical and geometrical averages of the generalized functions is established . this logarithm generalization , as shown in ref . @xcite , is the one of non - extensive statistical mechanics @xcite . it turns out to be precisely the form proposed by montroll and badger @xcite to unify the verhulst ( @xmath31 ) and gompertz ( @xmath32 ) one - species population dynamics model . the @xmath33-logarithm leads exactly to the richards growth model @xcite : @xmath34 where @xmath35 , @xmath36 is the population size at time @xmath37 , @xmath38 is the carrying capacity and @xmath39 is the intrinsic growth rate . the solution of eq . ( [ eq : richard_model ] ) is the _ @xmath1-generalized logistic _ equation @xmath40 } = e_{-{\tilde q}}[-\ln_{\tilde q}(p_0^{-1})e^{-\kappa t } ] = e_{-{\tilde q}}[\ln_{-\tilde q}(p_0)e^{-\kappa t}]$ ] . the competition among cells drive to replicate and inhibitory interactions , that are modeled by long range interaction among these cells . these interactions furnish an interesting microscopic mechanism to obtain richards model @xcite . the long range interaction is dependent on the distance @xmath41 between two cells as a power law @xmath42 . these cells have a fractal structure characterized by a fractal dimension @xmath43 . here we call the attention to eq . ( 7 ) of ref . @xcite , namely @xmath44\}$ ] , where @xmath45^{1-\gamma / d_f}-1\right\}/[d_f(1-\gamma / d_f)]$ ] . here , @xmath46 is a constant related to geometry of the problem , @xmath47 is the mean intrinsic replication rate of the cells and @xmath48 is the interaction factor . using eq . ( [ eq : gen_log ] ) , one can rewrite it simply as : @xmath49/{d_f}$ ] . calling , @xmath50 , @xmath51 and @xmath52 , this equation is the richard s model [ eq . ( [ eq : richard_model ] ) ] with an effort rate @xmath53 . in this context the parameter @xmath33 acquires a physical meaning related to the interaction range @xmath54 and fractal dimension of the cellular structure @xmath43 . if the interaction does not depend on the distance , @xmath55 , and it implies that @xmath56 . this physical interpretation of @xmath33 has only been possible due to richards model underlying microscopic description . introduced by nicholson in 1954 @xcite , scramble and contest are types of intraspecific competition models that differ between themselves in the way that limited resources are shared among individuals . in scramble competition , the resource is equally shared among the individuals of the population as long as it is available . in this case , there is a critical population size @xmath57 , above which , the amount of resource is not enough to assure population survival . in the contest competition , stronger individuals get the amount of resources they need to survive . if there is enough resources to all individuals , population grows , otherwise , only the strongest individuals survive ( strong hierarchy ) , and the population maintains itself stable with size @xmath38 . from experimental data , it is known that other than the important parameter @xmath39 ( and sometimes @xmath38 ) , additional parameters in more complex models are needed to adjust the model to the given population . one of the most general discrete model is the @xmath0-ricker model @xcite . this model describes well scramble competition models but it is unable to put into a unique formulation the contest competition models such as hassel model @xcite , beverton - holt model @xcite and maynard - smith - slatkin model @xcite . our main purpose is to show that eq . ( [ eq : limite ] ) is suitable to unify most of the known discrete growth models into a simple formula . this is done in the following way . in sec . [ sec : loquistic ] , we show that the richards model [ eq . ( [ eq : richard_model ] ) ] , which has an underlying microscopic model , has a physical interpretation to the parameter @xmath33 , and its discretization leads to a generalized logistic map . we briefly study the properties of this map and show that some features of it ( fixed points , cycles etc . ) are given in terms of the @xmath1-exponential function . curiously , the map attractor can be suitably written in terms of @xmath33-exponentials , even in the logistic case . in sec . [ sec : generalized_theta_ricker ] , using the @xmath1-exponential function , we generalize the @xmath0-ricker model and analytically calculate the model fixed points , as well as their stability . in sec . [ sec : generalizedskellam ] , we consider the generalized skellam model . these generalizations allow us to recover most of the well - known scramble / contest competition models . final remarks are presented in sec [ sec : conclusion ] . to discretize eq . ( [ eq : richard_model ] ) , call @xmath58 and @xmath59^{\tilde q}$ ] , which leads to : @xmath60 where @xmath61 . we notice that @xmath1 keeps its physical interpretation of the continuous model . in eq . ( [ eq : loquistic ] ) , if @xmath62 and @xmath63 , with @xmath64 $ ] , one obtains the _ logistic map _ , @xmath65 , which is the classical example of a _ dynamic system _ obtained from the discretization of the verhulst model . although simple , this map presents a extremely rich behavior , universal period duplication , chaos etc . @xcite . let us digress considering the feigenbaum s map @xcite : @xmath66 , with @xmath67 , @xmath68 and @xmath69 . firstly , let us consider the particular case @xmath62 . if one writes @xmath70 , with @xmath71 being a constant , then : @xmath72 $ ] . imposing that @xmath73 leads to @xmath74 and calling @xmath75 , one obtains the logistic map with @xmath76 , so that @xmath77 . one can easily relate the control parameter of these two maps , making the maps equivalent . for arbitrary values of @xmath33 , there is not a general closed analytical form to expand @xmath78 and one can not simply transform the control parameters of eq . ( [ eq : loquistic ] ) to the feigenbaum s map . here , in general , these two maps are not equivalent . it would be then interesting , to study the sensitivity of eq . ( [ eq : loquistic ] ) with respect to initial conditions as it has been extensively studied in the feigenbaum s map @xcite . returning to eq . ( [ eq : loquistic ] ) , in the domain @xmath79 , @xmath80 ( non - negative ) , for @xmath81 . since @xmath82 is real only for @xmath83 , @xmath84 is real only for @xmath85 . the maximum value of the function is @xmath86 which occurs at @xmath87 i. e. , the inverse of the generalized euler s number @xmath88 [ eq . ( [ eq : eqtilde ] ) ] . for the generalized logistic map , @xmath89 , so that @xmath90 , it leads to the following domain for the control parameter @xmath91 : @xmath92 the map fixed points @xmath93 $ ] are @xmath94 the fixed point @xmath95 is stable for @xmath96 and @xmath97 is stable for @xmath98 , where @xmath99 notice the presence of the @xmath33-exponentials in the description of the attractors , even for the logistic map @xmath62 . the generalized logistic map also presents the rich behavior of the logistic map as depicted by the bifurcation diagram of fig . the inset of fig . [ fig1 ] displays the lyapunov exponents as function of the central parameter @xmath100 . in fig . [ fig2 ] we have scaled the axis to @xmath101/ ( \rho_{max}\tilde{q})$ ] , where @xmath102 is given by eq . ( [ eq : rho_max ] ) and we plotted the bifurcation diagram for @xmath103 and @xmath104 . we see that the diagrams display the same structure but each one has its own scaling parameters . the role of increasing @xmath33 is to lift the bifurcation diagram to relatively anticipating the chaotic phase . the period doubling region start at @xmath105^{1/\tilde{q}}$ ] , so that for @xmath106 , @xmath107 and @xmath108 . when @xmath109 , then @xmath110 . in fig . [ fig3 ] we show the histograms of the distribution of the variable @xmath111 . we see that as @xmath33 increases , the histograms have the same shape as the logistic histogram has , but it is crooked in the counter clock sense around @xmath112 . the _ @xmath0-ricker _ model @xcite is given by : @xmath113 } , \label{eq : theta_ricker}\ ] ] where @xmath114 . notice that @xmath115 is the relevant variable , where @xmath116 . in this way eq . ( [ eq : theta_ricker ] ) can be simply written as @xmath117 . for @xmath118 , one finds the standard _ model @xcite . for arbitrary @xmath0 , expanding the exponential to the first order one obtains the generalized logistic map [ eq . ( [ eq : loquistic ] ) ] which becomes the logistic map , for @xmath118 . the @xmath0-ricker , ricker and quadratic models are all scramble competion models . if one switches the exponential function for the @xmath119-generalized exponential in eq . ( [ eq : theta_ricker ] ) , one gets the _ generalized @xmath0-ricker model _ : @xmath120 = \frac{\kappa_1 x_i}{\left[1 + \tilde{q } r \left ( \frac{x_i}{\kappa}\right)^{\theta } \right]^{1/\tilde{q } } } \ ; . \label{eq : generalized_theta_ricker_model}\ ] ] to obtain standard notation , write @xmath121 and @xmath122 , so that @xmath123 @xcite . the generalized model with @xmath118 , leads to the _ hassel _ model @xcite , which can be a scramble or contest competition model . one well - known contest competition model is the _ beverton - holt _ model @xcite , which is obtained taking @xmath124 . for @xmath125 , one recovers the ricker model and for @xmath126 , one recovers the logistic model . it is interesting to mention that the beverton - holt model @xcite is one of the few models that have the time evolution explicitly written : @xmath127 $ ] . from this equation , one sees that @xmath128 , for @xmath129 and @xmath130 for @xmath131 . using arbitrary values of @xmath0 in eq . ( [ eq : generalized_theta_ricker_model ] ) , for @xmath125 one recovers the @xmath0-ricker model , and for @xmath132 , the _ maynard - smith - slatkin _ model @xcite is recovered . the latter is a scramble / contest competition model . for @xmath126 , one recovers the generalized logistic map . the trivial linear model is retrieved for @xmath133 . in terms of the relevant variable @xmath134 , eq . ( [ eq : generalized_theta_ricker_model ] ) is rewritten as : @xmath135 where @xmath136 and we stress that the important parameters are @xmath137 , @xmath33 and @xmath114 . ( [ eq : final ] ) is suitable for data analysis and the most usual known discrete growth models are recovered with the judicious choice of the @xmath1 and @xmath0 parameters as it shown in table [ tabela ] . some typical bifurcation diagrams of eq . ( [ eq : final ] ) are displayed in fig . [ figbdtrm ] . .summary of the parameters to obtain discrete growth models from eq . ( [ eq : final ] ) . in the competition type column , _ s _ and _ c _ stand for scramble and contest models , respectively . the symbol @xmath138 stands for arbitrary values . [ cols="<,^,^ , < " , ] we have shown that the @xmath33-generalization of the exponential function is suitable to describe discrete growth models . the @xmath1 parameter is related to the range of a repulsive potential and the dimensionality of the fractal underlying structure . from the discretization of the richard s model , we have obtained a generalization for the logistic map and briefly studied its properties . an interesting generalization is the one of @xmath0-ricker model , which allows to have several scramble or contest competition discrete growth models as particular cases . equation ( [ eq : final ] ) allows the use of softwares to fit data to find the most suitable known model throughout the optimum choice of @xmath1 and @xmath0 . furthermore , one can also generalize the skellam contest model . only a few specific models mentioned in ref . @xcite are not retrieved from our generalization . actually , we propose a general procedure where we do not necessarily need to be tied to a specific model , since one can have arbitrary values of @xmath1 and @xmath0 . the authors thank c. a. s. terariol for fruitful discussions . asm acknowledges the brazilian agency cnpq ( 303990/2007 - 4 and 476862/2007 - 8 ) or support . rsg also acknowledges cnpq ( 140420/2007 - 0 ) for support . ale acknowledges cnpq for the fellowship and fapesp and mct / cnpq fundo setorial de infra - estrutura ( 2006/60333 - 0 ) .	here we show that a particular one - parameter generalization of the exponential function is suitable to unify most of the popular one - species discrete population dynamics models into a simple formula . a physical interpretation is given to this new introduced parameter in the context of the continuous richards model , which remains valid for the discrete case . from the discretization of the continuous richards model ( generalization of the gompertz and verhuslt models ) , one obtains a generalized logistic map and we briefly study its properties . notice , however that the physical interpretation for the introduced parameter persists valid for the discrete case . next , we generalize the ( scramble competition ) @xmath0-ricker discrete model and analytically calculate the fixed points as well as their stability . in contrast to previous generalizations , from the generalized @xmath0-ricker model one is able to retrieve either scramble or contest models . complex systems , population dynamics ( ecology ) , nonlinear dynamics 89.75.-k , 87.23.-n , 87.23.cc , 05.45.-a	4,874	269
the intensity anisotropy pattern of the cmbr has already been measured to an extraordinary precision , which helped significantly to establish the current cosmological paradigm of a flat universe with a period of inflation in its first moments and the existence of the so called dark energy @xcite . the polarization anisotropies of the cmbr are an order of magnitude smaller than the intensity anisotropies and provide partly complementary information . the polarization pattern is divided into two distinct components termed e- and b - modes which are scalar ( pseudoscalar ) fields . the e - modes originate from the dynamics due to the density inhomogeneities in the early universe . the b - modes are caused by lensing of the e - modes by the matter in the line of sight and by gravitational waves in the inflationary period in the very early universe and are expected to be at least one order of magnitude smaller than the e - modes . the status of the e - mode measurements is summarized in figure [ emodes ] from which it becomes obvious that the measurements are consistent with the theoretical model but not yet giving meaningful constraints . of special importance and interest are the b - modes expected from gravitational waves in the inflationary epoch , since a detection would allow unique access to the very first moments of the universe . the size of this contribution can not be predicted by theory , but is parametrized by the tensor - to - scalar ratio , @xmath1 @xcite . interesting inflationary energy scales of the order of the grand unifying theory ( gut ) scale of 10@xmath2 gev correspond to an @xmath1 of @xmath310@xmath0 , which would give rise to detectable signals of a few 10 nk . the tiny signal requires unprecedented sensitivity and control of systematics and foregrounds . by now receivers have reached sensitivities close to fundamental limits , so that the sensitivity will only be increased with the number of receivers . recent developments at the jet propulsion laboratory ( jpl ) led to the successful integration of the relevant components of a polarization - sensitive pseudo - correlation receiver at 90 and 40 ghz in a small chip package . this opened the way to future inexpensive mass production of large coherent receiver arrays and led to the formation of the q / u imaging experiment ( quiet ) collaboration . experimental groups from 12 international institutes have joined the experiment and are working on the first prototype arrays which are planned for deployment for 2008 in chile . a w - band ( 90 ghz ) array of 91 receivers and a q - band ( 40 ghz ) array of 19 receivers will be deployed on new 1.4 m telescopes mounted on the existing platform of the cosmic background imager ( cbi ) in the atacama desert at an altitude of 5080 m. it is foreseen to expand the arrays for a second phase of data taking ( 2010++ ) to arrays with 1000 receivers . for the expansion it is planned to mount more 1.4 m telescopes on the platform and relocate the 7 m crawford hill antenna from new jersey to chile to also access small angular scales . a sketch of one receiver and its components can be seen in figure [ receiver ] . the incoming radiation couples via a feedhorn to an orthomode transducer ( omt ) and from that to the two input waveguides of the chip package . the chip contains a complete radiometer with high electron mobility transistors ( hemts ) implemented as monolithic microwave integrated circuits ( mmics ) , phase shifters , hybrid couplers and diodes . the outputs of the four diodes of the radiometer provide measurements of the stokes parameters q and u and fast ( 4khz ) phase switching reduces the effects of the 1/f drifts of the amplifiers . for 10@xmath4 of the receivers the omt will be exchanged by a magic tee assembled in a way that the receivers measure temperature differences between neighboured feeds . the signals from the diodes are processed by a digital backend , sampling at 800 khz with subsequent digital demodulation . this allows unique monitoring of high - frequency noise as well as the production of null - data sets with out - of - phase demodulation giving a valuable check of possible subtle systematics . the receiver arrays together with the feedhorns are assembled in large cryostats and the chip radiometers are kept at 20 k to ensure low noise from the hemts . for a single element a bandwidth of 18(8 ) ghz and a noise temperature of 45 ( 20 ) k is aimed for at 90 ( 40 ) ghz , leading to expected sensitivites in chile of 250 ( 160 ) @xmath5k@xmath6 per element . a prototype array of 7 elements with one omt mounted on top of one chip radiometer is shown on the right hand side of figure [ receiver ] . the hexagonal prototype arrays of 91 and 19 elements are being assembled from similar subarrays . the omts were built in cost - effective split - block technique and the corrugated horn arrays were produced as platelet arrays where 100 plates with feed - hole patterns are mounted together by diffusion bonding . the increase in sensitivity is a necessary but not yet sufficient condition for the successful measurement of b - modes as the signal of interest is smaller than the one from astrophysical foregrounds . the diffuse emission ( synchrotron , dust ) from our galaxy and extragalactic sources produces polarized signals of which the distribution and characteristics are not yet known to the precision required for a full removal . multifrequency observations are mandatory to study the foreground behaviour and enable the clean extraction of the cmbr polarization anisotropies . quiet in its observations will use two frequencies which frame the frequency where the contamination from foregrounds in polarization is expected to be minimal , around 70 ghz . also , it will coordinate the patches to be observed with other polarization experiments to gain additional frequency information . fields were selected in which minimal foreground contamination is expected . the b - modes from gravitational waves will suffer from yet another foreground ( which in intself is of scientific interest ) which is the lensing of e - modes into b - modes . using the observations at small angular scales quiet will be able to determine a lensing correction and , with that , be able to remove that contribution properly . while currently ongoing cmbr experiments ( bicep , quad ) are running with tens of receivers all future experiments are aiming for large arrays with several hundreds of receivers , all of them ( but quiet ) using bolometers . figure [ comparison ] visualizes the main parameters of quiet in comparison to other ongoing and planned cmb experiments ( no interferometers are shown ) . some of the experiments have their main focus on observations of the sunyaev zeldovich effect ( marked accordingly ) and not polarization observations , but may still upgrade their detector arrays for polarization sensitivity . the parameters of the future experiments were taken from recent papers and talks about the various efforts , but since some of the technologies are not yet fully established and not all of the experiments are completely funded , it is clear that some of the parameters may change in the course of the production . both the left and middle plot display beam size versus frequency for the different experiments while the size of the squares indicates different parameters of the experiments . since some experiments ( quiet , polarbear ) are planned to operate in different phases , they have several squares at the same position . in the left panel the square area is proportional to the total sensitivity of the experiments , which means the smaller the square the more sensitive the experiment . as can be seen the next generation of cmb experiments will achieve the desired level of a few nk sensitivity . except for the space - based mission planck and the balloon experiment spider all ground - based experiments focus their sensitivity on small fractions of the sky . in this way it is possible to avoid regions of high foreground contamination and also gain a higher signal - to - noise ratio in the maps , which helps characterizing foregrounds and systematics . in order to compare the sensitivity on a map the middle figure displays squares which are in size proportional to the sensitivity in @xmath5k / square degree . the right figure then shows the corresponding white noise level for the different polarization experiments as a function of multipole l in comparison to the different polarization power spectra . as can be seen the white noise power is for planck a factor of 100 higher than for the ground - based experiments , which means the noise on a quiet map is about one order of magnitude lower than on the maps expected from planck . the main sensitivity of planck for the measurement of b - modes from gravitational waves comes from the reionization peak at low multipoles of l while the ground - based experiments like quiet will constrain @xmath1 from measuring at the maximum of the gravitational wave signal at @xmath7=100 , corresponding to an angular scale of 2 degrees . quiet is complementary to other experiments in many different ways : * quiet is the only ground - based effort using coherent receivers and thus dealing with different systematics than the bolometric systems . * it is the only experiment to measure the stokes parameters q and u simultaneously in one pixel which provides a good handle on several systematic effects . * the array at 40 ghz complements the high frequencies of the bolometer arrays and thus allows to account for the contamination from synchrotron radiation which dominates at low frequencies . * by using different telescope sizes quiet will be able to measure both large and small angular scales with the same receivers . = 0.36 ( lensing contribution not shown).,width=449 ] already in phase i quiet will be able to measure the e - mode spectrum to an unprecedented precision . the expected e- and b - mode power spectra for the phase ii of quiet with 1000 elements are shown in figure [ powspec ] . only the sensitivity of the w - band ( 90 ghz ) arrays from the 1.4 m telescopes was used assuming that the q - band sensitivity is used for foreground removal . the results were derived by including several real - data effects : a realistic observing strategy has been simulated and the method used in capmap to remove ground - pickup by mode - removal in single scans has been applied @xcite . the simulations also incorporate effects from e - b leakage where the b - mode measurement is degraded due to the e - mode signal leakage into the b - mode spectrum due to the finite size of the observed patch @xcite . additionally , the errors include a marginalization over the power in adjacent @xmath7-bins and for b - modes also over e power . the expected precision on cosmological parameters assuming initial adiabatic conditions is summarized in table [ cosmpar ] . note that these estimates had been performed before the publication of wmap results , but do agree well with the published wmap parameter errors . from the table one can see that quiet will improve the wmap parameter errors to a size competitive to the expected precision of planck . adding the quiet measurements to planck will only bring a small improvement in most of the parameters . however , quiet will already in phase i be able to constrain the tensor - to - scalar ratio together with planck to a level significantly smaller than planck is expected to . adding quiet phase ii will bring the limit on @xmath1 down to the level of 10@xmath0 . 3cmllllll & a & b & c & d & e + @xmath8 & 6 & 4 & 1 & 1 & 1 + @xmath9 & 8 & 7 & 4 & 2 & 2 + @xmath10 & 15 & 14 & 8 & 4 & 3 + @xmath11 & 34 & 23 & 14 & 7 & 6 + @xmath12 & 4 & 2 & 1 & 1 & 1 + @xmath13 & 1.35 & 0.021 & 0.009 & 0.042 & 0.009 + + + we are entering an era where probing gut scale physics is possible . a number of experiments are in preparation for seeing the signature of inflation in the b - modes . of these quiet is the only one using coherent detectors . a convincing discovery of the tiny signal will need consistent measurements from complementary techniques and observing frequencies . already within the next years quiet will reach the sensitivity to probe , together with other experiments , interesting levels of @xmath1 .	a major goal of upcoming experiments measuring the cosmic microwave background radiation ( cmbr ) is to reveal the subtle signature of inflation in the polarization pattern which requires unprecedented sensitivity and control of systematics . since the sensitivity of single receivers has reached fundamental limits future experiments will take advantage of large receiver arrays in order to significantly increase the sensitivity . here we introduce the q / u imaging experiment ( quiet ) which will use hemt - based receivers in chip packages at 90(40 ) ghz in the atacama desert . data taking is planned for the beginning of 2008 with prototype arrays of 91(19 ) receivers , an expansion to 1000 receivers is foreseen . with the two frequencies and a careful choice of scan regions there is the promise of effectively dealing with foregrounds and reaching a sensitivity approaching 10@xmath0 for the ratio of the tensor to scalar perturbations . [ 1999/12/01 v1.4c il nuovo cimento ]	3,112	242
not only are block copolymers promising materials for nano - patterned structures @xcite , drug delivery @xcite , and photonic applications @xcite , but they are also the ideal system for studying the influence of molecule architecture on macromolecular self - assembly @xcite . because of the ongoing interest in novel macromolecular organization , theoretical predictions based on heuristic characterization of molecular architecture offer crucial guidance to synthetic , experimental , and theoretical studies . though the standard diblock copolymer phase diagram @xcite was explained nearly a quarter of a century ago , the prediction and control of phase boundaries is fraught with subtle physical effects : weak segregation theory provides an understanding of the order - disorder transition @xcite , strong segregation theory ( sst ) predicts most of the ordered morphologies @xcite , and numerically exact , self - consistent field theory ( scft ) @xcite can resolve the small energetic differences between a variety of competing complex phases . in previous work , we argued that in diblock systems , that as the volume fraction of the inner block grows , ab interfaces are deformed into the shape of the voronoi polyhedra of micelle lattice , and therefore , the free - energy of micelle phases can be computed simply by studying properties of these polyhedra . in particular , we predicted that as volume fraction of inner micelle domain grows the a15 lattice of spheres should minimize the free energy as long as the hexagonal columnar phase ( hex ) did not intervene @xcite . we corroborated this prediction by implementing a spectral scft @xcite for branched diblock copolymers : in this paper we probe the regime of validity of our analytic analysis through both strong segregation theory and scft . though there is extremely small variation in the energy between different interfacial geometries , so too is the variation in energy between different stable phases . thus , we compare these two approaches not only by the phase diagram but also through the details of the ordering in the mesophases . since our original _ ansatz _ hinged on the ( minimal ) area of the interface between the incompatible blocks , we will focus strongly on the shape and structure of this interface . we will explore in detail the relationship between molecule architecture and the polyhedral distortion of the ab interface induced by the lattice packing of micelles to study hexagonal columnar phases . our results motivate the search for a stable a15 phase which we find in scft . in order to render the sphere - like phases stable in comparison to the hex phase , we are obliged to consider asymmetric diblocks ; while symmetric , linear diblock copolymers with an a and a b block have equivalent `` inside - out '' bulk morphologies when the a volume fraction @xmath1 is replaced with the b volume fraction @xmath2 , copolymers with branched or otherwise asymmetric architectures have no such symmetry , and therefore , tend to favor morphologies with one of the two components on the outside " a of curved ab interface ( i.e. on the outside of micelles ) . indeed , our previous scft analysis of branched diblocks is consistent with these findings . because of the challenge of accounting for all the competing effects , in section ii we implement a full scft for diblocks with the ab@xmath0 architecture to explore the mean field behavior of mitkoarm melts . in section iii we develop a strong - segregation theory approach for the hexagonal columnar phase which allows us to parameterize a large class of configurations and to explicitly assess the accuracy of the unit - cell approximation ( uca ) , which assumes the lattice voronoi cell to be perfectly cylindrical ( or spherical for three - dimensional lattices ) . our calculation builds on the kinked - path " calculation of milner and olmsted @xcite , and allows us to explore the influence of the hexagonal micelle lattice on the cylindrical morphology . we find that the shape of the voronoi cell of the lattice strongly influences the shape of the ab interface . in section iv we compare the predictions of the full scft calculation to the sst calculation in order to assess the accuracy of the latter . in addition , we demonstrate how the sst results of section iii can be used to compute an accurate phase boundary for transitions between lamellar ( lam ) to hex configurations . we briefly discuss the inverse phases ( where the b blocks are on the inside ) in section v. finally , we conclude in section vi . approximate self - consistent field theory calculations have explored the mean field phase behavior of linear diblocks with asymmetric monomer sizes @xcite which were confirmed through numerically exact scft @xcite . milner developed sst , applicable in the @xmath3 limit ( @xmath4 is the flory - huggins parameter for a and b monomers and @xmath5 is the degree of polymerization of the copolymers ) , for melts of a@xmath0b@xmath6 miktoarm star copolymers which also incorporates asymmetric monomer sizes @xcite . generally , the results of all of these calculations show that equilibrium morphologies which have blocks with stronger effective spring constants ( i.e. more arms or smaller statistical segment lengths ) on the outside of curved interfaces are favored over a much larger region of the phase space than in the symmetric diblock case . the details of the calculation implemented here will be reported elsewhere as a specific case of more general scft calculation for multiply - branched diblocks @xcite . the method is an extension of matsen and schick s spectral scft calculation for melts of linear @xcite and starblock copolymers @xcite . given the space group of a copolymer configuration , the mean field free - energy can be computed to arbitrary accuracy . the results of these scft calculations are accurate to the extent that mean field theory is correct and composition fluctuations can be ignored . the contributions of these fluctuations tends to zero in the @xmath7 limit @xcite , and therefore we can expect to capture the equilibrium results observed in the ps - pi miktoarm star experiments , for which @xmath8 @xcite . we consider a melt of fixed volume and number of copolymers , with each molecule composed of @xmath5 total monomers . the volume fraction of the a - type monomer is @xmath1 . in general , we could accomodate monomer asymmetry by allowing for two different statistical segment lengths , @xmath9 and @xmath10 , for the a and b species , respectively . in this case , each statistical segment length can be scaled appropriately so that the physical packing length " , @xmath11 , is fixed to the proper value for each of the two chemical species @xcite . thus , without loss of generality we define a common segment density for the two monomer types , @xmath12 . moreover , for our scft calculations in this section we will restrict ourselves to the case @xmath13 . the asymmetry of the copolymers we study arises entirely from their architecture . each copolymer is composed of one block of pure a monomer joined to @xmath14 blocks of pure b monomer at a common junction point ( see figure [ fig : figure1 ] ( a ) ) . we computed the full phase behavior for @xmath15 for @xmath16 . to achieve a numerical accuracy of @xmath17 for our free - energy calculations we employ up to 712 basis functions . this allows a precision in the phase boundary calculations which is better that @xmath18 for @xmath1 and @xmath19 for @xmath20 . the computed scft phase diagrams are shown in figures [ fig : figure2 ] and [ fig : figure3 ] . the first notable feature of these phase diagrams is that they are not symmetric about @xmath21 , as is the case for symmetric diblocks phase boundaries are shifted to the right . this indicates that phases with the @xmath14 b blocks on the outside of curved interfaces are favored . by adopting an interface with non - zero mean - curvature , the configuration relaxes the blocks on the outside of the ab interface at the expense of an increase in stretching of the inner blocks @xcite . due to the additional asymmetry introduced by the molecular architecture , the stability of phases with the b blocks on the outside of highly curved interfaces is enhanced . moreover , this effect is generally amplified by further increasing the number of b blocks in the molecule . if we look , for instance , at the boundary between flat and curved interfaces which separates the gyroid ( gyr ) from the lamellar ( lam ) phase , we see that for @xmath22 the transition occurs at @xmath23 , @xmath24 , @xmath25 , @xmath26 and @xmath27 for @xmath28 , 2 , 3 , 4 , and @xmath29 , respectively . likewise , the inverted " morphologies , with b blocks on the inside of curved interfaces , are suppressed because curving the interface inward towards the b domain introduces excess stretching in these blocks . this pushes the phase boundaries to greater @xmath1 for the inverse structures as well . we also note the appearance of a stable a15 phase of spherical micelles ( see figure [ fig : figure1 ] ( b ) ) . the a15 phase is also observed in experiments and simulations of another soft molecular system , namely neat mixtures of low molecular weight dendrons @xcite . the stability of this phase has been attributed to the fact that the area of the a15 voronoi cell is the minimal among the voronoi cells of three dimensional periodic structures ( for a given number density ) @xcite . we have argued that this minimal area makes the a15 lattice stable in the phase diagram of a 3-generation , multiply - branched copolymer @xcite : in the ( extreme ) limit that the ab interface adopts the flat shape of the voronoi cell of the micelle lattice , the a15 phase is stable over other sphere phases , such as bcc and fcc . thus , the appearance of the a15 phase in these phase diagrams is evidence of a highly distorted ab interface when b blocks compose the outer domains of micelles . due to melt incompressibilty , copolymer configurations must fill the lattice voronoi cell so that the monomer density is uniform @xcite . the tension of the outer block chains stretching towards the corners of the cell pulls on the interface , distorting it towards the polyhedral shape of the voronoi cell . of course , the interfacial tension of the ab interface and the tension of the inner blocks will frustrate this distortion , and the interface will actually distort into a rounded polyhedra of the same symmetry as the voronoi cell . in the next sections we will return to this point and will show , nonetheless , that the approximation of flat - faced ab interfaces improves as the number of outer arms grows . indeed , from figures [ fig : figure2 ] and [ fig : figure3 ] it is clear that increasing the asymmetry increases the stability of the a15 phase ; the window of a15 stability increases further in both the larger @xmath1 and smaller @xmath1 directions . since increasing @xmath14 , the number of b blocks per molecule , stabilizes sphere - like phases over cylindrical phases for larger @xmath1 the a15-hex boundary should move to larger @xmath1 . moreover , as @xmath14 grows the effective spring constant in the outer blocks grows as @xmath30 : for fixed number of b monomers @xmath31 , there are @xmath14 entropic springs with spring constant @xmath32^{-1}$ ] . this increased resistance to stretching will lead to a greater polyhedral distortion of the ab interface . this interfacial distortion is apparent in the oblate shape of the a domains of a15 phase in figure [ fig : figure1 ] ( b ) , corresponding to the oblate shape of voronoi cells of lattice sites on the faces of the @xmath33 unit cube . for larger values of @xmath14 interfaces approach the shape of voronoi cell at smaller values of @xmath1 and thus the bcc - a15 transition occurs at lower @xmath1 . of course , for the inverted micelles we expect the opposite to be true ; the tension of the highly stretched inner domains will prefer a spherical interface , ignoring the shape of the voronoi cell . therefore , we would not expect that the inverse a15 phase is stable at large @xmath1 and we do not find it within scft . at low segregations , near the order - disorder transition ( odt ) , the predicted phase behavior is dramatically altered from the case of linear diblocks . generally , the odt is shifted up to higher @xmath20 , or lower temperature . for symmetric diblocks the critical point , indicating a mean field , second - order disordered ( dis ) to lamellar transition , occurs at @xmath34 and @xmath21 @xcite . the critical points of the miktoarm star copolymer melts are shifted , for example , @xmath35 and @xmath36 for @xmath37 and @xmath38 and @xmath39 for @xmath40 . the upward @xmath20 shift in the odt is consistent with other scft calculations for conformationally asymmetric copolymers @xcite and indicates that chain fluctuations near the odt are systematically altered by molecular asymmetry . finally , we note that neither the cubic double - diamond nor the hexagonal - perforated lamellar phases are stable in these systems as with symmetric , linear diblocks . in order to compute the strong segregation phase behavior of ab@xmath0 miktoarm star copolymer melts , we compute the dis - bcc , bcc - hex , and hex - lam phase boundaries at @xmath41 @xcite . due to the numerical difficulties of considering the gyr and a15 phases for large @xmath20 we do not include these phases in our stability analysis . additionally , we ignore the window of stable closed - packed spheres , fcc , which is predicted by scft calculations to occur near the odt . since we expect that the free - energy differences between the sphere phases will be relatively small at these strong segregations @xcite our calculation should capture the change from spheres to cylinders . likhtman and semenov developed a general sst calculation to assess the stability of bicontinuous phases in the limit @xmath3 . their calculation shows that the gyr phase is unstable when chain fluctuations are ignored @xcite . it is not known whether the bicontinuous gyr phase is stable for finite but large @xmath20 even within the well studied scft phase diagram of symmetric , linear diblocks @xcite . recent experiments on melts of fluorinated pi - pee diblocks indeed suggest that this morphology is an equilibrium phase for @xmath42 @xcite , and we expect that the same may be true for more asymmetric copolymers , such as these miktoarm stars . although we can not prove its stability with our scft calculations , we expect that the gyr phase appears at compositions intermediate to the stable hex and lam regions . in figure [ fig : figure4 ] we compare our scft results for ab@xmath6 copolymers to the experimental results on polyisoprene ( pi)-polystyrene ( ps ) a@xmath43b@xmath43 @xcite , ab@xmath43 @xcite , ab@xmath44 @xcite , and ab@xmath45 @xcite miktoarm star copolymer melts . milner showed that sst phase boundaries depend only on @xmath1 , and the asymmetry parameter , @xmath46 , where @xmath47 and @xmath48 are the number of a and b chains per molecule @xcite . since we are considering symmetric monomers , @xmath13 , @xmath49 , and @xmath50 for our ab@xmath0 copolymers , we have simply @xmath51 . we see from figure [ fig : figure4 ] that scft very accurately captures the observed phase behavior of miktoarm star copolymer melts . therefore , ignoring composition fluctuations , as scft dictates , does not alter the phase behavior at this level of segregation . moreover , as the data suggest , the effect of the molecular asymmetry seems to saturate for @xmath52 , and phase boundaries do not change significantly as a function of @xmath1 for further asymmetry @xmath14 . it has been suggested that the saturation of phase boundaries occurs when the spheres or cylinders of the inner block reach close packing , _ i.e. _ @xmath53 for hex and @xmath54 for the body - centered cubic ( bcc ) lattice @xcite . in the next section we employ sst to explore how the symmetry of the micelle lattice frustrates their self - assembly , and moreover , how this frustration is related to the saturation of phase boundaries for @xmath55 . the appearance of a stable a15 phase suggests that the shape of the ab interface is strongly affected by the lattice symmetry for highly asymmetric diblock melts . tension in the outer block chains maintains a uniform outer domain thickness and consequently distorts the interface into the shape of the voronoi cell . however , previous studies have demonstrated that for symmetric ab diblocks the shape of the interface , and therefore , the calculation of the free - energy are insensitive to the shape of the voronoi cell @xcite . the free energy is dominated by the tension of the ab interface , and so the minimal - area cylinder is favored . because of this , a unit - cell approximation ( uca ) is often taken for the domain shape of the micelles . for instance , in the columnar and spherical phases , the voronoi cell is approximated by a perfect cylinder or sphere , respectively . however , packing these cylinders or spheres into a space - filling lattice leaves voids in the interstices which is incompatible with the incompressible melt state . therefore , the uca can only provide an estimate for the free - energy of these morphologies . in fact , this estimate is a lower - bound to the true free - energy since distorting the round , approximate unit cells can only raise the free energy either by stretching the outer block or distorting the interface . as our results in the last section suggest , the `` packing frustration '' @xcite between the surface tension and the stretching is highly dependent on molecular asymmetry . it is also unlikely that the uca can capture the saturation as a function of the number of blocks , @xmath14 ; the close - packing limitation can not be captured in the uca since the inner volumes can fill 100% of the approximate unit - cell without overlap @xcite . in order to quantitatively explore the role of packing frustration in miktoarm star copolymer melts , we build upon the sst calculation of the hex phase free energy for miktoarm star copolymers @xcite . olmsted and milner developed a kinked - path " _ ansatz _ for the extension of the copolymer chains , which allows the configuration to maintain a cylindrical ab interface while satisfying the constraints of melt incompressibility ( see figure [ fig : figure5 ] ) . here , we extend that calculation allowing the interface to adopt a more general class of interfaces , allowing us to systematically explore the effect of packing frustration in this morphology . within sst we ignore chain fluctuations , in addition to the composition fluctuations absent from the full scft . we assume that each chain extends only along its classical trajectory , and these paths must be consistent with the constant monomer concentration of the melt state . following @xcite we divide the hexagonal unit cell into infinitesimal wedges which extend along the direction of the chain paths . due to incompressibility we must have wedges with a volume fraction of a domain @xmath1 and a volume fraction of b domain @xmath2 . unless the ab interface adopts the same shape as the voronoi cell , the chains , and consequently the wedges must bend towards the cell corners in order to distribute the volume of the chains evenly . each chain starts with an a block that extends from the center of the micelle radially to the ab interface at an angle , @xmath56 ( see figure [ fig : figure5 ] ( b ) ) . this path extends along the vector , @xmath57 , which parameterizes the interface as function of @xmath56 . from that point on the interface , the outer blocks extend to some other point , at angle @xmath58 , on outer wall of the unit cell , along the vector @xmath59 . the vector which extends from the center of the micelle to the outer wall of the unit cell at @xmath58 is given by @xmath60 . we can parameterize the kinking of the wedges by the function @xmath61 which maps the chain position at the interface to the position at the wall of the cell . therefore , the extension of the b portion of the wedge is given by @xmath62 . the hexgonal unit cell is parameterized for @xmath63 by @xmath64 where @xmath65 is the cross - sectional area of the micelle and @xmath66 is the radial unit vector . because the configuration is strongly segregated and the inner domain is composed only of a monomers , we consider a class of interfaces which encloses a constant area fraction , @xmath1 , of the total micelle area . in particular , we would like to consider a class of interfaces which vary from area - minimizing and uniformly curved ( circle ) to stretch - minimizing and polygonal ( hexagon ) . such a one parameter family of interfaces is : @xmath67 when @xmath68 this is the circle and when @xmath69 it traces out an affinely shrunken version of the hexagonal unit cell ( [ eq : r ] ) . for all other values of @xmath70 $ ] , the shape interpolates the two extremes ( figure [ fig : figure6 ] ) . by construction the enclosed area of the interface is @xmath71 for all @xmath72 . we compute the free energy as a function of @xmath72 and minimize in search of a variational ground state . we expect the ground state to favor the minimal area , round interface ( @xmath68 ) in the low @xmath1 limit because differences in stretching to the walls and corners of the voronoi cell are small on the scale of the radius of gyration of the outer blocks , @xmath73^{1/2 } $ ] . in the other limit , @xmath74 , differences in chains stretching towards the corners and walls becomes large on this length scale , and the hexagonal interface ( @xmath69 ) will be preferred . we refer to the former case as the round - interface limit ( ril ) and the latter as the polyhedral - interface limit ( pil ) . of course , there is no guarantee that the true ground state belongs to this class of configurations . nonetheless , minimizing the free energy over @xmath72 will provide a variational upper - bound on the ground state free energy which is lower than either that from the straight - path calculation or the circular interface calculation . the essence of the kinked - path approximation is that each wedge must locally satisfy the volume constraint for the diblock so that : @xmath75\times \frac{d}{d \theta } \left[{\bf r}(\theta)+{\bf r}_a(\theta)\right ] = \\ ( 1-f ) { \bf r}_a(\theta)\times \frac{d}{d \theta } { \bf r}_a(\theta)\end{gathered}\ ] ] given the parameterizations introduced in eqs . ( [ eq : r ] ) and ( [ eq : r_alpha ] ) we find that @xmath76 where @xmath77 . it is easy to verify that @xmath78 has the appropriate limits : when @xmath68 , eq . ( [ eq : tantheta ] ) reduces to the results of ref . @xcite , and when @xmath79 , @xmath80 , which is the straight - path _ ansatz_. the free energy has three parts , arising from the interfacial energy , the stretching of the a blocks and the stretching of the b blocks : @xmath81 the interfacial energy per molecule ( in units of @xmath82 ) is given by the effective surface tension , @xmath83 @xcite , times the area of the ab interface divided by the number of chains per micelle : @xmath84 where @xmath65 is the cross sectional area of the micelle and , @xmath85 is the area of the ab interface measured in units of the area of circular interface enclosing the same volume . we compute the stretching free - energy using the self - consistent parabolic brush potential for melts @xcite as is shown in ref . for the inner a domain we have @xmath86 where @xmath87 is the reduced stretching moment of the a domain , introduced in ref . @xcite . the stretching term due to the outer b domain is given by @xmath88 where the stretching moment of the b blocks is given by the integral @xmath89 with @xmath90 and @xmath91 with @xmath92 . combining these terms we have the total free - energy per molecule and minimizing over the dimension of the micelle , @xmath93 , we compute the micelle free - energy per chain in units of @xmath82 : @xmath94^{1/3 } \ , \ ] ] where @xmath95 and @xmath96 is proportional to @xmath97and is independent of composition @xcite . the inner domain stretching , @xmath98 , and the interfacial area , @xmath99 , are minimized by circular interface at @xmath68 , while the outer domain stretching , @xmath100 , is always minimized for @xmath101 ( though we are only considering @xmath70 $ ] ) . as @xmath74 the only finite energy configuration is the hexagonal interface with @xmath102 . minimizing over alpha for @xmath103 and @xmath104 we find the optimum energy interface configurations and show the results in figure [ fig : figure6 ] . in the low @xmath1 limit a round interface is favored due to the relatively lower tension in the b chains . for larger @xmath1 , the stretching term of the b chains ( [ eq : f(alpha ) ] ) begins to dominate , distorting the interface towards the hexagon . because of the @xmath105 increase in the effective spring of the outer chains , it follows that for larger asymmetry the onset of this polyhedral distortion occurs at lower compositions @xmath1 . for high molecular asymmetries , @xmath106 , and a volume fractions larger than @xmath107 , we find equilibrium shapes with @xmath108 . in this case , the interface is bowed in , away from the walls , and pulled out towards the corners . however , because it is the only finite energy configuration , each solution approaches @xmath102 as @xmath74 . we expect that the true ground state prefers an interface which asymptotically approaches a hexagonal shape , rather than overshooting " @xmath69 before returning to the @xmath74 limit . the bowing inward of the interface is due to the oversimplified representation of the interface which does not allow the corners of the interface to relax smoothly towards the corners of the voronoi cell . nevertheless , this simple representation of the interface shape proves sufficient to capture the effect of the packing frustration introduced by the hexagonal symmetry of the micelle lattice . in order to make contact with the results of scft and experiment , we need to estimate the value of @xmath72 for an arbitrary interface . we introduce a numerical measure of the distortion from the circular shape . for copolymer melts in the large @xmath20 limit the ratio of the interfacial thickness to the domain size scales as @xmath109 @xcite ; thus , in the strong segregation limit the interface becomes infinitely thin on the scale of the domain . we find the location of the interface by the contour for which the volume fraction of a is 0.5 . one of these contours is shown in the inset of figure [ fig : figure6 ] . using this contour we measure the minimum and maximum distance from the center of the micelle to the interface , @xmath110 and @xmath111 , respectively . as a measure of the distortion from a circular interface , we introduce @xmath112 which is similar to the normalized amplitude of the @xmath113 modulation of the interface @xcite . since @xmath114 for the circular interface and @xmath115 for a hexagonal interface , we may define @xmath116 in the two limits @xmath117 and @xmath118 and it can be shown that @xmath119 is never greater than @xmath120 for the class of interfaces parameterized by @xmath121 . in particular , @xmath122 , and @xmath123 for @xmath124 , and @xmath125 , respectively . by using scft for linear diblock copolymers , matsen and bates showed that @xmath126 for @xmath1 as large as @xmath127 @xcite , and they concluded that there is negligible distortion so that the effect of packing frustration can be ignored for the hex phase . however , we find that even for the @xmath28 diblock case , interfacial distortion becomes appreciable for @xmath128 . while the hex phase is unstable at these compositions for ab diblocks , experiments for melts of abc triblocks yield stable hex phases with a composite ab inner domain volume fraction of about @xmath129 @xcite . for these triblocks , the interface separating the outermost c domain and the b domain is found to be very nearly hexagonal , demonstrating that packing - frustration in the hex phase is indeed amplified as the volume fraction of the inner domain is increased . indeed , this frustration can be relieved by adding c - type homopolymer which aggregates in the corners of the hexagonal unit cell , allowing the bc interface to relax to uniform curvature @xcite . we plot the free - energy of the optimal configuration as a function of composition for the various asymmetries in figure [ fig : figure7 ] . for lower compositions the free - energy is nearer the uca lower bound . as the composition increases the free - energy approaches the pil upper - bound . clearly , for large @xmath130 and large @xmath1 the free - energy of our configuration is best approximated by the straight - path upper bound @xcite . this suggests that for asymmetric copolymers , the ab interface of the hex micelle is significantly deformed from the uniformly - curved configuration over compositions ranges where this phase competes for stability . we review the consequences of this conclusion for the phase calculation in the next section . we compare the results of our sst calculation for asymmetric miktoarm star copolymer melts to the more exact " results of our scft calculations for the hex phase . since the sst approximation results from an asymptotic expansion of the full scft partition function in the @xmath3 limit @xcite , we expect that sst and scft results should agree provided that we have used the approximately correct shape of the ab interface . strictly speaking we should consider our sst results to be an upper bound on the free - energy of the hex phase since we did not minimize over all possible ground state interface and chain configurations . nevertheless , we argue that this bound is sufficiently close to the true ground state to capture the phase behavior of asymmetric miktoarm stars . we use the correspondence between @xmath72 and @xmath131 for @xmath121 as a basis for direct comparison of the predicted values from sst and the `` measured '' values of @xmath131 from scft . a comparison is made in figure [ fig : figure6 ] for @xmath37 and @xmath40 . despite our simplified representation of interface shape , @xmath121 , and the crude means of measuring the equilibrium scft shape parameter , the predicted @xmath72 and computed @xmath131 compare remarkably well . in particular , the scft results confirm that increasing @xmath1 and @xmath130 increases the equilibrium shape distortion . there is a systematic tendency in sst to overestimate the hexagonal distortion for low inner domain volume fractions . this can be attributed to the inadequacy of our interface parameterization , @xmath121 , at low distortions where the ab interface is likely best represented as a constant radius with a superposed @xmath132 modulation @xcite . nevertheless , the agreement at larger @xmath1 can be taken as evidence that we have accurately accounted for packing frustration for our sst hex phase calculation . the effect of interfacial distortion on the location of the predicted lam - hex phase boundary for miktoarm star copolymers is shown in figure [ fig : figure8 ] . there we show ( with dashed lines ) the calculated phase boundaries using both the lower- and upper - bound approximations which assume that the interface and voronoi cell are either both circular ( _ i.e. _ the uca ) or both hexagonal ( _ i.e. _ the pil ) , respectively @xcite . note that there is no difference between the round and polyhedral voronoi cell for the lam phase , and therefore , no approximation is necessary for this geometry . since the uca calculation underestimates the hex free - energy it overestimates the composition at which the hex phase becomes unstable against the lam phase . likewise , the upper - bound calculation underestimates the location of the lam - hex phase boundary . we expect , therefore , that the true lam - hex sst phase boundary should lie between these two approximate boundaries . though the upper - bound of the round - interface limit is close to the uca lower - bound for symmetric diblocks @xcite , the latter can not be used to predict the phase behavior for asymmetric diblock melts in sst , as experimental results suggest @xcite . this discrepancy is illustrated in figure [ fig : figure8 ] : while scft predicts that the effect of increasing asymmetry saturates for @xmath106 , the uca predicts that the phase boundaries move to larger @xmath1 as @xmath130 increases . the solid line in figure [ fig : figure8 ] shows the sst phase lam - hex phase boundary computed with the results of the previous section . for @xmath133 the ab interface is nearly undistorted and circular ; thus , the free - energy is best approximated by the lower - bound , uca calculation . hence , our lam - hex phase boundary follows the uca phase boundary for low asymmetry . however , for @xmath134 the effect of interfacial distortion becomes important , and the free - energy is closely approximated by the upper - bound , pil calculation . therefore , for large asymmetry our sst lam - hex boundary tends toward the hexagonal - interface approximation phase boundary . this transition from the round to the hexagonal interface approximation is also reflected in the scft boundary . indeed , our sst and scft lam - hex phase boundaries agree well in this entire parameter range ( @xmath135 ) . similarly , we see that hex - bcc boundary computed from scft tends to follow the uca boundary for @xmath136 and follows the polyhedral - interface approximation boundary for @xmath137 , confirming our picture of the effect of packing frustration on asymmetric copolymers . again , we know that interfacial distortion must be important along the phase boundary between sphere and columnar phases because the a15 , with its minimal interfacial area , is the favored sphere phase there . it would be interesting to pursue a one - parameter class of interfaces that interpolated between a spherical and polyhedral ab interface . such a calculation would not only predict the spherical - cylindrical micelle phase boundary but it would also elucidate the transition from bcc spheres with nearly spherical interfaces at small @xmath1 to a15 sphere - like micelles with polyhedral interfaces for larger @xmath1 . we can also see that within the polyhedral - interface approximation calculations , the effect of increasing asymmetry begins to saturate for @xmath138 , as seen in experiment @xcite . again , this stems from the relative @xmath139 increase in stretching energy of the b blocks over the a blocks . in the large @xmath130 limit the a block stretching can be ignored , and ( [ eq : f(alpha ) ] ) can be approximated by @xmath140^{1/3}$ ] , where we have assumed that @xmath141 . the same is true for the free - energy of other morphologies , namely @xmath142 , where @xmath143 lam , a15 , bcc , _ etc . _ and @xmath144 is some function specific to the morphology which depends only on composition @xcite . because all free - energies scale the same way with @xmath130 , the location of the transition between any two phases depends only on composition and is independent of @xmath130 in this limit . it should be noted that the effect of increasing asymmetry saturates for the uca bounds as well , though this occurs very close to @xmath145 for both the lam - hex and hex - bcc boundaries . the location of phase boundaries computed using the polyhedral - interface approximation saturates near @xmath146 and @xmath147 , for the hex - bcc and lam - hex boundaries , respectively . while the agreement between , experiment , our scft calculations and our sst analysis for @xmath148 is improved over previous efforts , the sst results appears still to overestimate the effect of increasing asymmetry . we should note , however , that sst is an asymptotic expansion of the full scft near @xmath3 , and at finite values of @xmath20 we need to assess the importance of higher order corrections . it is known that lowest order , strong - segregation results are @xmath149 $ ] and that the leading corrections which can distinguish between phases are @xmath150 . one of these corrections is associated with fluctuations of chain ends and junction points and are thus proportional to @xmath151 @xcite . the other is associated with a proximal layer near the interface , where the stretching energy of the chains deviates from the predictions of the parabolic brush potential @xcite . at @xmath41 this correction can be as high as @xmath152 , and will surely effect the relative free - energies of competing phases . further errors arise from the implicit assumption of a parabolic chemical potential used to compute the free - energy associated with molten polymer brushes @xcite . it has been shown that this approximation is equivalent to allowing for a negative chain end - density near the surface of a convex brush ; nevertheless , it has been shown that relaxing the constraint of a non - negative end - distribution hardly perturbs the free - energy of a two - dimensional convex brush @xcite . however small , this parabolic chemical - potential underestimates the constrained stretching free - energy of domains on the outside of cylindrical and spherical domains . a correction to the stretching energy of b domains at large asymmetries of 1% leads to a correction of the full free - energy of order 0.3% . while this seems small , the difference between upper- and lower - bound sst free - energies is only 3.6% and 6.8% , for the hex and bcc phases , respectively . since even small corrections to the free - energy lead to significant changes in the predicted phase behavior , we expect approximations of this order may be relevant . we have applied the sst analysis here to the hex phase where the a domain composes the interior of the micelle and the multiple b blocks compose the outer domain . these asymmetric configurations have the stiffer " blocks on the outside of micelle , hence , they are responsible for appreciable interfacial distortion . however , we could also use our sst results to determine the phase behavior of the inverse micelles , where the multiple b blocks composed the inner domain . assuming that all b chains extend radially from the center of the micelle to the ab interface , we can compute the inverse micelle free - energy by the substitutions @xmath153 and @xmath154 in eq . ( [ eq : f(alpha ) ] ) . we find , however , that such a calculation produces overestimates of the free - energy . moreover , the assumption that the b chains stretch only radially is inconsistent with scft results for the inverse phases . figure [ fig : figure9 ] ( a ) shows a real - space a monomer distribution for the hex phase with @xmath155 , @xmath156 and @xmath157 . figure [ fig : figure9 ] ( b ) plots the distribution of a monomer along a horizontal line extending across the micelle for @xmath157 and the inverse phase at @xmath158 . clearly , both configurations are strongly segregated with the volume fraction of the a monomer , @xmath159 , constant outside the inner domain ( @xmath160 for @xmath157 and @xmath161 for @xmath158 ) . however , while the volume - fraction of b component is zero inside the @xmath157 micelle , the volume fraction a component is never zero inside the inverse micelle at @xmath158 . this shows that in the inverse micelle , the miktoarm star junction points are not strictly confined to a narrow ab interface . rather , junction points are located somewhere within the inner domain and some of the chains must bend back towards the outer domain instead of stretching radially towards the center of the micelle . thus the simple sst analysis of section iii does not apply to inverse phases since the assumption that junction points are confined to the ab interface is clearly violated . in order to account for this we would parameterize the distribution of junction points in the micelle , and minimize the free - energy over the distribution . moreover , we would need to find the optimal configuration of b blocks , bending inwards and outwards in the micelle . such a calculation introduces many more degrees of freedom ; minimizing over these is likely very difficult . since scft provides numerically exact results for even very large segregations , it is a much better approach to finding the inverse micelle configurations to determining the phase behavior . we have implemented sst and scft calculations which elucidate the coupling between the molecular asymmetry of diblock copolymers and packing frustration introduced by the micelle lattice . in particular , we find that the shape of the ab interface is highly sensitive to molecular asymmetry , in the case of our miktoarm star copolymers , the number of b blocks per molecule . for linear diblocks , or otherwise nearly symmetric architectures , we show that the ab interface of micelles in the hex phase is rather insensitive to the shape of voronoi cell and maintains a nearly constant mean curvature shape . however , for ab@xmath0 miktoarm star copolymers with @xmath162 the interface is highly distorted towards the hexagonal shape of the voronoi cell in regions where the hex phase competes for stability . the effect of this distortion is to shift the predicted hex - lam phase boundary to lower compositions than is predicted by approximating the voronoi cell as a perfect cylinder . as a consequence of the importance of interfaces , we expect the a15 phase of spherical micelles to be stable for highly asymmetric copolymers due to the minimal area of its voronoi cell amongst lattices in three - dimensions . this prediction is borne out by our numerical scft calculations which compute the full phase behavior of ab@xmath0 miktoarm star copolymers for @xmath16 . indeed , we find that as molecular asymmetry is increased , the stability of the a15 phase is enhanced indicating that the polyhedral approximation for the ab interface is more valid as the copolymer architecture becomes more asymmetric . it is worth noting that the a15 phase of spherical micelles has yet to be identified experimentally in melts of highly asymmetric diblock copolymers . pochan , gido and coworkers report the appearance of an equilibrium cubic phase of spheres in ps - pi ab@xmath43 copolymer melts in regions of the @xmath163 phase space where we might expect a15 to be stable @xcite . however , they note that the small - angle x - ray scattering data from these melts can not distinguish between a simple - cubic or bcc micelle lattice ; nor does the data rule out the possibility of the a15 lattice which contains all of the reflections of the bcc lattice . experiments on sphere phases of more asymmetric miktoarm star copolymers in parameter ranges where a15 should be stable tend to yield poorly ordered arrangements of micelles @xcite . because the a15 and bcc phase are nearly degenerate ( with a free - energy difference on the order of 0.2% ) , the system may be kinetically trapped in some sort of glassy intermediate state in these parameter ranges . despite these difficulties , however , both the robustness of the minimal voronoi cell area principle and the scft calculations strongly indicate that the a15 phase of spherical micelles is an equilibrium phase of copolymer melts of sufficient molecular asymmetry . it is a pleasure to acknowledge stimulating discussions with b. didonna , g. fredrickson , s. gido , and v. percec . this work was supported by nsf grants dmr01 - 29804 , dmr01 - 02549 , and int99 - 10017 , by the state of pennsylvania under the nanotechnology institute , and by the university of pennsylvania research foundation . all boundaries are computed for @xmath156 with the exception of the low-@xmath1 bcc - hex and hex - lam boundaries for @xmath40 , 4 and 5 . for @xmath40 these boundaries are computed for @xmath164 , and for @xmath155 and 5 these boundaries are computed at @xmath165 . as @xmath3 the phase boundaries become independent of the degree of segregation , @xmath20 , @xcite and thus , we expect that the phase boundaries computed at lower segregation should very closely approximate the higher segregation results . when considering close - packed lattices of spherical micelles , hexagonally close - packed and fcc lattices are known to have degenerate free - energies within mean field theory @xcite . here , we consider only the free - energy of the phase of @xmath166 symmetry , the fcc phase .	we study ab@xmath0 miktoarm star block copolymers in the strong segregation limit , focussing on the role that the ab interface plays in determining the phase behavior . we develop an extension of the kinked - path approach which allows us to explore the energetic dependence on interfacial shape . we consider a one - parameter family of interfaces to study the columnar to lamellar transition in asymmetric stars . we compare with recent experimental results . we discuss the stability of the a15 lattice of sphere - like micelles in the context of interfacial energy minimization . we corroborate our theory by implementing a numerically exact self - consistent field theory to probe the phase diagram and the shape of the ab interface .	12,424	182
the dynamics studied in this paper owes its origin to a work of bullett @xcite and to a series of articles motivated by @xcite most notably @xcite . the object of study in @xcite is the dynamical system that arises on iterating a certain relation on @xmath9 . this relation is the zero set of a polynomial @xmath10 $ ] of a certain form such that : * @xmath11 and @xmath12 are generically quadratic ; and * no irreducible component of @xmath13 is of the form @xmath14 or @xmath15 , where @xmath16 . the form of @xmath17 above is such that , if @xmath18 denotes the biprojective completion of @xmath13 in @xmath19 and @xmath20 denotes the projection onto the @xmath21th factor , then the set - valued maps @xmath22 are both @xmath23-valued ( counting intersections according to multiplicity ) . in @xcite , this set - up was extended to polynomials @xmath10 $ ] of arbitrary degree that induce relations @xmath24 such that the first map given by is @xmath25-valued and the second map is @xmath26-valued , @xmath27 . it would be interesting to know whether such a correspondence exhibits an equidistribution property in analogy to brolin s theorem ( * ? ? ? * theorem 16.1 ) . this problem is the starting point of this paper . the reader will be aware of recent results by dinh and sibony @xcite that , it would seem , should immediately solve the above problem . _ however , key assumptions in the theorems of @xcite fail to hold for many interesting correspondences on @xmath1 . _ we shall discuss what this assertion means in the remainder of this section . we will need several definitions to be able to state our results rigorously , which we postpone to section [ s : results ] . on the dynamics of multivalued maps between complex manifolds : results of perhaps the broadest scope are established in @xcite . we borrow from @xcite the following definition . [ d : holcorr ] let @xmath28 and @xmath29 be two compact complex manifolds of dimension @xmath30 . we say that @xmath31 is a _ holomorphic @xmath30-chain _ in @xmath32 if @xmath31 is a formal linear combination of the form @xmath33 where the @xmath34 s are positive integers and @xmath35 are distinct irreducible complex subvarieties of @xmath32 of pure dimension @xmath30 . let @xmath36 denote the projection onto @xmath37 . we say that the holomorphic @xmath30-chain _ @xmath31 determines a meromorphic correspondence of @xmath28 onto @xmath29 _ if , for each @xmath38 in , @xmath39 and @xmath40 are surjective . @xmath31 determines a set - valued map , which we denote as @xmath41 , as follows : @xmath42 we call @xmath41 a _ holomorphic correspondence _ if @xmath43 is a finite set for every @xmath44 . [ rem : explcorr ] it is helpful to encode holomorphic correspondences as holomorphic chains . circumstances arise where , in the notation of , @xmath45 . for instance : even if we start with a holomorphic correspondence on @xmath1 determined by an _ irreducible _ variety @xmath46 , composing @xmath47 with itself ( see section [ s : keydefns ] ) can result in a variety that is not irreducible and some of whose irreducible components occur with multiplicity@xmath48 . suppose @xmath49 is a compact khler manifold of dimension @xmath30 ( @xmath50 denoting the normalised khler form ) and @xmath0 is a meromorphic correspondence of @xmath51 onto itself . one of the results in @xcite says , roughly , that if @xmath52 , where @xmath53 and @xmath54 are the dynamical degrees of @xmath0 of order @xmath55 and @xmath30 respectively , then there exists a probability measure @xmath2 satisfying @xmath56 , such that @xmath57 when @xmath58 , the assumption @xmath52 translates into the assumption that the ( generic ) number of pre - images under @xmath0 is _ strictly larger _ than the number of images under @xmath0 , both counted according to multiplicity . when @xmath59 , the assumption @xmath52 is a natural one however , none of the techniques in the current literature are of help in studying correspondences @xmath0 for which @xmath60 , even when @xmath61 ( in this paper @xmath62 will denote the khler form associated to the fubini study metric ) . why should one be interested in the dynamics of a correspondence @xmath63 for which @xmath60 ? the work of bullett and collaborators suggest several reasons in the case @xmath64 . thus , _ we shall focus on correspondences on @xmath1 _ ( although parts of our results hold true for riemann surfaces ) . a mating of two monic polynomials on @xmath9 is a construction by douady @xcite that , given two monic polynomials @xmath65 $ ] of the same degree , produces a continuous branched covering @xmath66 of a topological sphere to itself whose dynamics emulates that of @xmath67 or of @xmath17 on separate hemispheres . for certain natural choices of pairs @xmath68 , one can determine in principle see ( * ? ? ? * theorem 2.1 ) when @xmath66 is semiconjugate to a rational map on @xmath1 . in a series of papers @xcite , bullett and collaborators extend this idea to matings between polynomial maps and certain discrete subgroups of the mbius group or certain hecke groups . the holomorphic objects whose dynamics turn out to be conjugate to that of matings in this new sense are holomorphic correspondences on @xmath1 . such correspondences are interesting because they expose further the parallels between the dynamics of kleinian groups and of rational maps . it would be interesting to devise an ergodic theory for such matings . _ in all known constructions where a holomorphic correspondence @xmath0 of @xmath1 models the dynamics of a mating of some polynomial with some group , @xmath69_. in this context , to produce an invariant measure and , especially , to give an explicit prescription for it would require that the techniques in @xcite be supplemented by other ideas . we now give an informal description of our work ( rigorous statements are given in section [ s : results ] ) . since we mentioned brolin s theorem , we ought to mention that an analogue of brolin s theorem follows from and certain other results in @xcite when @xmath70 . to be more precise : there exists a polar set @xmath71 such that @xmath72 where @xmath2 is as in with @xmath64 . this means that we have extremely precise information about the measure @xmath2 . our first theorem ( theorem [ t : d_1more ] ) uses this information to show that the support of @xmath2 is disjoint from the normality set of @xmath0 , where `` normality set '' is the analogue of the fatou set in the context of correspondences . when @xmath0 ( a holomorphic correspondence on @xmath1 ) satisfies @xmath73 , there is no reason to expect . indeed , consider these examples : @xmath74 , in which case @xmath75 ; or the holomorphic correspondence @xmath76 determined by the @xmath77completion of the zero set of the rational function @xmath78 , in which case @xmath79 . when @xmath73 , we draw upon certain ideas of mcgehee @xcite . we show that if @xmath0 admits a repeller @xmath80 in the sense of mcgehee , which extends the concept of a repeller known for maps having certain properties , then there exists a neighbourhood @xmath81 and a probability measure @xmath2 satisfying @xmath82 , such that @xmath83 a rigorous statement of this is given by theorem [ t : d_1less ] . the condition that @xmath0 admits a repeller is very natural , and was motivated by the various examples constructed by bullett _ et al_. we take up one class of these examples in section [ s : example ] and show that the conditions stated in theorem [ t : d_1less ] hold true for this class . observe that differs from in that it does not state that @xmath84 is polar ( or even nowhere dense ) , but this is the best one can expect ( see remark [ rem : care ] below ) . the measure @xmath2 appearing in and is not , in general , invariant under @xmath0 in the usual measure - theoretic sense ; it is merely @xmath85-invariant . we direct the reader to the paragraph preceding corollary [ c : d1_less ] for more details . however , equidistribution is a phenomenon that arises in many situations ( see , for instance , the work of clozel , oh and ullmo @xcite , which involves correspondences in a different context ) and is revealing from the dynamical viewpoint . but if , for a holomorphic correspondence @xmath0 on @xmath1 , @xmath86 , then we can show that there exists a measure that _ is _ invariant under @xmath0 in the usual sense . this is the content of corollary [ c : d1_less ] below . in this section , we isolate certain essential definitions that are somewhat long . readers who are familiar with the rule for composing holomorphic correspondences can proceed to section [ ss : normality ] , where we define the normality set of a holomorphic correspondence . there are significant differences _ at the level of formalism _ between our definition of the normality set and those that have appeared in the literature earlier . this definition , which is also a bit involved , thus has a subsection , section [ ss : normality ] , dedicated to it . let @xmath51 be a complex manifold of dimension @xmath30 . for any holomorphic @xmath30-chain @xmath31 on @xmath51 , we define the _ support _ of @xmath31 , assuming the representation , by @xmath87 consider two holomorphic correspondences , determined by the @xmath30-chains @xmath88 in @xmath89 . the @xmath30-chains @xmath90 , @xmath91 have the alternative representations @xmath92 where the primed sums indicate that , in the above representation , the irreducible subvarieties @xmath93 , are _ not necessarily distinct _ and are repeated according to the coefficients @xmath94 . we define the holomorphic @xmath30-chain @xmath95 by the following two requirements : @xmath96 where the @xmath97 s are the distinct irreducible components of the subvariety on the right - hand side of , and @xmath98 is the generic number @xmath99 s as @xmath100 varies through @xmath97 for which the membership conditions on the right - hand side of are satisfied . finally , we define the @xmath30-chain @xmath101 if @xmath102 and @xmath103 determine holomorphic correspondences on @xmath51 , then so does @xmath104 . it requires a certain amount of intersection theory to show this , but the interested reader is directed to section [ s : basic ] for an _ elementary _ proof of this fact when @xmath105 . the @xmath26-fold iterate of @xmath31 will be denoted by @xmath106 . given a @xmath30-chain @xmath31 on @xmath51 ( @xmath51 as above ) , we may view @xmath107 as a relation of @xmath51 to itself . in certain sections of this paper , we will need to make essential use of mcgehee s results from @xcite on the dynamics of closed relations on compact spaces . let @xmath102 and @xmath103 be as above . then , the composition of the two relations @xmath108 and @xmath109 in the classical sense ( denoted here by @xmath110 ) which is the sense in which the term is used in @xcite is defined as @xmath111 it is easy to see that for holomorphic correspondences @xmath112 in what follows , we shall adopt a notational simplification . given a @xmath30-chain @xmath31 that determines a holomorphic correspondence , and there is no scope for confusion , we shall simply denote @xmath41 by @xmath0 . let @xmath0 be a holomorphic correspondence on a compact riemann surface @xmath113 . the motivation for the concept of the _ normality set of @xmath0 _ is quite simple . however , we will need some formalism that enables good book - keeping . in this subsection , we will use the representation for a holomorphic correspondence @xmath31 . the set of integers @xmath114 will be denoted by @xmath115}$ ] . given @xmath116 , we say that @xmath117}^n$ ] ( see for the meaning of @xmath118 ) is _ a path of an iteration of @xmath0 starting at @xmath119 , of length @xmath120 _ , or simply an _ @xmath120-path starting at @xmath119 _ , if @xmath121 next , given any two irreducible subvarieties @xmath122 and @xmath123 in the decomposition of @xmath31 in the sense of , we define @xmath124 in general , for any multi - index @xmath125}^j$ ] , @xmath126 , we recursively define @xmath127 in all discussions on the normality set of @xmath0 , we shall work with only those @xmath120-paths @xmath128 that satisfy * _ for each @xmath129 , @xmath130 is an irreducible subvariety of @xmath131 for every sufficiently small ball @xmath132 . _ an @xmath120-path will be called an _ admissible @xmath120-path _ if it satisfies @xmath133 . now fix a @xmath134 and @xmath116 , and set @xmath135 we will denote an element of @xmath136 either by @xmath137 or by @xmath138}^n$ ] , depending on the need . observe that if @xmath137 is an admissible @xmath120-path , @xmath139 , then there is a _ unique _ irreducible component of @xmath140 to which @xmath141 belongs , @xmath142 . hence , if @xmath137 is an admissible @xmath120-path , let us write @xmath143 finally , let @xmath144 , where @xmath145 is a compact riemann surface , denote the desingularization of @xmath146 . we now have the essential notations needed to define the normality set . the definitions that follow are strongly influenced by the notion introduced by bullett and penrose @xcite . yet , what we call a `` branch of an iteration '' will look vastly different from its namesake in @xcite . this is because , for our purposes , we need to attach more detailed labels to all the maps involved than what the notation in @xcite provides . i.e. , the difference is largely in formalism . the one departure that we make from the bullett penrose definition is that , in defining a holomorphic branch of an iteration along @xmath137 , we assume that @xmath137 is admissible . the only purpose of this restriction is that , in proving theorem [ t : d_1more ] , we will require precise accounting of all the branches involved , but _ we do not want the simple motivation for the normality set to be obscured by too much book - keeping paraphernalia . _ the normality set of @xmath0 is the analogue of the fatou set . the need for the extra formalism in definition [ d : prenset ] is summarised by the following : * the analogue for the family of all iterates of a map must be the collection of local maps defined by the germ of @xmath147 at @xmath148 , _ whenever this makes sense _ ( around a chosen @xmath119 ) for every @xmath149 and for every @xmath116 . * if one of the germs in @xmath150 has a singularity ( at some point in @xmath151 ) or does not project injectively under @xmath152 , then the above notion of `` local maps '' _ will not _ make sense . we must replace maps by parametrisations in that event . let @xmath153 and @xmath154 denote the following projections : @xmath155 where @xmath156 . we now have all the notations for the key definition needed on the way to defining the normality set of @xmath0 . a schematic drawing is presented on the following page to give an impression of the various objects occurring in the following definition . [ d : prenset ] let @xmath113 be a compact riemann surface and let @xmath0 be a holomorphic correspondence on @xmath113 . let @xmath116 , @xmath134 , and let @xmath157 . write @xmath158 . we call the list @xmath159 a _ holomorphic branch of an iteration of @xmath0 along @xmath137 _ if @xmath137 is an admissible @xmath120-path , @xmath160 and @xmath161 are as described above , @xmath162 is a connected neighbourhood of @xmath119 and @xmath163 are holomorphic mappings defined on a planar domain @xmath164 containing @xmath165 such that , for each @xmath129 : * @xmath166 . * @xmath167 is a finite - sheeted ( perhaps branched ) covering map onto @xmath168 , and @xmath160 maps the latter set homeomorphically onto @xmath169 . * the set @xmath170the irreducible component of @xmath171 containing @xmath141 , where we define @xmath172 , & \text{if $ j\geq 2$. } \end{cases}\ ] ] note that @xmath173 is a ( perhaps branched ) covering map . * @xmath174 , provided @xmath175 . [ rem : paramet ] as @xmath137 is admissible , @xmath169 in condition(3 ) is well - defined and the set @xmath168 is a smooth patch parametrising it ( not necessarily diffeomorphically ) ; see , for instance , ( * ? ? ? * chapter 7 ) . note that if @xmath0 is a non - constant rational map on @xmath1 then @xmath176 where @xmath164 is a small disc around @xmath165 , satisfy all the conditions in the above definition . this is the situation that definition [ d : prenset ] generalises . having defined holomorphic branches , we can give the following definition . [ d : nset ] let @xmath113 be a compact riemann surface and let @xmath0 be a holomorphic correspondence on @xmath113 . a point @xmath134 is said to belong to the _ normality set of @xmath0 _ , denoted by @xmath177 , if there exists a connected neighbourhood @xmath162 of @xmath119 and a _ single _ planar domain @xmath164 containing @xmath165 , which depends on @xmath119 , such that * for each @xmath178 and each @xmath179 , there exists a holomorphic branch @xmath180 of an iteration of @xmath0 along @xmath137 with @xmath181 for every @xmath182 . * the family @xmath183 is a normal family on @xmath164 . the set @xmath177 is open , although it is not necessarily non - empty . if @xmath184 and @xmath162 is the neighbourhood of @xmath119 as given by definition [ d : nset ] , then it is routine to show that @xmath185 . we need to present some formalisms before we can state our first result . given a holomorphic correspondence on @xmath51 , @xmath186 , determined by a holomorphic @xmath30-chain @xmath31 , its _ adjoint correspondence _ is the meromorphic correspondence determined by the @xmath30-chain ( assuming the representation for @xmath31 ) @xmath187 note that , in general , @xmath188 may not determine a holomorphic correspondence . however , when @xmath189 , it is easy to see that _ any meromorphic correspondence of @xmath51 is automatically holomorphic_. thus , if @xmath190 is a holomorphic correspondence on @xmath1 , then so is @xmath191 . in the abbreviated notation introduced in section [ s : keydefns ] , we shall henceforth write : @xmath192 given a holomorphic @xmath30-chain @xmath31 on @xmath89 , @xmath31 detemines a current of bidimension @xmath193 via the currents of integration given by its constituent subvarieties @xmath38 . we denote this current by @xmath194 $ ] . if @xmath0 is the holomorphic correspondence determined by @xmath31 , we _ formally _ define the action of @xmath0 on currents @xmath195 on @xmath51 of bidegree @xmath196 , @xmath197 , by the prescription : @xmath198\right),\ ] ] where , as usual , @xmath20 denotes the projection of @xmath89 onto the @xmath21th factor . this prescription would make sense for those currents for which the pullback by @xmath199 makes sense and the intersection of this new current with @xmath194 $ ] also makes sense . that this is the case is easy to see when @xmath195 is a smooth @xmath196 form ( hence a current of bidegree @xmath196 on @xmath51 ) . the reader is referred to ( * ? ? ? * section 2.4 ) for details . if @xmath200 is a finite , positive borel measure , then the intersection with @xmath194 $ ] in makes sense . here , @xmath200 is viewed as a current of bidegree @xmath193 . let us work out @xmath201 for a specific example that is central to this paper . let @xmath202 and let @xmath203 be the dirac mass at @xmath204 . the prescription is interpreted as @xmath205 , \pi_1^\varphi \right\rangle \ : = & \ \sum_{j=1}^n m_j\langle(\left.\pi_2\right\|_{\gamma_j})^(\delta_x ) , \pi_1^\varphi \rangle \notag \\ = & \ \sum_{j=1}^n m_j\langle \delta_x , \big(\left.\pi_2\right\|_{\gamma_j}\big ) _ ( \pi_1^\varphi ) \rangle , \label{e : pullbackdirac1}\end{aligned}\ ] ] where each summand in the last expression is just the way we define the pullback of a current under a holomorphic mapping ( in this case , @xmath206 ) of an analytic space that is submersive on a zariski open subset . if @xmath207 is a zariski - open subset of @xmath51 such that @xmath208 is a covering space for each @xmath129 , then , for @xmath209 , @xmath210 is just the sum of the values of @xmath67 on the fibre @xmath211 for any @xmath212 . thus , when @xmath209 , equals the quantity @xmath213(x ) \qquad x\in { \omega}.\ ] ] for any fixed _ continuous _ function @xmath214 , @xmath215 $ ] extends continuously to each @xmath216 . we shall denote this continuous extension of the left - hand side of also as @xmath215 $ ] . in other words , @xmath217 can be defined as a measure supported on the set @xmath218 , and @xmath219(x ) \ ; \ ; \forall x\in x , \ ; \ ; \forall \varphi\in { \mathcal{c}}(x).\ ] ] the arguments preceding continue to be valid if , in , @xmath203 is replaced by @xmath200 , a finite , positive borel measure on @xmath51 . the push - forward of a current @xmath195 by @xmath0 is defined by the equation @xmath220 whenever the latter makes sense . we define two numbers that are essential to the statement of our theorems . with @xmath0 as above , let @xmath221 denote the generic number of preimages under @xmath0 of a point in @xmath1 , counted according to multiplicity . what we mean by `` counted according to multiplicity '' @xmath222 being any zariski - open set of the form discussed prior to the equation , and the @xmath223-chain @xmath31 having the representation is the number @xmath224 which is independent of the choice of @xmath225 . in other words , @xmath221 is the _ topological degree of @xmath0 _ , often denoted as @xmath226 . define @xmath227 . we will first consider a holomorphic correspondence @xmath0 of @xmath1 such that @xmath228 . a very special case of a result of dinh and sibony ( ? ? ? * corollaire 5.3 ) is that there exists a probability measure @xmath2 such that @xmath229 where @xmath62 denotes the fubini study form on @xmath1 , treated as a normalised area form . let us call this measure the _ dinh sibony measure associated to @xmath0_. since equidistribution is among the main themes of this paper , we should mention that for a generic @xmath230 , @xmath2 is the asymptotic distribution of the iterated pre - images of @xmath3 . more precisely : [ l : clarify ] let @xmath0 be a holomorphic correspondence on @xmath1 such that @xmath231 and let @xmath2 be the dinh sibony measure associated to @xmath0 . there exists a polar set @xmath71 such that for each @xmath232 @xmath233 consequently , @xmath234 . the above follows by combining with another result from @xcite . we shall make this clearer for the convenience of readers who are unfamiliar with @xcite through a few remarks at the beginning of section [ s : proof - d_1more ] . however , fact [ l : clarify ] will have _ no role to play _ herein except to set the context for our first result . it establishes that the following theorem is a partial generalisation of brolin s observation on the support of the brolin measure @xmath235 ( associated to a polynomial map @xmath67 on @xmath9 ) . [ t : d_1more ] let @xmath0 be a holomorphic correspondence on @xmath1 and assume that @xmath231 . let @xmath2 be the dinh sibony measure associated to @xmath0 . then , @xmath236 is disjoint from the normality set of @xmath0 . the principle underlying the proof of the above theorem is as follows . for @xmath237 , we shall show that one can apply marty s normality criterion in such a manner as to deduce that the volumes of any compact @xmath238 with respect to the measures induced by @xmath239 are bounded independent of @xmath26 . the result follows from this , together with the fact that @xmath240 . the situation is _ very different _ when @xmath241 . to repeat : we should not expect asymptotic equidistribution of preimages in general , even when @xmath242 , as the holomorphic correspondence @xmath0 whose graph is the @xmath77completion of the zero set of the rational function @xmath78 illustrates . we require some dynamically meaningful condition for things to work . it is this need that motivates the next few definitions . let @xmath51 be a compact hausdorff space and let @xmath243 be a relation of @xmath51 to itself such that @xmath244 . for any set @xmath245 , we write @xmath246 we define the @xmath26th iterated relation by @xmath247 where the composition operation @xmath248 is as understood from above . it is useful to have a notion of omega limit sets in the context of iterating a relation analogous to the case of maps . this definition is provided by mcgehee in ( * ? ? ? * section 5 ) . following mcgehee , for a subset @xmath245 , let us write @xmath249 ( with the understanding that @xmath250 is the diagonal in @xmath89 ) . omega limit set of @xmath195 under @xmath67 _ , denoted by @xmath251 , is the set @xmath252 we say that a set @xmath253 is an _ attractor _ for @xmath67 if @xmath254 and there exists a set @xmath162 such that @xmath255 and such that @xmath256 . these concepts motivate the following two definitions in the context of holomorphic correspondences . [ d : repeller ] let @xmath0 be a holomorphic correspondence on a riemann surface @xmath51 given by the holomorphic @xmath223-chain @xmath31 . a set @xmath253 is called an _ attractor _ for @xmath0 if it is an attractor for the relation @xmath107 in the sense of @xcite ( i.e. , as discussed above ) . a set @xmath257 is called a _ repeller _ for @xmath0 if it is an attractor for the relation @xmath258 . we must note here mcgehee calls the relation on @xmath51 induced by @xmath258 the _ transpose _ of @xmath107 , and our @xmath258 is @xmath259 in the notation of @xcite . [ d : strongrep ] let @xmath0 be as above and let @xmath257 be a repeller for @xmath0 . we say that @xmath257 is a _ strong repeller _ for @xmath0 if there exists a point @xmath260 and an open set @xmath261 such that for each @xmath262 , there exists a sequence @xmath263 such that * @xmath264 ; and * @xmath265 as @xmath266 . the term _ strong attractor _ has an analogous definition . we call @xmath267 a _ critical value _ if there exists an irreducible component @xmath38 such that at least one of the irreducible germs of @xmath38 at some point in @xmath268 is either non - smooth or does _ not _ project injectively under @xmath199 . we are now in a position to state our next result . [ t : d_1less ] let @xmath0 be a holomorphic correspondence on @xmath1 for which @xmath241 . assume that @xmath0 has a strong repeller @xmath257 that is disjoint from the set of critical values of @xmath0 . then , there exist a probability measure @xmath2 on @xmath1 that satisfies @xmath234 and an open set @xmath81 such that @xmath269 it may seem to the reader that could be stronger , since the theorem does not state that @xmath84 is polar ( or even nowhere dense ) . however , given that @xmath241 , this is very much in the nature of things . in this regard , we make the following remark . [ rem : care ] if @xmath0 is as in theorem [ t : d_1less ] , we _ can not conclude , in general , that the set @xmath84 is polar . _ the following example constitutes a basic obstacle to @xmath84 being even nowhere dense . let @xmath270 be any polynomial whose filled julia set has non - empty interior . consider the holomorphic correspondence @xmath0 determined by @xmath271 here , @xmath272 . note that @xmath273 is a strong repeller . however , @xmath274 can not contain any points from the filled julia set of @xmath270 . [ rem : obvbutess ] the reader might ask why one should even address correspondences for which @xmath275 in theorem [ t : d_1less ] , since one gets an invariant measure for such an @xmath0 by merely applying the results in @xcite to @xmath5 . this is a reasonable question and gives rise to corollary [ c : d1_less ] below but it misses two vital points : * finding invariant measures is not an end in itself in holomorphic dynamics . the goal is to find measures that are sufficiently well - adapted to the correspondence in question that they enable the study of various geometric features of its dynamics . * in many respects , especially concerning the equidistribution - of - preimages phenomenon , correspondences @xmath0 that satisfy @xmath275 behave in ways that are similar to the behaviour of correspondences @xmath276 for which @xmath277 . in studying such phenomena , it makes sense to study both cases as a unified whole . we will not elaborate on @xmath150 at this juncture . but theorem [ t : d_1more ] is an illustration of this point . regarding @xmath278 : consider the polynomial @xmath270 described in remark [ rem : care ] but , this time , also assume that the leading coefficient , say @xmath279 , satisfies @xmath280 . write @xmath281 , let @xmath282 and let @xmath0 be the correspondence given by @xmath31 . once again , @xmath273 is a strong repeller for @xmath0 . it takes a little more effort to show that , with suitable choices for the coefficients of the lower - degree terms , @xmath283 has non - empty interior . as in the example in remark [ rem : care ] , @xmath0 admits a large set @xmath284 such that the iterated inverse images of any @xmath285 equidistribute , but whose complement is _ not polar . _ this is the similarity that @xmath278 of remark [ rem : obvbutess ] alludes to . however , in contrast to remark [ rem : care ] , here @xmath286 . from the perspective of studying the problem of equidistribution of _ inverse _ images , theorem [ t : d_1less ] is , to the best of our knowledge , the first theorem concerning the equidistribution problem for holomorphic correspondences @xmath0 such that @xmath241 . we must point out that , in general , given @xmath228 , the dinh sibony measure is _ not _ an invariant measure . ( it is what one calls an @xmath85-invariant measure . ) yet there is much that one can do even with this weak sense of invariance which is what is implied by @xmath150 above . a borel measure @xmath200 is said to be _ invariant under @xmath0 _ if its push - forward by @xmath0 preserves ( compensating for multiplicity if @xmath0 is not a map ) measure of all borel sets i.e. , if @xmath287 . thus , under the condition @xmath241 , we can actually construct measures that are invariant under @xmath0 : [ c : d1_less ] let @xmath0 be a holomorphic correspondence on @xmath1 for which @xmath241 . 1 . if @xmath288 , there exists a measure @xmath2 that is invariant under @xmath0 . 2 . suppose @xmath286 . if @xmath0 has a strong attractor that is disjoint from the critical values of @xmath5 , then there exists a measure @xmath2 that is invariant under @xmath0 . the proof of theorem [ t : d_1less ] relies on techniques developed by lyubich in @xcite . given our hypothesis on the existence of a repeller @xmath257 , one can show that there exists a compact set @xmath289 such that @xmath290 and @xmath291 . this allows us to define a perron frobenius - type operator @xmath292 , where @xmath293 with @xmath294 being the operator given by with @xmath289 replacing @xmath295 . our proof relies on showing that the family @xmath296 satisfies the conditions of the main result in @xcite . this goal is achieved , _ in part _ , by showing that for each @xmath297 there are , for each @xmath178 , sufficiently many holomorphic branches of @xmath26-fold iteration of the correspondence @xmath5 . the examples in remarks [ rem : care ] and [ rem : obvbutess ] have some very special features . one might ask whether there are plenty of holomorphic correspondences @xmath0 on @xmath1 with @xmath241 _ without _ such rigid features as in the correspondences in remarks [ rem : care ] and [ rem : obvbutess ] that satisfy the conditions stated in theorem [ t : d_1less ] . one might also ask whether any of the correspondences alluded to in section [ s : intro ] satisfy the conditions in theorem [ t : d_1less ] . the reader is referred to section [ s : example ] concerning these questions . in the next section , we shall establish a few technical facts which will be of relevance throughout this paper . the proofs of our theorems will be provided in sections [ s : proof - d_1more ] and [ s : proof - d_1less ] . we begin by showing that the composition of two holomorphic correspondences on @xmath1 , under the composition rule , produces a holomorphic correspondence . one way to see this is to begin with how one computes @xmath298 if one is given exact expressions for @xmath299 and @xmath76 . let @xmath300 be the graph of @xmath301 , @xmath302 , and consider the representations given by . fix indices @xmath21 and @xmath303 such that @xmath304 and @xmath305 . it follows that there exist irreducible polynomials @xmath306 $ ] such that @xmath307 see , for instance , @xcite . now , given any polynomial @xmath308 $ ] , set @xmath309 then , there is a choice of projective coordinates on @xmath1 such that @xmath310,[w_0:w_1])\in { \mathbb{p}^1}\times{\mathbb{p}^1 } : z_0^{d_z(p_1)}w_0^{d_w(p_1)}p_1(z_1/z_0,w_1/w_0)=0\ } , \notag \\ { \gamma^\bullet}_{2,\,l } \ = \ \{([z_0:z_1],[w_0:w_1])\in { \mathbb{p}^1}\times{\mathbb{p}^1 } : z_0^{d_z(p_2)}w_0^{d_w(p_2)}p_2(z_1/z_0,w_1/w_0)=0\}. \notag\end{aligned}\ ] ] with these notations , we are in a position to state our first proposition . [ p : nolines ] let @xmath311 and @xmath312 be irreducible subvarieties belonging to the holomorphic @xmath223-chains @xmath102 and @xmath103 respectively . let @xmath313 and @xmath314 be the defining functions of @xmath315 and @xmath316 respectively . 1 . let @xmath317 , where @xmath318 denotes the resultant of two univariate polynomials . let @xmath319 denote the biprojective completion in @xmath19 of @xmath320 . then @xmath321 . @xmath319 has no irreducible components of the form @xmath322 or @xmath323 , @xmath324 . let us write @xmath325 . since two polynomials @xmath326 $ ] have a common zero if and only if @xmath327 , @xmath328 hence , as @xmath47 is the biprojective completion of @xmath329 in @xmath19 , @xmath330 follows . to prove @xmath331 , let us first consider the case when @xmath332 $ ] . then , it suffices to show that @xmath333 has no factors of the form @xmath334 or @xmath335 . we shall show that @xmath333 has no factors of the form @xmath334 . an analogous argument will rule out factors of the form @xmath335 . to this end , assume that there exists an @xmath16 such that @xmath336 in @xmath337 $ ] . this implies @xmath338 thus , for each @xmath339 , the polynomial @xmath340 has a zero in common with @xmath341 $ ] . note that @xmath342 because , otherwise , @xmath343 , which would contradict the fact that @xmath344 is surjective , @xmath302 . thus , there exists an uncountable set @xmath345 and a point @xmath346 such that @xmath347 but this implies @xmath348 , i.e. that @xmath349 . this is impossible , for exactly the same reason that @xmath350 . hence @xmath333 has no factors of the form @xmath351 . note that , if we write @xmath352\in { \mathbb{p}^1}:z_1\neq 0\}$ ] , then , arguing as in the beginning of this proof , @xmath353 where @xmath354 is as defined in the beginning of this section . if we define @xmath355 $ ] by @xmath356 then we get @xmath357 in @xmath337 $ ] . thus , @xmath358\}\times{\mathbb{p}^1}$ ] is not an irreducible component of @xmath47 . by a similar argument , @xmath359\}$ ] is not an irreducible component of @xmath47 either . it is now easy to see that @xmath104 determines a holomorphic correspondence on @xmath1 . let us pick @xmath311 and @xmath312 as in proposition [ p : nolines ] and let @xmath360 be an irreducible component of @xmath361 . by the fundamental theorem of algebra , @xmath362 would fail to be surjective for some @xmath363 only if @xmath360 is of the form @xmath322 or @xmath323 , @xmath324 . this is impossible by part @xmath331 of proposition [ p : nolines ] . hence , we have the following : [ c : allokay ] let @xmath102 and @xmath103 be two holomorphic correspondences on @xmath1 . then @xmath104 is a holomorphic correspondence on @xmath1 . the next lemma will be useful in simplifying expressions of the form @xmath364 or @xmath365 . its proof is entirely routine , so we shall leave the proof as an exercise . [ l : iterate ] let @xmath51 be a compact complex manifold and let @xmath0 be a holomorphic correspondence on @xmath51 . then @xmath366 . the final result in this section is important because it establishes that the measures @xmath367 appearing in theorem [ t : d_1less ] are probability measures . the result below is obvious if @xmath0 is a map . we could not find a proof of any version of proposition [ p : degreeseq ] in the literature , _ nor could several colleagues to whom we wrote suggest a reference . _ since it appears to be a folk lemma , we provide a proof in the case of @xmath1 . [ p : degreeseq ] let @xmath0 be a holomorphic correspondence on @xmath1 . then @xmath368 . [ rem : moregeneral ] the proof of the above proposition with @xmath1 replaced by _ any _ compact complex manifold @xmath51 is almost exactly the one below . the only difference is that when @xmath369 those parts of the proof below that rely on part@xmath331 of proposition [ p : nolines ] will follow from the following : * _ let @xmath51 be a compact complex manifold and let @xmath311 and @xmath312 be as in . the projections @xmath152 and @xmath199 are surjective when restricted to each irreducible component of @xmath361 . _ since we have not given a proof of the above when @xmath369 doing so would be a considerable digression we state proposition [ p : degreeseq ] for @xmath370 only . we shall use induction . the above formula is a tautology for @xmath371 . assume that it is true for @xmath372 for some @xmath373 . let @xmath31 be the holomorphic @xmath223-chain that determines @xmath0 . following the representation , let us denote @xmath374 and let @xmath31 be precisely as in . then @xmath375 let @xmath376 be a zariski - open subset of @xmath1 such that @xmath377 , @xmath378 and @xmath379 are covering spaces for each @xmath380}\times{[{1}\,.\,.\,{n}]}$ ] . we can then find a zariski - open subset @xmath47 of @xmath381 such that * @xmath382 , @xmath383 is an at most finite union of lines of the form @xmath322 or @xmath323 , and for each @xmath380}\times{[{1}\,.\,.\,{n}]}$ ] , @xmath384 and @xmath385 @xmath386 . that there is a @xmath47 for which @xmath387 holds true for all pertinent @xmath388 follows easily from part@xmath331 of proposition [ p : nolines ] . ( we shall provide an argument below for a fact analogous to @xmath387 ; a simpler version that argument gives @xmath387 . ) let us write @xmath389 we now fix a @xmath380}\times{[{1}\,.\,.\,{n}]}$ ] . let us write @xmath390 . let @xmath97 s , @xmath391 , be the distinct irreducible components of @xmath392 . it follows from a standard intersection - theory argument ( note that @xmath97 is _ irreducible _ ) , the fact that @xmath393 , and from the structure of @xmath47 , that there is subset @xmath394 such that @xmath395 is a zariski - open subset of @xmath47 and @xmath396 for each @xmath391 and @xmath380}\times{[{1}\,.\,.\,{n}]}$ ] , where @xmath397 is as given immediately after the composition rule . consider the set @xmath398 this set is clearly a zariski - open subset of @xmath395 , but we claim that @xmath399 is a _ non - empty _ zariski - open set . to this end , consider the sets @xmath400 owing to the part@xmath331 of proposition [ p : nolines ] and the structure of @xmath47 , @xmath401 and @xmath402 are finite sets . hence @xmath403 then , owing to the property @xmath387 @xmath404 whence our claim . now , for each @xmath405 , define @xmath406 by the construction of @xmath399 , @xmath407 . then , by the construction of @xmath47 , @xmath408 let @xmath409 denote the set of all points in @xmath392 that belong to more than one @xmath97 . then , by @xmath410 recalling our definitions : @xmath411 now let @xmath412 be a zariski - open subset of @xmath1 such that @xmath413 are covering spaces for every @xmath391 and every @xmath380}\times{[{1}\,.\,.\,{n}]}$ ] . then , writing @xmath414 , we have by definition @xmath415 in view of and , the above equation simplifies as follows : @xmath416 . \notag\end{aligned}\ ] ] by induction , the result follows . we begin this section with a few remarks on fact [ l : clarify ] . for this purpose , we need to invoke one of the results of @xcite . we shall paraphrase it _ specifically in the context of correspondences on @xmath1 . _ we leave to the reader the task of verifying our transcription of ( * ? ? ? * thorme 4.6 ) to the present context . it might however be helpful for those readers who are unfamiliar with @xcite if we mention a couple of identities . we shall not define here the notion of _ intermediate degrees of @xmath0 of order @xmath417 _ ; the reader is referred to ( * ? ? ? * section 3.1 ) . ( the proof of the following fact just involves writing out the relevant definitions in two different ways . ) [ l : topdeg ] let @xmath49 be a compact khler manifold of dimension @xmath30 and assume @xmath418 . let @xmath419 be a holomorphic correspondence . let @xmath420 denote the intermediate degree of @xmath0 of order @xmath417 , @xmath421 . then : * @xmath422 . * @xmath423 . in what follows @xmath62 shall denote the fubini study form normalised so that @xmath424 . the key result needed is : [ r:1stres ] let @xmath425 , @xmath426 , be holomorphic correspondences of @xmath1 . suppose the series @xmath427 converges . then , there exists a polar set @xmath428 such that for each @xmath232 , @xmath429 for each @xmath430 and , in fact , this convergence is uniform on subsets that are bounded with respect to the @xmath431-norm . let us take the @xmath432 s of result [ r:1stres ] to be @xmath433 , @xmath434 since any function in @xmath435 can be approximated in the sup - norm by functions in @xmath436 , the weak@xmath437 convergence asserted by fact [ l : clarify ] follows from , result [ r:1stres ] and proposition [ p : degreeseq ] . one concern that ( perhaps ) arises when we use the results of @xcite is the way we compose two holomorphic correspondences . the composition rule presented in @xcite appears to be _ very different _ from that given by . if either @xmath102 or @xmath103 determines a meromorphic , _ non - holomorphic _ correspondence , then may not result in a meromorphic correspondence . this is the phenomenon that the composition rule in @xcite addresses . in our setting , this problem does not arise . in fact , the rule in @xcite is equivalent to when both @xmath102 and @xmath103 are holomorphic ; see ( * ? ? ? * section 4 ) for more details . we now present : we assume that @xmath237 ; there is nothing to prove otherwise . let us fix a @xmath184 . then , any @xmath179 is admissible , and by property(4 ) of definition [ d : prenset ] , assuming that @xmath438 , we get @xmath439 iterating this argument , we deduce the following : @xmath440 let us now _ fix _ a disc @xmath441 around @xmath442 such that @xmath443 ( where @xmath164 is as given by definition [ d : nset ] ) . define @xmath444}}\bigcap_{x\in \pi_1^{-1}\{z_0\}\cap{\gamma^\bullet}_j } \pi^0_1\circ\nu_{((z_0,\pi_2(x);\,j),1)}\circ\psi_{((z_0,\pi_2(x);\,j),1)}({\overline{\delta}}).\ ] ] clearly , there is a region @xmath445 , containing @xmath165 , such that @xmath446 where @xmath447 is our abbreviation for the subscript @xmath448 and @xmath449 is as described in property(3 ) of definition [ d : prenset ] . from the above fact and we deduce the following : @xmath450 we can deduce from definitions [ d : prenset ] and [ d : nset ] that @xmath451 . as @xmath452 has non - empty interior , it suffices to show that for any _ non - negative _ function @xmath453 with @xmath454 , @xmath455 . hence , let us pick some function @xmath453 as described . for any path @xmath456 , let us write @xmath457 we adopt notation analogous to that developed just before definition [ d : prenset ] . for any multi - index @xmath125}^j$ ] , @xmath126 , @xmath458 , let us define @xmath459 then , @xmath460 is a @xmath461-open neighbourhood of the point @xmath462 . furthermore , by our constructions in definition [ d : prenset ] , @xmath463 is covered by the sets @xmath460 as @xmath137 varies through @xmath464 . hence , by definition : @xmath465 it is routine to show that @xmath466 is a branched covering map onto its image . as @xmath467 maps @xmath468 homeomorphically onto @xmath469 , the topological degree of @xmath470 equals the the topological degree of @xmath466 . let us denote this number by @xmath471 . by the change - of - variables formula , we get : @xmath472 since @xmath460 , in general , has singularities , we discuss briefly what is meant above by `` change - of - variables formula '' . note that : * the magnitude of the form @xmath473 stays bounded on punctured neighbourhoods of any singular point of @xmath460 . * @xmath474 after at most finitely many punctures is the image of a zariski - open subset of @xmath468 under a @xmath471-to-1 covering map . given these facts , it is a standard calculation that the right - hand side of transforms to the last integral above . for each @xmath179 , let us write @xmath475 by the change - of - variables formula for branched coverings of finite degree , we get : @xmath476 the last expression arises from and the fact that @xmath454 . endow @xmath1 with homogeneous coordinates . in view of the argument made below , we may assume without loss of generality that @xmath477 does not contain _ both _ @xmath478 $ ] and @xmath479 $ ] ( if not , then for each @xmath26 and each @xmath179 for which this happens , we split the relevant integral below into a sum of integrals over a two - set partition of @xmath164 ) . write @xmath480,\ ] ] where @xmath481 and have no common zeros in @xmath164 , and define @xmath482\in \pi^n_n\circ\nu_{({\boldsymbol{\mathcal{z}}},n)}\circ\psi_{({\boldsymbol{\mathcal{z}}},n)}({d})$ } , \\ y_{({\boldsymbol{\mathcal{z}}},n)}/x_{({\boldsymbol{\mathcal{z}}},n ) } , & \text{if $ [ 1:0]\in \pi^n_n\circ\nu_{({\boldsymbol{\mathcal{z}}},n)}\circ\psi_{({\boldsymbol{\mathcal{z}}},n)}({d})$}. \end{cases}\ ] ] from the expression for the fubini study metric in local coordinates and from , we have the estimate @xmath483 since , by hypothesis , @xmath484 is a normal family , it follows by marty s normality criterion see , for instance , conway ( * ? ? ? * chapter vii/3 ) that the family @xmath485 is locally uniformly bounded . as @xmath445 , there exists an @xmath486 such that @xmath487 given @xmath178 , the number of summands in is at most @xmath488 . the equality is a consequence of proposition [ p : degreeseq ] and lemma [ l : iterate ] . from it and the last two estimates , it follows that @xmath489^n\!m^2 { \sf area}(\overline{g } ) \;{\longrightarrow}\ ; 0 \ \text{as $ n\to \infty$,}\ ] ] since , by hypothesis , @xmath70 . in view of ( * corollaire 5.3 ) applied to @xmath1 with @xmath490 , i.e. the limit , and by , we have @xmath491 in view of our remarks earlier , the theorem follows . as the paragraphs in section [ s : results ] preceding the statement of theorem [ t : d_1less ] would suggest , its proof relies on several notions introduced in @xcite . we therefore begin this section with a definition and a couple of results from @xcite . [ d : attrblock ] let @xmath51 be a compact hausdorff space and let @xmath243 be a relation of @xmath51 to itself such that @xmath244 . a set @xmath492 is called an _ attractor block _ for @xmath67 if @xmath493 . we recall that , given a relation @xmath67 and a set @xmath245 , @xmath494 is as defined in section [ s : results ] . [ r : mcg_block ] let @xmath51 be a compact hausdorff space and let @xmath243 be a relation of @xmath51 to itself such that @xmath244 . assume @xmath67 is a closed set . if @xmath289 is an attractor block for @xmath67 , then @xmath289 is a neighbourhood of @xmath495 . [ r : mcg_chooseblock ] let @xmath51 be a compact hausdorff space and let @xmath243 be a relation of @xmath51 to itself such that @xmath244 . assume @xmath67 is a closed set . if @xmath496 is an attractor for @xmath67 and @xmath47 is a neighbourhood of @xmath496 , then there exists a closed attractor block @xmath289 for @xmath67 such that @xmath497 and @xmath498 . we clarify that , given two subsets @xmath279 and @xmath289 of some topological space , @xmath289 is called a neighbourhood of @xmath279 here ( as in @xcite ) if @xmath499 . before we can give the proof of theorem [ t : d_1less ] , we need one more concept . for this purpose , we shall adapt some of the notations developed in section [ ss : normality ] . here , @xmath0 will denote a holomorphic correspondence on @xmath1 . firstly : given @xmath116 , we say that @xmath500}^n$ ] ( see for the meaning of @xmath118 ) is _ a path of a backward iteration of @xmath0 starting at @xmath501 , of length @xmath120 _ , if @xmath502 in analogy with the notation in section [ ss : normality ] , we set : @xmath503 next , we say that a point @xmath267 is a _ regular value _ of @xmath0 if it is not a critical value ( recall that we have defined this in section [ s : results ] ) . we can now make the following definition : [ d : invregbr ] let @xmath0 be a holomorphic correspondence on @xmath1 , let @xmath116 , and let @xmath504 . let @xmath505 . we call the list @xmath506 a _ regular branch of a backward iteration of @xmath0 along @xmath507 _ if : * @xmath508 are regular values . * for each @xmath129 , @xmath509 is a holomorphic function defined by @xmath510 where @xmath511 is a _ local _ irreducible component of @xmath512 at the point @xmath513 such that : @xmath150 @xmath511 is smooth ; @xmath278 @xmath199 restricted to @xmath511 is injective ; and @xmath514 @xmath515 when @xmath516 . the above is a paraphrasing for the scenario in which we are interested of the notion of a `` regular inverse branch of @xmath0 of order @xmath120 '' introduced by dinh in @xcite . the following is the key proposition needed to prove theorem [ t : d_1less ] . [ p : key - d_1less ] let @xmath0 be a holomorphic correspondence of @xmath1 having all the properties stated in theorem [ t : d_1less ] and let @xmath257 be a strong repeller that is disjoint from the set of critical values of @xmath0 . then , there exists a closed set @xmath517 such that @xmath518 and such that : * the operator @xmath519 , where @xmath294 is as defined in with @xmath289 replacing @xmath295 , maps @xmath520 into itself . * there exists a probability measure @xmath521 that satisfies @xmath522 and such that @xmath523-{\int_{\raisebox{1pt}{$\scriptstyle { { b}}$}}\!{\varphi}}\,d\mu_b\right\| \ = \ 0 \quad\forall \varphi\in { \mathcal{c}}(b;{\mathbb{c}}).\ ] ] let @xmath260 and let @xmath162 be an open set containing @xmath257 such that : * for each @xmath262 , there is a sequence @xmath263 such that @xmath524 for each @xmath26 , and @xmath265 as @xmath266 . * @xmath162 contains no critical values of @xmath0 . by result [ r : mcg_block ] and result [ r : mcg_chooseblock ] , we can find an open neighbourhood @xmath525 of @xmath257 such that @xmath526 and @xmath527 is a closed attractor block for the relation @xmath258 . repeating the last argument once more , we can find a closed attractor block , @xmath289 , for @xmath258 such that @xmath528 by the above chain of inclusions and by the definition of the term `` attractor block '' , it follows that the operator @xmath529 maps @xmath520 into itself . * claim 1 . * _ for each fixed @xmath530 , @xmath531\}_{n\in { \mathbb{z}}_+}$ ] is an equicontinuous family . _ it is easy to see that @xmath257 is a closed _ proper _ subset . we can thus make a useful observation : * we can choose @xmath525 so that @xmath532 is non - empty . hence , we can choose coordinates in such a way that we may view @xmath525 as lying in @xmath9 , and that @xmath533 . we shall work with respect to these coordinates in the remainder of this proof . let us pick a point @xmath501 in @xmath289 ( which , by construction , is a regular value ) and let @xmath534 be a small disc centered at @xmath501 such that @xmath535 . let us fix an @xmath536 and consider a path @xmath537 . recall that , by construction : @xmath538 we can infer from that there exists a regular branch @xmath506 of a backward iteration of @xmath0 along @xmath507 . to see why , first note that , as @xmath501 is a regular value and @xmath535 , we get : 1 . there is an open neighbourhood @xmath539 of @xmath501 containing only regular values . 2 . writing @xmath540 and defining @xmath541any one of the irreducible components of the ( local ) complex - analytic variety @xmath542 , the function @xmath543 3 . the open set @xmath544 and hence contains only regular values . the assertion @xmath545 follows from the fact that @xmath546 and that the latter contains no critical values . let us now , for some @xmath156 , @xmath547 , _ assume _ the truth of the statements @xmath548 , @xmath549 and @xmath550 , which are obtained by replacing all the subscripts @xmath165 and @xmath223 in @xmath551 , @xmath552 and @xmath545 ( _ except _ the subscript in @xmath152 ) by @xmath553 and @xmath30 , respectively . now , @xmath554 follows from @xmath550 . defining @xmath555 in exact analogy to @xmath556 , and writing @xmath557 the holomorphicity of @xmath558 follows from @xmath554 and our definition of a regular value of @xmath0 . thus @xmath559 holds true . we get @xmath560 by appealing once again to and using the fact that @xmath561 . by induction , therefore , a regular branch of a backward iteration of @xmath0 along @xmath507 exists . we thus conclude the following : * @xmath562 ; and * @xmath563 ( see @xmath564 above ) . recall that @xmath507 was arbitrarily chosen from @xmath565 and that the arguments in the last two paragraphs hold true for any choice of @xmath511 , @xmath566 , and for any @xmath116 . thus , by montel s theorem , we infer the following important fact : the family @xmath567 pick a @xmath530 and let @xmath568 . as @xmath289 is compact , there exists a number @xmath569 such that : @xmath570 we pick a @xmath571 . by taking @xmath572 in the discussion in the previous paragraph , we infer from the normality of the family @xmath573 that we can find a sufficiently small number @xmath574 such that : @xmath575 now , for each @xmath571 write : @xmath576 if @xmath577 is a path of backward iteration ( with @xmath578 being the above @xmath579 ) , basic intersection theory tells us that the local intersection multiplicity of @xmath512 with @xmath580 at @xmath581 equals the number of distinct branches @xmath582 one can construct according to the above inductive prescription ( this number is greater than @xmath223 if @xmath512 has a normal - crossing singularity at @xmath581 ) . from this , and from the iterative construction of the @xmath583 s above , it follows that : @xmath584 \ = \ d_1(f)^n \;\;\ ; \forall { \zeta}\in b.\ ] ] from , and , we get : @xmath585(\xi)-{\mathbb{a}}_b^n[\varphi]({\zeta})\| \ = \ & \left\|\sum_{(\psi,{\boldsymbol{\mathcal{w}}})\in { \mathscr{i\!\!b}}(n,{\zeta})}\!\!\!d_1(f)^{-n } ( \varphi\circ\psi({\zeta})-\varphi\circ\psi(\xi))\right\| \notag \\ \leq \ & d_1(f)^{-n}\!\!\!\sum_{(\psi,{\boldsymbol{\mathcal{w}}})\in { \mathscr{i\!\!b}}(n,{\zeta})}\!\!\!\|\varphi\circ\psi({\zeta})-\varphi\circ\psi(\xi)\| \ < \ { \varepsilon}\notag \\ & \forall\xi \in b \ \text{such that $ \|\xi-{\zeta}\| < r({\varepsilon},\xi)$ and $ \forall n\in { \mathbb{z}}_+$. } \notag\end{aligned}\ ] ] the above holds true for each @xmath571 . this establishes claim 1 . in what follows , the term _ unitary spectrum _ of an operator on a complex banach space will mean the set of all eigenvalues of the operator of modulus @xmath223 , which we will denote by @xmath586 . it follows from the above claim and from @xcite that @xmath587 . * claim 2 : * _ @xmath588 , and the eigenspace associated with @xmath223 is @xmath9 . _ the ingredients for proving the above claim are largely those of @xcite . however , to make clear the role that the properties of @xmath257 play , we shall rework some of the details of lyubich s argument . let us fix a @xmath589 and let @xmath590 be an associated eigenfunction . let @xmath591 be such that @xmath592 . by definition @xmath593 the above equality would fail if , for some @xmath594 occurring above , @xmath595 . furthermore , if , for some @xmath594 occurring above , @xmath596 and @xmath597 do not lie on the same ray through @xmath442 , then cancellations would lead to ( recall the fact and that @xmath598 ) @xmath599 which is a contradiction . thus @xmath600 . iterating , we get @xmath601 since , by construction , @xmath602 , there exists an @xmath603 , such that @xmath604 . therefore , owing to , the sequence @xmath605 is a convergent sequence . as @xmath606 ( by definition ) , this implies that @xmath607 . observe that , having determined that @xmath608 , also gives @xmath609 note that @xmath610 } = { \mathbb{a}}[\overline{\varphi } ] \ \forall\varphi\in { \mathcal{c}}(b;{\mathbb{c}})$ ] . hence , @xmath611 and @xmath612 are also eigenvectors of @xmath613 associated to @xmath608 . thus , we have the following analogue of : @xmath614 where @xmath615 stands for either a point of global maximum or a point of global minimum of @xmath611 . using the above as a starting point instead of and repeating , with appropriate modifications , the argument that begins with and ends at , we get : @xmath616 similarly , we deduce that : @xmath617 combining the above with , we conclude that , for any eigenvector @xmath618 associated to @xmath608 , @xmath619 . this establishes claim 2 . to complete this proof , we need the following : [ r : lyubich_abs ] let @xmath620 be a complex banach space . let @xmath621 be a linear operator such that @xmath622 is a relatively - compact subset of @xmath620 for each @xmath623 . assume that @xmath624 and that @xmath223 is a simple eigenvalue . let @xmath625 be an invariant vector of @xmath279 . then , there exists a linear functional @xmath200 that satisfies @xmath626 and @xmath627 , and such that @xmath628 for each @xmath623 . take @xmath629 in the above theorem . note that @xmath630\|$ ] is bounded by @xmath631 for @xmath434 thus , in view of claims 1 and 2 , @xmath529 satisfies all the hypotheses of result [ r : lyubich_abs ] . hence ( recall that the function that is identically @xmath223 on @xmath289 is an eigenvector of @xmath529 ) there is a regular complex borel measure @xmath632 on @xmath289 such that @xmath633 , and @xmath634-{\int_{\raisebox{1pt}{$\scriptstyle { { b}}$}}\!{\varphi}}\,d\mu_b\right\| \ = \ 0 \quad\forall \varphi\in { \mathcal{c}}(b;{\mathbb{c}}).\ ] ] it is clear from the above equation that @xmath632 is a positive measure . hence it is a probability measure on @xmath289 . let @xmath289 be any closed set having the properties listed in the conclusion of proposition [ p : key - d_1less ] . let @xmath632 be the probability measure associated to this @xmath289 . we claim that @xmath2 is given by defining : @xmath635 we must show that @xmath2 does not depend on the choice of @xmath289 . the proof of this is exactly as given in ( * ? ? ? * theorem1 ) . we fix a point @xmath636 . so , @xmath637 for any choice of @xmath289 . thus , @xmath638(z ) & & ( \text{by proposition~\ref{p : key - d_1less } } ) \notag \\ & = \ \lim_{n\to \infty}d_1(f)^{-n}{\boldsymbol{\lambda}}^n[\varphi](z ) , & & ( \text{since $ { { } ^\dagger\!f}^n(z)\subset { { } ^\dagger\!f}^n({\mathcal{r}})\subset { \mathcal{r}}$ } ) \notag\end{aligned}\ ] ] where @xmath639 is as described in the passage following . the last line is independent of @xmath289 . hence the claim . by the above calculation , we also see that @xmath640 . thus , @xmath2 is a probability measure . let @xmath162 be the open set described at the beginning of the proof of proposition [ p : key - d_1less ] . we now define : @xmath641 we see from the proof of proposition [ p : key - d_1less ] that , owing to our hypotheses , @xmath642 is non - empty . hence @xmath643 is a non - empty open set that contains @xmath257 . let @xmath297 . there exists a @xmath644 such that @xmath645 . a close look at the essential features of its proof reveals that this @xmath289 has all the properties listed in the conclusion of proposition [ p : key - d_1less ] . consider any @xmath646 . we now apply lemma [ l : iterate ] to get @xmath647(z ) & & ( \text{from \eqref{e : pullbackdirac2 } and lemma~\ref{l : iterate } } ) \notag \\ = \ & { \mathbb{a}}_b^n[\varphi](z ) & & ( \text{since $ { { } ^\dagger\!f}^n(z)\subset { { } ^\dagger\!f}^n(b^\circ)\subset b$ } ) \notag \\ & { \longrightarrow}\ { \int_{\raisebox{1pt}{$\scriptstyle { { { \mathbb{p}^1}}}$}}\!{\varphi}}\,d\mu_f \;\ ; \text{as $ n\to \infty$ } , \label{e : quasi - eqd}\end{aligned}\ ] ] and this holds true _ for any @xmath297_. the last line follows from our observations above on @xmath2 . now note that , by construction , @xmath648 for each @xmath297 . therefore , in view of equation , it follows from that @xmath82 . we are now in a position to provide : recall that , by definition , for any borel measure @xmath200 , @xmath649 . thus , the proof of corollary [ c : d1_less ] involves , in each case , applying one of the results above to @xmath5 . the proof @xmath330 follows from fact [ l : clarify ] applied to @xmath5 . in view of definitions [ d : repeller ] and [ d : strongrep ] and the hypothesis of part @xmath331 , @xmath5 satisfies the conditions of theorem [ t : d_1less ] ( i.e. , with @xmath5 replacing @xmath0 ) . thus , @xmath331 follows from theorem [ t : d_1less ] . the purpose of this section is to provide concrete examples that illustrate some of our comments in sections [ s : intro ] and [ s : results ] about the extent to which our results apply to interesting correspondences on @xmath1 . we shall begin by showing that it is easy to construct examples of holomorphic correspondences on @xmath1 that satisfy all the conditions stated in theorem [ t : d_1less ] , but , unlike the examples discussed in remarks [ rem : care ] and [ rem : obvbutess ] , have `` large '' repellers . after this , we shall discuss one of the classes of holomorphic correspondences studied by bullett and collaborators . for each correspondence @xmath0 in this class , @xmath650 , and we shall show that theorem [ t : d_1less ] and corollary [ c : d1_less]@xmath331 are applicable to these examples . choose a complex polynomial @xmath651 with @xmath652 such that its julia set @xmath653 and such that no critical values of @xmath651 lie in @xmath654 . it follows see , for instance , @xcite that there is a compact set @xmath289 such that @xmath655 and avoids the critical values of @xmath651 , and such that @xmath656 . next , choose a polynomial @xmath657 with @xmath658 and having the following properties : next , we define : @xmath662 , [ w_0:w_1 ] ) : w_0^{{\rm deg}(q)}z_1-w_0^{{\rm deg}(q)}z_0\,q(w_1/w_0 ) = 0\ } , \notag \\ \gamma_2 \ & : = \ \{([z_0:z_1 ] , [ w_0:w_1 ] ) : w_1z_0^{{\rm deg}(p)}-w_0z_0^{{\rm deg}(p)}p(z_1/z_0 ) = 0\}. \notag\end{aligned}\ ] ] the projective coordinates are so taken that @xmath478 $ ] stands for the point at infinity . we set @xmath663 and let @xmath0 denote the correspondence determined by @xmath31 . clearly @xmath664 . in this type of construction , we will have @xmath288 in general . however , apart from satisfying the rather coarse properties @xmath150 and @xmath278 above , @xmath651 and @xmath657 can be chosen with considerable independence from each other . thus , this construction will also produce holomorphic correspondences @xmath0 that satisfy @xmath286 . by construction , @xmath665 . in other words , @xmath289 an attractor block for the relation @xmath258 . therefore , it follows from result [ r : mcg_block ] and definition [ d : repeller ] that @xmath0 has a repeller @xmath290 ; this repeller is just @xmath666 . it is easy to see that @xmath667 ( the reader may look up ( * ? ? ? * theorem 5.1 ) for a proof ) . the above implies that @xmath668 thus , the correspondence @xmath0 defined above has a repeller that is disjoint from the set of critical values of @xmath0 . owing to @xmath150 and @xmath278 above , for each @xmath669 , there exists a point @xmath524 such that @xmath670 . hence , @xmath257 is a strong repeller . unlike the examples in remarks [ rem : care ] and [ rem : obvbutess ] , @xmath257 is `` large '' in a certain sense . the ideas developed in the last section are of relevance to the correspondences alluded to in section [ s : intro ] introduced by bullett and his collaborators . we shall examine one such class of correspondences . we shall not elaborate here upon what precisely is meant by the mating between a quadratic map on @xmath1 and a kleinian group . the idea underlying this concept is simple , but a precise definition requires some exposition . we will just state here ( rather loosely ) that such a mating provides a holomorphic correspondence @xmath0 on @xmath1 , and partitions @xmath1 into an open set and two - component closed set ( denoted below by @xmath671 ) both totally invariant under @xmath0 such that the action of the iterates of @xmath0 on @xmath672 resembles the action of the given kleinian group on its regular set , and the iterates of distinguished branches of @xmath0 and @xmath5 on the components of @xmath671 , respectively resemble the dynamics of the given quadratic map on its filled julia set . we refer the reader to the introduction of @xcite or to @xcite . the example we present here is that of a holomorphic correspondence on @xmath1 that realises a mating between certain faithful discrete representations @xmath673 in @xmath674 of @xmath675 and a quadratic map @xmath676 . in this discussion , @xmath673 and @xmath677 will be such that : the fact that is pertinent to this discussion is that the mating of the above two objects is realisable as a holomorphic correspondence @xmath0 on @xmath1 . this is the main result of the article @xcite by bullett and harvey . for such an @xmath0 , @xmath650 . ( in fact , ( * ? ? ? * theorem 1 ) establishes the latter fact for a much larger class of maps @xmath678 . for simplicity , however , we shall limit ourselves to the assumptions above . ) let @xmath31 denote the holomorphic @xmath223-chain that determines the correspondence provided by ( * ? ? ? * theorem 1 ) . let us list some of the features of @xmath0 that are relevant to the present discourse ( we will have to assume here that readers are familiar with @xcite ) : * there exists a closed subset @xmath681 that is totally invariant under @xmath0 and is the disjoint union of two copies @xmath682 and @xmath683 of a homeomorph of a closed disc . * there exist two open neighbourhoods @xmath684 and @xmath685 of @xmath683 such that * * @xmath686 ; * * @xmath687 is the union of the graphs of two functions @xmath688 , @xmath689 ; * * @xmath690 ; * * there is a quasiconformal homeomorpism of @xmath684 onto an open neighbourhood @xmath691 of @xmath679 that carries @xmath683 onto @xmath679 , @xmath692 onto the julia set of @xmath678 , is conformal in the interior of @xmath683 , and conjugates @xmath693 to @xmath694 . * * @xmath695 . * there exist two open neighbourhoods @xmath696 and @xmath697 , @xmath698 , of @xmath682 and a pair of functions @xmath699 such that @xmath700 and the analogues of the properties @xmath701@xmath702 obtained by swapping the `` @xmath703 '' and the `` @xmath704 '' superscripts hold true . * @xmath705 and @xmath706 . we must record that property @xmath707 is stated in @xcite without some of the features stated above . however , it is evident from sections 3 and 4 of @xcite that ( under the assumptions stated at the beginning of this subsection ) there is a hybrid equivalence , in the sense of douady hubbard , between @xmath708 and @xmath709 . the julia set of @xmath678 , @xmath710 , is a strong repeller for @xmath678 in the sense of definition [ d : strongrep ] . furthermore ( see @xcite , for instance ) there exists a neighbourhood basis of @xmath710 , @xmath711 , say , such that @xmath712 also note that , from properties @xmath713@xmath702 , the generic number of pre - images of any @xmath714 under @xmath0 equals @xmath23 . from this and , we can deduce that there exists a closed set @xmath289 such that @xmath715 , @xmath716 and @xmath717 since @xmath716 , it is not hard to show ( see ( * ? ? ? * theorem 5.4 ) , for instance , for a proof ) that the right - hand side above equals @xmath718 . then , by definition , we have : furthermore , as @xmath710 is a strong repeller for the map @xmath678 , invoking property @xmath719 ( along with the fact that the generic number of pre - images of any @xmath714 under @xmath0 equals @xmath23 ) gives us : now , let @xmath721 be the quasiconformal homeomorphism such that @xmath722 . assume @xmath692 contains a critical value of @xmath693 . by the construction described above , and by property @xmath719 , @xmath692 is totally invariant under @xmath693 . hence , there exists a point @xmath723 such that @xmath724 . there is a small connected open neighbourhood @xmath276 of @xmath578 such that @xmath725 and @xmath726 contains two distinct pre - images under @xmath693 of each point in @xmath684 belonging to a sufficiently small deleted neighbourhood of @xmath727 . as @xmath728 and @xmath657 is a homeomorphism , an analogous statement holds true around @xmath729 . by the inverse function theorem , this is impossible because @xmath710 contains no critical points of @xmath678 . the last statement is a consequence of hyperbolicity see , for instance ( * ? ? ? * theorem 3.13 ) . hence , our assumption above must be false . finally , by property @xmath702 , any critical values of @xmath730 lie away from @xmath683 . thus , we conclude our discussion of the present example by observing that , in view of property @xmath732 above and an argument analogous to the one that begins with the equation , with @xmath0 replacing @xmath5 and @xmath682 replacing @xmath683 , we can also show that corollary [ c : d1_less]@xmath331 applies to this example . * acknowledgements . * a part of this work was carried out during a visit by shrihari sridharan to the indian institute of science in 2012 . he thanks the department of mathematics , indian institute of science , for its support and hospitality . gautam bharali thanks shaun bullett for his helpful answers to several questions pertaining to the example in section [ ss : kleinian ] .	this paper is motivated by brolin s theorem . the phenomenon we wish to demonstrate is as follows : if @xmath0 is a holomorphic correspondence on @xmath1 , then ( under certain conditions ) @xmath0 admits a measure @xmath2 such that , for any point @xmath3 drawn from a `` large '' open subset of @xmath1 , @xmath2 is the weak@xmath4-limit of the normalised sums of point masses carried by the pre - images of @xmath3 under the iterates of @xmath0 . let @xmath5 denote the transpose of @xmath0 . under the condition @xmath6 , where @xmath7 denotes the topological degree , the above phenomenon was established by dinh and sibony . we show that the support of this @xmath2 is disjoint from the normality set of @xmath0 . there are many interesting correspondences on @xmath1 for which @xmath8 . examples are the correspondences introduced by bullett and collaborators . when @xmath8 , equidistribution can not be expected to the full extent of brolin s theorem . however , we prove that when @xmath0 admits a repeller , the above analogue of equidistribution holds true .	25,461	351
the main asteroid belt is a relic from the formation of the solar system . although much of its mass has been lost , it retains a great deal of information about solar system history and presents us with a laboratory in which we can study collisional processes that once operated throughout the circumsolar disk in which earth and the other planets were formed . one of the most straightforward observables constraining such processes is the asteroid belt s size - frequency distribution ( sfd ; bottke et al . the current main belt s sfd can be successfully modeled as the result of 4.5 billion years of collisional evolution @xcite . while such models fit the ` collisional wave ' set up by 100 km asteroids able to survive unshattered through the age of the solar system , they can not be observationally tested in the 100 meter size range . objects in this size range are very interesting , because they supply most near - earth asteroids and meteorites by shattering one another and/or migrating inward via yarkovsky and resonance effects @xcite . modern 8 - 10 meter telescopes can detect them , but monitoring them over many nights to determine an orbit requires a prohibitively large time investment for such powerful telescopes ( e.g. , 710 nights ; gladman et al . thus their distances and sizes remain unknown , and detailed analyses are confined to larger objects @xcite or use only rough statistical distances @xcite . we present a method to obtain precise distances to main belt asteroids ( mbas ) using only two nights of observations . distances translate directly into absolute magnitudes and hence to sizes given a reasonable assumption for the albedo distribution . this method , which we refer to as rotational reflex velocity ( rrv ) , will greatly increase the efficiency of surveys aimed at probing collisional evolution in the solar system by measuring the sfds for extremely small mbas . we demonstrate rrv distance determination using a data set from the 0.9-meter wiyn telescope , which we have analyzed using digital tracking @xcite in order to enhance our sensitivity to faint asteroids . digital tracking is a method for detecting faint moving objects that was first applied to the kuiper belt ( e.g. bernstein et al . 2004 ) , and very recently has begun to be applied to asteroids @xcite . although the rrv distances we calculate herein are all based on our digital tracking analysis , the rrv method is equally useful for asteroids detected by more conventional means , or by other specialized methods such as those of @xcite and @xcite . suppose that at a given instant , an asteroid located a distance @xmath0 from an earth - based observer is moving with velocity @xmath1 , while the observer is moving with velocity @xmath2 ( e.g. , the orbital velocity of the earth ) . the angular velocity at which the observer sees the asteroid move relative to distant stars is given by : @xmath3 where the @xmath4 subscript indicates the vector component perpendicular to the line of sight , so that @xmath5 is the projection of the asteroid s relative velocity onto the plane of the sky . although @xmath2 can be accurately calculated for any earth - based observation , the velocity @xmath1 of a newly discovered asteroid is always unknown initially , and hence the distance can not be calculated by simply plugging the measured value of @xmath6 into equation [ eq : bot ] . given appropriate measurements , however , we can isolate the component of @xmath6 that reflects the observer s motion around the geocenter due to earth s rotation , and from this calculate the distance . this is the essence of the rrv method for distance determination . the velocity @xmath2 of an observer on the surface of the earth can be expressed as the sum of earth s orbital velocity @xmath7 and the velocity @xmath8 with which the earth s axial rotation carries the observer around the geocenter . neglecting the slight asphericity of the earth , @xmath9 , where @xmath10 is the observer s terrestrial latitude and @xmath11 is the earth s equatorial rotation velocity of 1674.4 km / hr . for convenience , we define @xmath12 as the asteroid s velocity relative to the geocenter : @xmath13 . the angular velocity @xmath14 that would be measured by an ideal observer located at the geocenter then depends only on @xmath12 and the distance , but the angular velocity @xmath15 that is measured by a real observer based on the earth s surface depends also on @xmath8 . the two angular velocities are given by : @xmath16 @xmath17 if we could measure @xmath14 , we could therefore calculate the distance : @xmath18 where we have dropped the @xmath4 subscript , because it will henceforward apply to all physical velocities in our calculations . now suppose that the asteroid is observed near midnight on two different nights , that the two observations are separated by exactly one sidereal day , and that the position and angular velocity @xmath15 are recorded for each observation . the angular distance the asteroid moved between the two observations will thus be accurately known ; call this @xmath19 . because exactly one full rotation of the earth elapsed between the two observations , the observer s position relative to the geocenter is the same for both of them . thus , the average geocentric angular velocity of the asteroid in between the two measurements is @xmath20 , where @xmath21 is the elapsed time between the observations : one sidereal day . let the measured values of @xmath15 on the first and second nights be @xmath22 and @xmath23 , and similarly let the perpendicular rotational velocities ( which are obtained by calculation , not measurement ) be @xmath24 and @xmath25 . we can then evaluate the difference between geocentric and observer angular velocities twice : the average of @xmath26 and @xmath27 will be a factor of @xmath28 more precise than a single measurement if the uncertainty on @xmath14 measurement is much smaller than on @xmath22 and @xmath23 . this is likely to be the case , since @xmath14 is based on a longer temporal baseline . the distance is then given by : @xmath29 so far we have assumed that @xmath12 and @xmath0 show no appreciable change over the 24 hour period of the measurements , so that @xmath26 and @xmath27 are effectively two measurements of the same quantity . we will now determine the errors that result when this assumption is violated . we will first consider changes in @xmath12 and @xmath0 that are linear in time : that is , when the first time derivatives @xmath30 and @xmath31 are significant but the second derivatives @xmath32 and @xmath33 are not . we will parameterize the change in @xmath12 by @xmath34 , and the change in @xmath0 by @xmath35 . for convenience , we will also use @xmath36 since the observations do not have to be taken exactly one sidereal day apart , and hence the earth s projected rotational velocity may vary . for well - optimized observations @xmath37 should always be close to zero . note that while @xmath38 and @xmath37 are unit - less , fractional changes , @xmath39 still has units of linear velocity . we then obtain : @xmath40 we substitute these into equation [ eq : av01 ] . retaining terms only to second order in the variational quantities @xmath39 , @xmath38 , and @xmath37 , after a good deal of algebra we find that the fractional error on the calculated distance is : @xmath41 for main belt asteroids @xmath38 is always very small ( i.e. @xmath42 ) . this renders @xmath37 innocuous even though it can reach 0.3 for poorly - planned observations ( section [ sec : nonopt ] ) . the velocity change @xmath39 approaches zero for asteroids at opposition , where the accuracy of equation [ eq : av01 ] can be better than @xmath43 , but even months from opposition the error remains only of order @xmath44 . equation [ eq : av01 ] is therefore very accurate under all conditions where the required input data could reasonably be obtained , and the uncertainties even of highly accurate measurements ( section [ sec : measprec ] ) will generally be larger than the equation s intrinsic error . we caution the reader against a much less accurate alternative to equation [ eq : av01 ] that may appear more attractive because it is obtained simply by applying equation [ eq : vel2 ] individually to each night and then averaging the resulting two distance measurements : @xmath45 despite its intuitive simplicity , this equation should never be used . its fractional error expansion contains the term @xmath46 , which has very large values for objects more than a few days from opposition , as illustrated by figure [ fig : reflexerrors ] . by contrast , for equation [ eq : av01 ] the precursors of the @xmath46 term and several other error terms cancel in the denominator , leaving only the very much smaller error terms given by equation [ eq : err02 ] . equation [ eq : av00 ] should therefore be avoided , and equation [ eq : av01 ] ( or its generalized form , equation [ eq : newav01 ] ) should always be used for rrv distances . for simplicity , the discussion in section [ sec : math ] assumed the observer could make instantaneous angular velocity measurements . in practice , of course , angular velocity measurements must be made over a period of time . this has very little effect on our mathematical derivation . the only change is that the projected velocities of the observer relative to the geocenter , @xmath24 and @xmath25 , should not be considered as instantaneous but rather as averages over the same period of time in which the angular velocities @xmath22 and @xmath23 were measured . positions and angular velocities of asteroids are measured in terms of right ascension ( ra ) and declination ( dec ) on the celestial sphere , and thus are two dimensional . for example , an observer could measure an asteroid as having an angular velocity of -37.36 arcsec / hr in ra ( the negative sign indicating westward rather than eastward motion ) and 10.32 arcsec / hr in dec , with the positive sign indicating the object is moving toward celestial north . in principle , independent versions of equation [ eq : av01 ] could be constructed for each dimension , with angular velocities in ra corresponding exactly to projected east - west motion of the observer and dec velocities corresponding to projected north - south motion . however , the true rotational velocity of the observer is always strictly east - west , and although its projection @xmath8 can have a north - south component , this component is usually small i.e. , less than 10% of the east - west component for observations with mean hour angle in the range @xmath47 targeting asteroids within @xmath48 of the celestial equator . as we will find below , for our test observations only the ra velocity components are large enough to supply useful distances . we expect this will generally be the case . in most asteroid surveys , many fields are observed , and each field is visited at least three times per night . each visit yields a single celestial position of each detected asteroid . two observations of each asteroid on each night will suffice for our purposes . using the traditional symbols @xmath49 and @xmath37 for ra and dec , suppose that on night 1 we obtain observations @xmath50 at time @xmath51 and @xmath52 at time @xmath53 , and similarly on night 2 . then we can also define mean positions and the mean time @xmath54 , @xmath55 , and @xmath56 , with the analogous quantities being calculated for night 2 . note that the times here are measured on a continuous sequence and not reset when the date changes . for simplicity , at present we will suppose further that the time of each measurement on night 2 is exactly one sidereal day later than the time of the corresponding measurement on night 1 : thus @xmath57 one sidereal day . the required angular velocities are then given by the following equations , where for simplicity we show only the ra component : @xmath58 @xmath59 @xmath60 where the denominator of the last equation is of course also equal to one sidereal day . n , roughly the latitude of mauna kea ) , but they are separated by an elapsed time of 3.989 hr , corresponding to 60@xmath61 of sidereal rotation . the rotational velocity of earth s surface at 20@xmath61 n lat is 1573.4 km / hr . the effect on the asteroid s measured positions is determined by the projected distance moved between the observations , which has eastward and northward components as labeled . the projected rotational velocity @xmath8 that should be used in the rrv distance calculation ( equation [ eq : av01 ] or equation [ eq : newav01 ] ) is therefore @xmath62 km / hr eastward and @xmath63 km / hr northward . [ fig : earth ] ] the only other quantities needed for the distance calculation are @xmath24 and @xmath25 , the observer s projected rotational velocities relative to the geocenter . as illustrated by figure [ fig : earth ] , the observer s projected motion relative to the geocenter traces out a segment of an ellipse whose axis ratio is the sine of the asteroid s declination . the projected distance of the observer from the geocenter is given by : @xmath64 where @xmath65 and @xmath66 are positive toward the east and north ; @xmath67 is the observer s latitude on the earth ; @xmath37 is the declination of the asteroid ; @xmath68 is the time of the observation ; @xmath69 is the time when the asteroid stands at zero hour angle for the observer ; and @xmath70 is the length of a sidereal day . note that the argument @xmath71 is the hour angle at which the observer sees the asteroid : at @xmath72 the asteroid crosses the meridian and stands closest to the observer s zenith and in the best sky position for observing . for observations at @xmath51 and @xmath53 , the projected linear velocity @xmath24 in the east - west ( @xmath65 ) and north - south ( @xmath66 ) directions is : @xmath73 @xmath74 figure [ fig : earth ] illustrates a specific example of two observations of an asteroid at -12@xmath61 dec , made from a single observing site at 20@xmath61 n lat on the earth s surface . the first observation is made when the asteroid is rising , and the second about four hours later , shortly after it has crossed the meridian . the rotational velocity of earth at 20@xmath61 n lat is 1573.4 km / hr and is of course strictly eastward , but projecting the velocity into a plane perpendicular to the line of sight reduces the absolute magnitude and introduces a north - south component ( except for asteroids exactly at 0@xmath61 dec ) . in this example the projected velocities east and north are 1410.3 km / hr and 106.2 km / hr . since celestial ra and dec correspond exactly to terrestrial longitude and latitude , the east - west physical velocity of equation [ eq : proj2x ] can be combined with the ra components of the angular velocities from equations [ eq : meas01][eq : meas03 ] and plugged directly into equation [ eq : av01 ] to yield the distance . we will now provide a simplified but basically realistic example . suppose the mba illustrated by figure [ fig : earth ] is being observed at opposition . it will be in its retrograde loop , and hence its motion in ra will be negative ( westward ) . the angular velocity @xmath75 measured by the observer will be faster to the west than the geocentric angular velocity @xmath14 , because on the night side of the earth the rotational velocity adds to the orbital velocity , making the asteroid fall back westward at a faster rate . arcsec / hr and @xmath77 arcsec / hr . the rotational reflex velocity , given by the difference @xmath78 , is then 1 arcsec / hr or @xmath79 rad / hr . taking the physical and angular velocities to be exactly the same on the second night for simplicity , equation [ eq : av01 ] reduces to : @xmath80 with equation [ eq : calc01 ] we have calculated the distance to an asteroid based on the known properties of earth s rotation . interestingly , @xcite have developed a closely analogous method for determining the distance to a kuiper belt object using the known properties of earth s _ orbit_. just as our derivation does not ( explicitly ) include the effects of earth s orbit , that of @xcite does not include the effects of earth s rotation . neglecting earth s orbit works in our case because , as we have demonstrated in section [ sec : math ] , the dominant errors due to orbital acceleration of both the earth and the asteroid cancel when the correct form of the distance equation is used . similarly , @xcite can safely neglect earth s rotation because the kuiper belt objects whose distances they aim to measure are sufficiently far away that their rrv signature is negligible . it follows that the method of @xcite should not be directly applied to measuring asteroid distances without some form of correction for the rrv signal , although ( as they point out ) it is useful for distinguishing true kuiper belt objects from asteroids that have similar angular velocities for a few days near their turnaround points . up to now we have assumed for simplicity that the observations obtained on successive nights are exactly one sidereal day apart . while it is not difficult to plan and execute observations that satisfy this criterion to within a few minutes , clouds or instrument failures could intervene , or one could be processing an archival data set that was not taken with rrv distances in mind . in such cases the mean times @xmath81 and @xmath82 may be separated by up to a couple of hours less or more than one sidereal day . a direct calculation of the geocentric angular velocity @xmath14 then becomes impossible . in its place , we must use instead the mean angular velocity across the two nights , defined by : @xmath83 we will define three additional quantities . the time @xmath84 is the moment exactly one sidereal day before @xmath82 ( in the ideal case , @xmath81 would have been equal to @xmath84 ) . the projected rotational velocity @xmath85 is the observer s mean projected rotational velocity over the time interval between @xmath81 and @xmath84 . finally , the fractional parameter @xmath86 is given by : @xmath87 note that @xmath86 goes to zero in the case of optimally - timed measurements . the distance formula analogous to equation [ eq : av01 ] becomes : @xmath88 the error terms for equation [ eq : newav01 ] are the same as those given in equation [ eq : err02 ] , with the addition of an @xmath89 term . for main belt asteroids , this will always be negligible . in closing our discussion of error in the case of non - optimal observation timing , we note that @xmath37 , the fractional change in projected rotational velocity from night 1 to night 2 , is likely to be considerably larger in this case , up to 0.3 . however , in the error terms for equation [ eq : newav01 ] it is always multiplied by @xmath38 , which is always of order @xmath44 or smaller for mbas , and thus should not constitute a major source of error . equation [ eq : newav01 ] is therefore the central formula for asteroid distance determination with rrv . in addition to our analytical error calculation ( equation [ eq : err02 ] ) , we have probed the errors of our approximations numerically using ephemerides of known objects from the jpl horizons ephemeris generator . note that no actual observations are involved in this test , only ephemeris positions and velocities for objects with well - known orbits . in figure [ fig : reflexerrors ] we plot the error of equation [ eq : newav01 ] as determined by this test . in contrast to our analytical error calculations , this test intrinsically includes the contributions of second and higher - order time derivatives of @xmath0 and @xmath12 , since it uses jpl horizons ephemerides for real objects . the plotted errors are affected by roundoff error in the ephemerides ( accurate only to 0.15 arcsec in ra ) , which introduces pseudo - noise such that only the lower envelope of the plotted points can be meaningfully compared with our analytical error estimates . nevertheless , the plots show that equation [ eq : newav01 ] produces accurate distances out to at least 60 days from opposition ; that accurate distances can be obtained from observations that are not optimally timed ; and that the second - order time derivatives @xmath32 and @xmath33 do not introduce significant error . figure [ fig : reflexerrors ] , although based on calculated ephemerides and not actual data , shows that relatively small position errors can create distance inaccuracies at the level of a few percent . we now consider the measurement precision that is necessary to yield distances to various levels of accuracy . in equation [ eq : calc01 ] , the difference between the night - to - night mean angular velocity in ra ( @xmath90 ) and the angular velocity in ra observed within a given night ( @xmath91 ) was one arcsec / hr , and this rotational reflex velocity corresponded to a distance of about 2 au . this is typical for observations obtained near opposition at terrestrial latitudes of 2030@xmath61 . the fractional uncertainty of the distance is equal to that of @xmath92 . this will generally be dominated by the uncertainty in @xmath91 , since it is calculated within a single night and is thus based on a smaller temporal baseline than @xmath90 . thus , if @xmath91 is measured with an accuracy of 0.1 arcsec / hr , we should expect the distance to have an accuracy of only 10% at 2 au . at 1 au , the rrv signature is twice as large : @xmath92 will therefore be about 2 arcsec / hr , and the calculation will be accurate to 5% . as we will now illustrate , one can usually obtain angular velocity measurements with better than 0.1 arcsec / hr uncertainties in practice . in our current paradigm , @xmath91 is based on two measurements of celestial positions separated by a time @xmath93 ( i.e. , equation [ eq : meas01 ] or [ eq : meas02 ] ) . if the uncertainty on ra for these measurements is @xmath94 , the uncertainty on @xmath91 will be : @xmath95 based on our experience , well - sampled ( e.g. @xmath96 arcsec / pixel ) images taken in @xmath97 arcsecond seeing can yield @xmath94 = 0.030.05 arcsec for bright objects ( e.g. , those at least several times brighter than the 10@xmath98 detection threshold ) . for faint objects near the detection limit , uncertainties of @xmath99 arcsec are more typical ) rather than being obtained from two positions . ] a temporal baseline of 5 hr is easy to obtain near opposition . thus , we can expect @xmath100 = 0.0080.014 arcsec / hr for bright objects and @xmath100 = 0.03 arcsec / hr for fainter objects . at 2 au , distances will therefore be accurate to about 3% for faint objects and 11.5% for bright objects , while at a distance of 1 au , the accuracy should be about 1.5% even for faint objects . these accuracies are sufficient for good analyses of the size statistics of small mbas , especially given that their albedos are unknown and must be treated as a statistical distribution spanning a factor of @xmath10110 @xcite . where the objects are faint or distant ( e.g. 35 au ) , or where extremely accurate distances are desired , it may be possible to obtain smaller values of @xmath94 by obtaining each ra measurement not from a single image but from the average over a set of images acquired close together in time . extending the same principle further , we can use the technique of digital tracking to obtain accurate measurements for objects too faint even to be detected in individual images , as we describe below . the observations we use to demonstrate the usefulness of rotational reflex distances for asteroids were obtained using digital tracking , as described in our companion paper @xcite . these data consist of 126 two - minute exposures of a single starfield that we acquired using the wiyn 0.9-meter telescope at kitt peak on the night of april 19 , and 130 identically acquired images from the following night . very briefly , a digital tracking search involves shifting and stacking such sets of images to reveal moving objects too faint to be detected on any individual frame . a separate trial stack is produced for each angular velocity vector in a finely - sampled grid that spans the full range of possible sky velocities for the target population . for mbas , a digital tracking data set can span only one night @xcite , hence we analyze our april 19 and april 20 observations independently . digital tracking has been used to great effect for kuiper belt objects ( e.g. allen et al . 2001 , fraser & kavelaars 2009 , and many others ) , but has not typically been used for faster - moving asteroids where a larger numbers of trial vectors must be probed . it is now computationally tractable even for asteroids , however , and produces a factor of @xmath102 increase in sensitivity over conventional methods . this sensitivity increase enables us to detect 215 asteroids within the 1-degree field of our test observations , despite using only a small , 0.9-meter telescope . we obtained precise angular velocity measurements on both nights for 197 of these asteroids , including 48 previously known objects with accurate orbits . these last allow us to test the accuracy of rrv distance measurements using real data . the geocentric angular velocity @xmath14 of an asteroid detected on two subsequent nights is determined in exactly the same way with digital tracking observations as with conventional data . the determination of the observed angular velocities @xmath15 on each individual night is different , however particularly because digital tracking enables the accurate measurement of asteroids that can not even be detected on an individual image . any asteroid detection in a digital tracking search occurs on a particular trial image stack , which corresponds to a particular angular velocity . thus , for example , our automated digital tracking routine might search for objects with angular velocities between -50 and -20 arcsec / hr in ra and -10 and + 20 arcsec / hr in dec , using a grid spacing of 0.2 arcsec / hr , and might detect a particular asteroid as a bright point source on a trial stack whose shifts correspond to an angular velocity of -41.2 arcsec / hr in ra and 11.8 arcsec / hr in dec . we improve on the relatively crude angular velocity measurement that is implicit in such a detection by probing a new , much more finely spaced grid of angular velocity vectors , using small postage - stamp images centered on the detected asteroid to make the search computationally tractable . for each trial stack in this finely sampled grid , we calculate the measured flux of the asteroid within a small aperture of radius roughly equal to the half - width of the point spread function ( psf ) of our image . we then perform a 2-d quadratic fit to the measured flux as a function of the trial angular velocity . the angular velocity at which the quadratic fit reaches its peak value then constitutes an accurate measurement of @xmath15 for that asteroid , and the uncertainty on @xmath15 is derived from the uncertainty of the quadratic fit , which is naturally larger for faint objects that are more affected by sky background noise . all the inputs required to calculate the distance using equation [ eq : newav01 ] are now available except @xmath8 . as in the case of discrete observations , equation [ eq : proj1 ] indicates how @xmath8 should be calculated , but there is one subtle aspect . the appropriate value of @xmath8 is not a projected distance divided by an elapsed time as it was in the discrete case ( figure [ fig : earth ] ) , but neither is it the time average of the projected velocities @xmath103 and @xmath104 . instead , the value of @xmath8 that corresponds to the angular velocity measured by digital tracking is given by the slopes of the best linear fits to @xmath65 and @xmath66 as functions of time . figure [ fig : proj1 ] illustrates the difference between linear fit velocities and average velocities . for our data , the linear fit velocities of the kitt peak observer were 1340.0 km / hr eastward and -4.8 km / hr northward on april 19 ; and 1337.3 km / hr eastward and -16.3 km / hr northward on april 20 ; by contrast the average velocities were 1286.6 km / hr eastward and -6.9 km / hr northward on april 19 and 1283.0 km / hr eastward and -14.6 km / hr northward on april 20 . in this case , the north - south velocities are negligible for practical purposes . our observations on 2013 april 19 span a temporal range of 5.72 hr . those obtained on april 20 have the same range , and are centered 23.968 hr later in time . as this interval is only 2 minutes longer than a sidereal day , the timing of our observations is almost exactly optimal as defined in section [ sec : opt ] : the parameter @xmath86 from equation [ eq : meas08 ] is only 0.0014 . we detected a total of 215 asteroids through digital tracking analysis of our observations in a field of view only slightly larger than 1 square degree . we reported positions and brightnesses for all of these objects to the minor planet center . of these 215 mbas , 202 were detected on both april 19 and april 20 , and 197 of these two - night objects had sufficiently accurate measurements on both nights for good distance calculations . of these , 48 were previously discovered objects with accurately known orbits , nine were previously discovered objects with poorly known orbits prior to our observations , and the remaining 140 were new discoveries for which we received designations from the minor planet center . @xcite provides further details . we have used equation [ eq : newav01 ] to calculate distances for all 197 objects with sufficient measurements . the 48 objects in this sample with well - known orbits allow us to test the accuracy of rotational reflex distances using real data . the mean absolute error of our calculated distances for these objects is 0.042 au , and the mean absolute fractional error is 2.2% . as expected , errors are smaller for more nearby objects . distances with 2% accuracy are more than sufficient to analyze the size statistics of mbas . nevertheless , even better results are possible , as we will see below . close investigation reveals some evidence of systematic error in the distance calculations described in section [ sec : initdist ] . the weighted average signed fractional error for the 48 known objects is @xmath105 : i.e. , the calculated distances are 1.8% too small on average , and the offset is 10@xmath98 significant . it might be suspected that this bias is due to using the wrong value for @xmath8 i.e. , that we were wrong to use linear fit velocities rather than time - averaged velocities ( section [ sec : angdigi ] ) . this is easily demonstrated not to be the case , however . substituting averaged values in place of linear fit values for @xmath8 changes the result too much and in the wrong direction : the mean systematic offset goes from -1.8% to -5.2% . the true cause of the bias is more subtle , and points to a further interesting measurement that can be extracted from our data . with digital tracking , our measurement of the single - night observed angular velocity @xmath15 is made over a period of time equal to the temporal span of our observations on each night . the instantaneous angular velocity over this interval is continuously changing , with the rotational reflex velocity itself being the dominant source of change . for example , the rotation of the earth causes the westward sky motions of the asteroids to be slightly faster near midnight than they are at the beginning and end of each night s observations ( see figure [ fig : proj1 ] ) . as we stated in section [ sec : angdigi ] and illustrated by figure [ fig : proj1 ] , we can account for the acceleration and curvature of an asteroid s observed motion by using a value for @xmath8 that is obtained by a linear fit to the observer s projected coordinates @xmath106 and @xmath107 . this assumes , however , that the angular velocity measurement is effectively an _ unweighted _ fit to angular position as a function of time . such is not necessarily the case . for example , the images taken near midnight , when the westward sky motion is at its fastest , may be more sensitive due to better seeing or lower atmospheric extinction . the digital tracking stack will then effectively weight them more highly and produce a mean angular velocity that is systematically too fast . this effect would produce a systematic underestimation of the asteroids geocentric distances : exactly what we observe for known asteroids in our data . the effective weighting of different images will not necessarily be the same for different asteroids . an asteroid that passes too close to a bright star near midnight will only be measured near the beginning and end of the night , resulting in a measured angular velocity slower than the true average value . asteroids can also exhibit significant brightness changes during the night due to their own rotation , which will change the effective weighting of different images in the final measurement of their angular velocities . if asteroids actually moved in straight lines at constant velocity , none of these weighting effects would introduce error into our angular velocity determinations . if we could _ predict _ the curvature and acceleration of each asteroid s track due to the earth s rotation , and apply small shifts to each image in our digital tracking stacks in order to linearize the motion , we would therefore remove the bias in our angular velocity determinations . we now describe how to do this . the angular distances between the position an observer measures for an asteroid and the position it would have if measured from the geocenter are simply a reflection of equation [ eq : proj1 ] , scaled by the distance : @xmath108 averaged over the span of a digital tracking integration , @xmath109 and @xmath110 contribute a constant offset to the measured position of an asteroid , and their first time derivatives @xmath111 and @xmath112 produce a constant offset ( i.e. the rrv signal itself ) to its measured angular velocity . neither of these constant offsets concerns us here : we wish to isolate and remove only the component that represents curvature and acceleration . thus , we create new functions @xmath113 and @xmath114 that behave exactly like @xmath115 and @xmath116 except that over the span of our digital tracking integrations , their means and the means of their first time derivatives are zero : @xmath117 where the notation @xmath118 denotes the time - average of the enclosed quantity over the span of a digital tracking integration . the left panel of figure [ fig : curve ] presents equation [ eq : curve2 ] in graphical form , plotting the curved track of an asteroid in @xmath119 space over the course of our april 19 observations . the calculation of @xmath120 and @xmath121 of course requires knowledge of the asteroid s distance @xmath0 , but the 2% accurate values from section [ sec : initdist ] are more than sufficient for this purpose . thus , we adjust the methodology described in section [ sec : angdigi ] for measuring precise angular velocities from digital tracking data : we apply additional shifts equal to @xmath122 to each postage stamp image before the final stack . this straightens the asteroid s track so that curvature can no longer influence the calculated angular velocity . the distance can be recalculated based on the resulting new , de - biased angular velocities . note that because equation [ eq : curve2 ] subtracts _ average _ reflex velocities , we must now use averaged rather than linear - fit values for @xmath8 ( section [ sec : angdigi ] and figure [ fig : proj1 ] illustrate how the velocities differ ) . compared to the calculation in section [ sec : initdist ] , our curvature correction reduces the mean absolute error of the distances for known objects from 0.042 au to 0.032 au , and the mean absolute fractional error from 2.2% to 1.6% . more significantly , the new calculation changes the weighted average signed fractional error from @xmath105 to @xmath123 . the systematic offset is thus reduced by a factor of six , and the new value of -0.3% is statistically consistent with zero . the left panel of figure [ fig : distmag ] illustrates the precision of our measured distances for known objects . we emphasize that these distance measurements were entirely calculated from first principles and known quantities describing the earth . we have used the known asteroids in our data to quantify our errors , but not to determine any of the parameters used in the distance calculation . beyond removing systematic errors in our measurements of asteroids mean angular velocities , we can in principle use equation [ eq : curve2 ] to _ measure _ the curvatures of asteroid tracks within just a single night s data . this is an extremely challenging measurement because the curvatures are so small : the rms curvature amplitude , defined as the root mean of @xmath124 , is less than 0.1 arcsec for an asteroid at 2 au . we attempt to measure it using our curvature - corrected postage - stamp analysis as described in section [ sec : two_precise ] , but instead of calculating @xmath125 and @xmath126 based on a previously - calculated approximate distance , we probe a range of rms curvature amplitudes for each asteroid . for each trial curvature amplitude we calculate @xmath125 and @xmath126 , apply the appropriate shifts to the postage stamp images , and perform a quadratic fit to flux as a function of angular velocity as described in section [ sec : angdigi ] . we compare the peak asteroid flux values produced by the quadratic fits for the different trial curvature amplitudes , and identify the curvature amplitude that yields the highest peak asteroid flux . this should be the curvature value that resulted in all of the individual asteroid images being most accurately registered . as the curvature amplitudes map uniquely to distances ( equations [ eq : curve1 ] and [ eq : curve2 ] ) , our identification of the best - fit amplitude constitutes a distance measurement , in principle . however , the uncertainty of curvature amplitudes measured from the current data set is large enough that we do not choose to calculate individual curvature - based distances . instead , we plot the best - fit curvature amplitudes as a function of the known rrv distances calculated in section [ sec : two_precise ] . a clear inverse relationship emerges , as predicted by equation [ eq : curve1 ] . the relationship gets tighter when we reject the minority of asteroids with out - of - range best - fit curvature values and all asteroids fainter than @xmath127 mag , which can not be measured as accurately . the 48 objects meeting these criteria in our april 19 observations are plotted in the right - hand panel of figure [ fig : curve ] , which shows the rms curvature amplitude as a function of geocentric distance in au , and compares the data with the predicted curve from equations [ eq : curve1 ] and [ eq : curve2 ] . although there is a mild systematic offset , the slope of the best linear fit to curvature as a function of @xmath128 for these 48 asteroids is 9@xmath98 significant , matches the predicted value to within 0.4@xmath98 , and has an intercept only 1.4@xmath98 away from the predicted value of zero . the rms error of the measured curvature amplitudes relative to their predicted values is only 0.03 arcsec . thus we have effectively performed astrometry of moving , 20th magnitude objects with a precision of 30 milli - arcsec using only an 0.9 m telescope . in principle , our success at measuring asteroid curvatures within a single night s data means that digital tracking can be used to measure the geocentric distances to asteroids based on observations from only one night . however , our current curvature measurements have uncertainties that are too large to yield useful distances for individual objects : we have been able to demonstrate the efficacy of our curvature - measuring methodology only by recourse to an ensemble of 48 asteroids with distances known by other means . with a larger telescope , more accurate curvature measurements of fainter asteroids would be possible , especially in good seeing . these could conceivably yield meaningful distance measurements for unknown main - belt asteroids over just a single night , albeit only for objects considerably brighter than the digital tracking detection limit . by contrast , the two - night rrv calculations of sections [ sec : initdist ] [ sec : two_precise ] yield precise distances even for faint objects just above the detection threshold . there is therefore little reason to use the curvature method for asteroids measured on more than one night . however , where only one night s data is available , distances based on digital - tracking curvature measurements could be very valuable . this is particularly true in at least two cases that may arise in future surveys . the first is the case of an neo survey where some objects perhaps due to their fast motion have inadvertently been measured on only one night . being much closer to the earth than mbas , neos will show much larger and more easily measurable curvature , which is likely to yield useful distances . such distances could enable the inclusion in a statistical analysis of objects that were otherwise unusable due to their one - night status . the second case is the one - night detection of a nearby neo moving at nearly the same space velocity as earth , such that its slow angular velocity mimics that of a much more distant object . this scenario is statistically rare but troubling , because such an object would likely be overlooked in an neo survey and yet could be an incoming earth - impactor . if the discovery survey used digital tracking , a curvature analysis would reveal the object s actual , very small geocentric distance . we note in closing that curvature measurements can in principle be performed on asteroids detected with conventional methods rather than digital tracking , but they will normally be less accurate because of the smaller number of images available . while our current data only suggest the possibility of curvature - based distance measurements from a single night , they allow precise two - night distance measurements from rotational reflex velocity ( section [ sec : two_precise ] ) . these precise distance measurements enable us to calculate the absolute magnitudes of each of our detected asteroids , and hence their physical diameters modulo the uncertainty in albedo . the histograms of these values are shown in the right - hand panel of figure [ fig : distmag ] . at least half of the asteroids we have measured are smaller than 1 km . by contrast , only a small fraction of known mbas are in this size range . the smallest objects we have detected are 300 meters in diameter under the assumption of low , 5% albedo , but could be as small as 150 meters if their albedo is 25% . the statistics of mbas in this size range are already known @xcite based on a sample several times larger than we present herein . given this , the detailed completeness analysis that would be required to convert our detections into a measurement of the sfd is not worthwhile . the aims of the current work and our companion paper @xcite are instead the validation of the rrv technique and of digital tracking for asteroids , respectively . similar observations using a larger telescope , however , would extend to much smaller objects and accurately measure the sfd of a previously unexplored size regime in the main belt . we have described how the reflex angular velocity of asteroids due to earth s rotation can be used to determine the distances to main belt asteroids based on only two nights of observations . we refer to this as the rotational reflex velocity ( rrv ) method for measuring asteroid distances ( equation [ eq : newav01 ] ) . the required approximations are accurate to about @xmath44 ( equation [ eq : err02 ] ) , and measurement uncertainties are typically 1 - 3% . such distances can be used to calculate precise size statistics of small main belt asteroids using a much smaller investment of time on a large telescope than has previously been required . accurate rrv distances may be calculated either from conventional asteroid - search observations , from data analyzed by the technique of digital tracking @xcite , or from any other specialized asteroid - observation technique ( e.g. milani et al . 1996 or gural et al . 2005 ) that accurately measures the celestial coordinates of the asteroids . we have tested rrv distance determination with a data set acquired on april 19 and 20 , 2013 using the wiyn 0.9-meter telescope at kitt peak . using measurements based on digital tracking , we have calculated distances to 197 asteroids in this data set . while the majority of these were new discoveries in our data , 48 of them are previously known objects with accurate orbits . these allowed us to test the accuracy of rotational reflex distances . a preliminary analysis yielded distances with a mean fractional error of only 2.2% . without digital tracking , our detections would be confined to substantially brighter asteroids ( or , the observations would require a larger telescope ) , but the accuracy of rrv distances for detected objects would likely be at least this good ( see section [ sec : measprec ] ) . we have identified a 1.8% systematic error in our preliminary rrv distances . we have linked this error to the curvature of the asteroids observed tracks that results from earth s rotation . this curvature , combined with the non - uniform sensitivity of images obtained under conditions of different seeing and airmass , causes a slight bias in the angular velocity measurements obtained from our digital tracking image - stacks . finding our initial distance measurements more than sufficient to predict curvature amplitudes , we have calculated the form and amplitude of the curvature for each of our asteroids . we have applied a curvature correction to our image stacks to linearize the asteroid motions . re - calculating the distances using angular velocities from curvature - corrected image stacks reduces the systematic bias from 1.8% to @xmath129% , and thus eliminates it as a significant effect . the corrected distances have a mean fractional error of only 1.6% for the 48 known objects in our data . note that such a curvature correction would not be needed for rrv distances based on asteroids detected on discrete images . the calculation of accurate rrv distances would therefore be simpler for conventional asteroid detections , although digital tracking can detect much fainter objects . besides correcting our digital tracking angular velocity measurements for the curvature due to earth s rotation , we have attempted to measure this curvature by creating several different image stacks for each asteroid at a range of different curvature amplitudes . as the curvature amplitude is proportional to @xmath130 , this constitutes a rotational reflex velocity measurement of an asteroid s distance based on just a single night s data . we find that we can indeed measure the curvature amplitudes for asteroids brighter than @xmath131 mag with a precision of about 30 milli - arcsec . the measured curvature values for 48 asteroids brighter than this limit show the expected @xmath130 dependence at 9@xmath98 significance , and the best - fit slope is within 0.4@xmath98 of the predicted value . while distance measurements based on single - night curvature amplitudes are too noisy to be useful for asteroids in our current data set ( and are not needed given our highly precise two - night values ) , single - night distances could be valuable in specific cases for future surveys . in particular , curvature measurements can identify nearby objects moving at slow angular velocities characteristic of a much more distant population . our precise distances from two - night rrv measurements allow us to calculate absolute magnitudes and hence approximate diameters for our newly discovered asteroids . our faintest objects have @xmath132 mag and hence diameters of 130300 meters depending on their albedo . while the current census of the main belt becomes substantially incomplete at a diameter of about 2 km , we have detected dozens of new asteroids in the 200500 meter size range with an 0.9 m telescope . based on observations at kitt peak national observatory , national optical astronomy observatory ( noao prop . i d : 2013a-0501 ; pi : aren heinze ) , which is operated by the association of universities for research in astronomy ( aura ) under a cooperative agreement with the national science foundation . this publication makes use of the simbad online database , operated at cds , strasbourg , france , and the vizier online database ( see @xcite ) . we have also made extensive use of information and code from @xcite . we have used digitized images from the palomar sky survey ( available from http://stdatu.stsci.edu/cgi-bin/dss_form ) , which were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . allen , r. l. , bernstein , g. m. , & malhotra , r. 2001 , , 549 , l241 bernstein , g. & kushalani , b. 2000 , , 120 , 3323 bernstein , g. m. , trilling , d. e. , allen , r. l. , brown , m. e. , holman , m. , & malhotra , r. 2004 , , 128 , 1364 bottke , w. f. jr . , durda , d. d. , nesvorn , d. , jedicke , r. , morbidelli , a. , vokrouhlick , d. , & levison , h. 2005 , icarus , 175 , 111 bottke , w. f. jr . , durda , d. d. , nesvorn , d. , jedicke , r. , morbidelli , a. , vokrouhlick , d. , & levison , h. f. 2005 , icarus , 179 , 64 de ela , g. c. & brunini , a. 2007 , , 466 , 1159 farinella , p. , froeschl , chr . , froeschl , cl . , gonczi , r. , hahn , g. , morbidelli , a. , & valsecchi , g. b. 1994 , nature , 371 , 314 fraser , w. c. & kavelaars , j. j. 2009 , , 137 , 72 gladman , b. j. , davis , d. r. , neese , c. , jedicke , r. , williams , g. , kavelaars , j. j. , petit , j - m . , scholl , h. , holman , m. , warrington , b. , esquerdo , g. , & tricarico , p. 2009 , icarus , 202 , 104 gural , p. s. , larsen , j. a. , & gleason , a. e. 2005 , , 130 , 1951 heinze , a. n. , metchev , s. , & trollo , j. 2015 , , submitted masiero , j. r. , mainzer , a. k. , bauer , j. m. , grav , t. , nugent , c. r. , & stevenson , r. 2013 , , 770 , 7 milani , a. , villani , a. , & stiavelli , m. 1996 , earth , moon , and planets , 72 , 257 ochsenbein , f. , bauer , p. & marcout , j. 2000 , , 143 , 23o pravec , p. , harris , a. w. , kusnirk , p. , galad , a. , & hornoch , k. 2012 , icarus , 221 , 365 press , w. h. , teukolsky , s.a . , vetterling , w. t. , & flannery , b. p. 1992 , numerical recipes in c ( second edition ; new york , ny : cambridge university press ) yoshida , f. , nakamura , t. , watanabe , j. 2003 , pasj , 55 , 701 yoshida , f. & nakamura , t. 2007 , p&ss , 55 , 1113 zhai , c. , shao , m. , nemati , b. , werne , t. , zhou , h. , turyshev , s. g. , sandhu , j. , hallinan , g. , & harding , l. k. 2014 , , 792 , 60	we present a method for calculating precise distances to asteroids using only two nights of data from a single location far too little for an orbit by exploiting the angular reflex motion of the asteroids due to earth s axial rotation . we refer to this as the rotational reflex velocity method . while the concept is simple and well - known , it has not been previously exploited for surveys of main - belt asteroids . we offer a mathematical development , estimates of the errors of the approximation , and a demonstration using a sample of 197 asteroids observed for two nights with a small , 0.9-meter telescope . this demonstration used digital tracking to enhance detection sensitivity for faint asteroids , but our distance determination works with any detection method . forty - eight asteroids in our sample had known orbits prior to our observations , and for these we demonstrate a mean fractional error of only 1.6% between the distances we calculate and those given in ephemerides from the minor planet center . in contrast to our two - night results , distance determination by fitting approximate orbits requires observations spanning 710 nights . once an asteroid s distance is known , its absolute magnitude and size ( given a statistically - estimated albedo ) may immediately be calculated . our method will therefore greatly enhance the efficiency with which 4-meter and larger telescopes can probe the size distribution of small ( e.g. 100 meter ) main belt asteroids . this distribution remains poorly known , yet encodes information about the collisional evolution of the asteroid belt and hence the history of the solar system .	14,480	376
dislocation behavior in solids under dynamic conditions ( e.g. shock loading @xcite ) has recently attracted renewed attention , @xcite partly due to new insights provided by molecular dynamics studies . @xcite whereas theoretical investigations mainly focused on the stationary velocities that regular or twinning dislocations can attain as a function of the applied stress ( possibly intersonic or even supersonic with respect to the longitudinal wave speed @xmath0),@xcite one other major concern is to establish an equation of motion @xcite ( eom ) suitable to instationary dislocation motions towards or from such high velocities , and which is computationally cheap . this would be an important step towards extending dislocation dynamics ( dd ) simulations @xcite to the domain of high strain rates , in order to better understand hardening processes in such conditions . the key to instationary motion of dislocations lies in the inertia arising from changes in their long - ranged displacement field , which accompany the motion . these retarded rearrangements take place at finite speed , through wave emission and propagation from the dislocation . as a consequence , dislocations possess an effective inertial mass,@xcite which has bearings on the process of overcoming dynamically obstacles such as dipoles , etc . @xcite inertial effects are non - local in time , and are related to effective `` viscous '' losses . for small velocities where the eom is linear,@xcite this relation takes the form of the kramers - krnig relations between the reactive and dissipative parts of the causal mass kernel.@xcite one major ingredient of the eom should thus be the effective visco - inertial force exerted on the dislocation by its own wave emission.@xcite an eom results from balancing it by the applied stress , and by drags of various origins.@xcite eoms with effective masses , but which ignore retardation ( e.g. , ref . ) , can not truly capture visco - inertial effects . previous works on these questions having mainly been confined to the linear regime , their influence in the relativistic domain remains largely unexplored in spite of analytical progresses , partly due to the complexity of the formalism ( especially for edge dislocations ) . hereafter , eshelby s eom for screws with a rigid core,@xcite valid at small velocities , is first re - examined , and cast under a simple form which suggests a straightforward regularization procedure for finite core effects . this allows us to appeal to previous results for point dislocations valid at high velocities.@xcite we then build in an heuristic way an eom for accelerated or decelerated screw and edge dislocations in the drag - dominated subsonic regime , that consistently accounts for saturation effects at velocities comparable to the sound speed . results from the equation are compared to quasi - exact calculations from a numerical method of the phase - field type . having in mind applications to dd simulations , the scope of the study is limited to continuum theory , so that dispersion effects due to the atomic lattice,@xcite or to the influence of the peierls potential,@xcite are not explicitly considered . within the peierls - nabarro model in isotropic elasticity,@xcite and with the usual @xmath1 ansatz for the relative displacement @xmath2 of the atoms on both sides of the glide plane , eshelby computed the visco - inertial force @xmath3 experienced by a screw dislocation of burgers vector @xmath4 , centered on position @xmath5 at time @xmath6 , moving with a velocity @xmath7 small compared to the shear wave speed @xmath8:@xcite @xmath9^{1/2}}\\ & & { } + m_0\int_{-\infty}^t \hspace{-1em}{\rm d}\!\tau \frac{t_{\rm s}^2}{\left[(t-\tau)^2+t_{\rm s}^2\right]^{3/2 } } \frac{\rm d}{{\rm d}\tau}\left(\frac{x(t)-x(\tau)}{t-\tau}\right)\nonumber.\end{aligned}\ ] ] the dislocation is assumed to have a _ rigid _ core of half - width @xmath10 . then @xmath11 is the time of shear wave propagation over the core width . the mass per unit dislocation length @xmath12 depends on the shear modulus @xmath13 . in ref . ( and in ref . as well ) , an incorrect factor @xmath14 is present in front of the second integral , and has been removed here . this factor is of no important physical consequence , save for different values of the linear response kernels ; see below . that ( [ eq : eshforce ] ) is correct can be verified as follows . starting from eshelby s expression of the force as a double integral in eq . ( 26 ) of ref . , and expanding it to linear order in the velocity @xmath15 or in @xmath16 , the following expression is easily obtained : @xmath17\right\},\end{aligned}\ ] ] where @xmath18 . using integrations by parts over @xmath19 , each of eq . ( [ eq : eshforce ] ) and ( [ eq : flin1 ] ) can be put under the following irreducible form : @xmath20^{3/2}},\ ] ] which shows them to coincide . by the same token , we check that ( [ eq : eshforce ] ) can be further simplified as : @xmath21^{1/2}}\frac{\rm d}{{\rm d}\tau}\left(\frac{x(t)-x(\tau)}{t-\tau}\right).\ ] ] by fourier transforming @xmath22 [ under the form ( [ eq : canonical ] ) ] and by writing @xmath23\,x(\omega),\ ] ] we identify effective mass @xmath24 and viscosity @xmath25 kernels . @xcite their expression in closed form involves the modified bessel and struve functions @xmath26 , @xmath27 and @xmath28 : [ eq : masskercf ] @xmath29\right\}\end{aligned}\ ] ] to leading orders in the pulsation @xmath30 , [ eq : massker ] @xmath31 where @xmath32 is euler s constant . moreover , we observe that @xmath33 result ( [ eq : massker ] ) coincides to leading order with eshelby s , @xcite as @xmath34 . the mass increase with wavelength as @xmath35 implies very different behaviors for , e.g. , quasi - static and shock loading modes , since the latter involves a wider frequency range . we note that @xmath36 as @xmath34 , since losses should be absent from the model in the stationary subsonic regime . @xcite the non - analytical behavior of the kernels at @xmath37 ( due to @xmath38 ) , and its associated non - locality in time has been emphasized in ref . . the finite `` instantaneous '' viscosity ( [ eq : viscoinfty ] ) stems from the first term in the r.h.s . of ( [ eq : canonical ] ) , and is responsible for a velocity jump @xmath39 undergone by the dislocation when subjected to a jump @xmath40 in the applied force.@xcite from ( [ eq : viscoinfty ] ) we deduce : @xmath41 the velocity jump ( [ eq : vjump ] ) increases with core width . it was first predicted by eshelby from his equation,@xcite and can be understood as follows for a screw dislocation along the @xmath42 axis : the force jump @xmath40 is due to a shear stress jump @xmath43 attaining simultaneously all the points of the whole glide plane ( e.g. , as the result of shear loading applied on faces of the system containing the plane , parallel to the latter ) . neglecting material inertia of the atoms on both sides of the dislocation plane , the medium undergoes an elastic strain jump @xmath44 , determined by a material velocity jump @xmath45 . the latter is equilibrated through outward emission of a shear wave with velocity @xmath8 . on the other hand , the slope of the displacement function near the core is @xmath46 , so that @xmath45 is related to the dislocation velocity jump @xmath39 by @xmath47 . combining these relationships yields ( [ eq : vjump ] ) , up to a numerical constant factor . the same argument applies to other types of dislocations . in case of several relaxation waves ( e.g. , longitudinal and shear waves for an edge dislocation ) , that of lowest celerity controls the amplitude of the velocity jump . it should be borne in mind , however , that accounting for material inertia from the atoms on both sides of the glide plane results in an instantaneous inertial force of order @xmath48 to be added to ( [ eq : eshforce]).@xcite by balancing the forces , it is seen that this force should spread the velocity jump over a short rise time @xmath49 no expression analogous to ( [ eq : eshforce ] ) is available for edge dislocations . however , clifton and markenscoff computed the force acting on a _ point _ screw or edge dislocation moving with any subsonic velocity in an isotropic medium , that jumps instantaneously at instant @xmath50 from rest to a constant velocity @xmath51.@xcite a generalization to anisotropic media is available.@xcite to maintain its velocity constant , this dislocation must be subjected , at time @xmath52 , to the time - decaying force @xmath53 where the function @xmath54 depends on its character and on anisotropy . @xcite we now construct heuristically a force for accelerated motion by interpreting such a motion as a succession of infinitesimal velocity jumps . assuming that , for instationary motion , @xmath51 in ( [ eq : fcm ] ) can be interpreted as @xmath15 , the elementary force that would arise from the elementary jump @xmath55 at @xmath50 is : @xmath56\delta v(\tau)$ ] @xmath57 @xmath58 . then , the total force experienced by the dislocation results from integrating such elementary forces over past history : @xmath59 comparing ( [ eq : newforce0 ] ) to ( [ eq : flin2 ] ) shows , firstly , that the relevant `` accelerations '' at linear order are different . however , we remark that @xmath60/(t-\tau)\}\to \dot{v}(\tau)$ ] as @xmath61 , and moreover that for a screw dislocation , @xmath62 . @xcite hence , since we interpret @xmath51 in ( [ eq : fcm ] ) as @xmath15 , the numerator of the integrand in ( [ eq : newforce0 ] ) is correct at least for small velocities and for small times @xmath61 . its relevance for large velocities is demonstrated below through comparisons to full - field calculations . next , integral ( [ eq : newforce0 ] ) is singular at @xmath63 , due the point - dislocation hypothesis at the root of ( [ eq : fcm ] ) . however , using ( [ eq : flin2 ] ) as a physical motivation , we propose a regularization consisting in replacing the kernel @xmath64 in ( [ eq : newforce0 ] ) by @xmath65^{1/2}$ ] where @xmath66 , the counterpart of @xmath67 in ( [ eq : eshforce ] ) , is some time characteristic of sound propagation over a core diameter . in sec . [ sec : appli ] , @xmath66 is chosen alternatively proportional to @xmath11 and to @xmath68 in the case of edge dislocations for illustrative purposes , whereas @xmath66 is proportional to @xmath67 for screws . the proportionality factor , 1/2 in all cases , is justified below . from a physical point of view , inertia is controlled by the slowest wave so that better results are expected using @xmath8 for all types of dislocations . given eshelby s rigid - core hypothesis in ( [ eq : flin2 ] ) , and the approximations made , it would be pointless to refine this treatment . another kind of regularization is used in ref . ( p. 195 ) , which consists in replacing the upper bound @xmath6 of integral ( [ eq : newforce0 ] ) by @xmath69 ( in ref . , the integrand assumes that @xmath70 ) . with the above regularization the force eventually reads : @xmath71^{1/2}}\dot{v}(\tau).\ ] ] its fourier transform for small velocities where @xmath72 yields , in terms of modified bessel and struve functions of order 0 , [ eq : masskercfapprox ] @xmath73\nonumber\\ & = & \frac{\pi}{2}\|\omega\|+o(t_{\rm s}\|\omega\|^2)\\ \label{veshcfapproxinf}\eta(\|\omega\|\to\infty)/g'(0)&=&1/t_0.\end{aligned}\ ] ] the approximation therefore preserves the logarithmic character of the mass , and the viscosity , to leading order . the mass is slightly decreased , the constant @xmath74 in ( [ mesh ] ) being absent . this difference is insignificant given the approximations made . in the limit of small velocity for a screw dislocation , our approximation amounts to retaining in ( [ eq : eshforce ] ) the first integral only . in order to recover a correct velocity jump for screws , we must take @xmath75 since the instantaneous viscosity ( [ veshcfapproxinf ] ) is different from ( [ eq : viscoinfty ] ) . this `` calibration '' is used in the next section for screws and ( somewhat arbitrarily ) for edges as well . in the stationary limit , the visco - inertial force ( [ eq : newforce1 ] ) vanishes . for @xmath76 , the asymptotic velocity should be determined by a viscous drag force , mainly of phonon origin,@xcite @xmath77 , where @xmath78 is the viscosity . this force is modified ( in the context of the peierls - nabarro model ) by the relativistic contraction of the core , into @xmath79 . for subsonic velocities , @xmath80 , where:@xcite @xmath81^{1/2},\ ] ] with @xmath82 , is an effective viscosity - dependent core contraction factor , such that the core length in the laboratory frame reads : @xmath83 . the purely relativistic contraction factor @xmath84 is , with @xmath85:@xcite @xmath86 with this drag , and introducing the applied stress @xmath87 , the eom finally reads : @xmath88^{1/2}}+f_{\rm drag}(v(t))=b\sigma_a,\ ] ] where @xmath89 , and where:@xcite @xmath90/v,\quad\text{for edge dislocations}.\end{aligned}\ ] ] this is our main result . by construction , it reproduces the asymptotic velocities of ref . . we checked numerically that the replacement of @xmath10 by @xmath91 in @xmath66 does not change by more than a few percent the overall results described in the following section . since this change in @xmath66 would bring in nothing useful , we choose to use @xmath10 in @xmath66 in the following section . upper : relationship between dimensionless applied stress @xmath92 and asymptotic velocity @xmath93 provided by the png code , @xcite for an accelerated edge dislocation in the stationary regime of an accelerated dislocation ( dots ) , compared to that predicted by rosakis s model i@xcite ( lines ) for different viscosity parameters @xmath94 , for a screw dislocation in the subsonic regime . lower : normalized velocity - dependent core width @xmath95 measured under same conditions . @xmath96 is the rayleigh velocity.,width=302 ] setting @xmath97 , eq . ( [ eq : pdpeq ] ) is solved numerically for edge and screw dislocations , in an implicit way with a time step @xmath98 small enough . results are compared with numerical points obtained with the _ peierls - nabarro - galerkin _ ( png ) approach @xcite used here as a benchmark . this method is less noisy than molecular dynamics , allows for full - field dynamic calculations of the displacement and stress fields in the whole system , accounting for wave propagation effects , and allows for better flexibility . we can thus , e.g. , control boundary conditions by applying analytically computed forces , so as to prevent image dislocations from perturbing the simulation window . firstly , to check the accuracy of the benchmark , asymptotic velocities of screw and edge dislocations were compared to the stationary predictions of rosakis model 1.@xcite in the png method , the permanent lattice displacement field ( which is part of the full atomic displacement , @xmath99 ) is relaxed by means of a landau - ginzburg equation , with viscosity parameter @xmath100 . an exact correspondence holds between this viscosity and rosakis s viscosity parameter @xmath94 , namely @xmath101 , as can be shown by specializing to one dimension the general field equations of ref .. a @xmath32-potential @xmath102 , with @xmath103 , is used . the material is an elastically cubic material , with elastic moduli taken such that @xmath104 to insure isotropy . due to the elastic correction made to the @xmath32-surface potential in order to remove its quadratic elastic part,@xcite the core at rest is a bit larger in the png results than in the peierls - nabarro solution . the time dependent core width @xmath105 is measured from the numerical simulations by using @xmath106 ^ 2 $ ] ( the value corresponding to a core of the arctan type ) . two - dimensional calculations are carried out using a simulation box of size @xmath107 @xmath108 , with a unique horizontal glide plane along @xmath109 . eight nodes per burgers vector are used in both directions . forces are applied on the top and bottom sides so as to induce shear on the unique glide plane . free boundary conditions are used on sides normal to the @xmath109 axis . measurements are done near the center of the box , where the mirror attracting forces these sides generate on the dislocation , are negligible . the box is wide enough so that the dislocation accelerates and reaches its terminal velocity . comparisons between png results and rosakis model are displayed in fig . [ fig : rosakis ] for different viscosities @xmath94 , in the case of an edge dislocation . the core scaling factor @xmath110 and the asymptotic velocity @xmath93 , are directly measured from simulations under different applied stresses @xmath111 , and compared to theory . @xcite the png asymptotic velocities were found to be @xmath112 systematically lower than the theoretical results . this correction is accounted for in the figure . the overall agreement is excellent . it is emphasized that core contraction effects in the viscous drag [ eq . ( [ eq : contract ] ) ] are required in order to obtain a good match . velocities vs. time for accelerated screw dislocations : white dots , png code ; solid , ref . ; dots ( in left curve only ) , eq . ( [ eq : eshforce ] ) ; dash - dots , linear approximation to ( [ eq : pdpeq ] ) ; dashes , fully relativistic equation ( [ eq : pdpeq]).,width=321 ] velocities vs. time for accelerated edge dislocations : white dots , png code ; solid , ref . ; dash - dots , linear approximation to ( [ eq : pdpeq ] ) ; dashes , fully relativistic equation ( [ eq : pdpeq ] ) . curves obtained from ( [ eq : pdpeq ] ) are duplicated , using either @xmath66 computed with @xmath8 ( upper ) , or with @xmath0 ( lower ) , see text.,width=321 ] next , comparisons in the accelerated regime are made with eq . ( [ eq : pdpeq ] ) and with other models . fig . [ fig : step_vis ] displays , as a function of time , the velocity of a screw dislocation accelerated from rest by a constant shear stress @xmath87 applied at @xmath113 . low and high shear stresses are examined . these stresses lead to terminal asymptotic velocities @xmath114 and @xmath115 , computed from ( [ eq : pdpeq ] ) . the results displayed are obtained : ( i ) with the png approach ( white dots ) ; ( ii ) with eq . ( [ eq : pdpeq ] ) using fully `` relativistic '' expressions of @xmath116 and @xmath110 ( dashes ) ; ( iii ) with linear small - velocity approximations of @xmath116 , but with the full expression of @xmath110 , in order to emphasize the importance of relativistic effects in the retarded force ( dash - dots , for the case @xmath117 ) ; ( iv ) with a previous eom,@xcite using a typical cut - off radius @xmath118 nm in the logarithmic core term ( solid ) corresponding to a typical dislocation density of @xmath119/m@xmath120 . the result arising from using ( [ eq : eshforce ] ) in the eom is also displayed for the lowest speed ( dots , left figure only ) . figure [ fig : step_coin ] presents similar curves for an edge dislocation . for the latter , @xmath66 is taken either as @xmath121 or as @xmath122 , @xmath68 , thus providing two limiting curves . the curves with @xmath67 provide the best matches , consistently with the above observation that the wave of lowest velocity @xmath123 should provide the main contribution to inertia . at low and high speeds , good agreement is obtained between png points and eq . ( [ eq : pdpeq ] ) , provided that fully `` relativistic '' expressions are used for @xmath124 ( especially for edge dislocations ) ; otherwise , inertia is strongly underestimated . in all the curves , the relativistic expression of the non - linear viscous terms was used . moreover , variations of the core width with velocity,@xcite implicitly present in png calculations , and ignored in the expression of @xmath66 used in the visco - inertial term of ( [ eq : pdpeq ] ) , are not crucial to accelerated or to decelerated motion ( see fig . [ fig : deceler ] ) ; still , the core width shrinks by 20% during the acceleration towards @xmath125 . on the other hand , retardation effects in the effective mass are crucial : curves with non - local inertial forces are markedly different from the solid ones using the masses of ref . , computed at constant velocity the version of the png code used here does not include the above - mentioned effects of material inertia in the glide plane , so that the full - field velocity curves indeed display what resembles a velocity jump , like the eom . owing to ( [ eq : risetime ] ) , this lack of accuracy solely concerns the time interval between the time origin and the first data point : hence we can consider that the velocity jump is a genuine effect , and not an artefact , at least from the point of view of full - field calculations in continuum mechanics . however , we should add that , to our knowledge , this effect has not been reported so far in molecular dynamics simulations . velocities vs. time for a decelerated screw dislocation . comparison between the png method ( white dots)@xcite and equation ( [ eq : pdpeq ] ) with ( dashes ) or without ( dot - dash ) fully relativistic expressions.,width=321 ] figure [ fig : deceler ] displays the velocity of screw and edge dislocations decelerated from the initial velocity @xmath126 . comparisons between eoms and png calculations are then harder to make than in the accelerated case . indeed , the non - relativistic ( resp . relativistic ) theoretical curves from eq.([eq : pdpeq ] ) ( dashed - dot ) [ resp . ( dashed ) ] are obtained by assuming that an applied stress abruptly vanishes at @xmath113 . this induces a negative velocity jump in the curves . this jump is larger if non - relativistic expressions are used , which demonstrates in passing the higher inertia ( i.e. `` mass '' ) provided by relativistic expressions . the same loading was tried in the png calculations as well , but led to non - exploitable results due to multiple wave - propagation and reflection phenomena . therefore , png curves for decelerated motion were obtained instead using a somewhat artificial loading : the medium was split in a zone of constant stress , separated from a zone of zero stress by an immobile and sharp boundary . the dislocation is then made to accelerate in the zone of constant stress . due to the finite core width , the boundary is crossed in a finite time @xmath127 , which explains the smoothed decay of the velocity in the png data points . this type of loading can not be realistically implemented within the framework of eq.([eq : pdpeq ] ) because the dislocation core is not spatially resolved . hence , though the curves strongly suggest that relativistic effects are as important in deceleration as in acceleration , and that ( [ eq : pdpeq ] ) reproduces well the png points , the comparison between the latter and theoretical curves should be taken here with a grain of salt . on the other hand , the eom of ref . ( solid ) is once again clearly imprecise . as a final remark , we expect our neglecting of retardation effects in the nonlinear viscous term of ( [ eq : pdpeq ] ) to induce an underestimation of damping effects . this may explain why the png curves decay faster than that from eq . ( [ eq : pdpeq ] ) . an empirical relativistic equation of motion for screw and edge dislocations , accounting for retardation effects in inertia , eq . ( [ eq : pdpeq ] ) , has been proposed . we compared it , together with another available approximate eom , to a quasi - exact numerical solution of a dynamical extension of the peierls - nabarro model , provided by the _ peierls - nabarro galerkin _ code.@xcite the latter was beforehand shown to reproduce very well the asymptotic velocities of rosakis s model 1@xcite in the subsonic regime . the best matches with full - field results were found with our eom , both for accelerated and for decelerated motion , thus illustrating quantitatively the importance of retardation and of relativistic effects in the dynamic motion of dislocations . to these effects , our eom provides for the first time a satisfactory approximation for high velocities in the subsonic range . our comparisons rule out the use of masses computed at constant velocity . one of the restrictions put forward by eshelby to his eom was its limitation to weakly accelerated motion , mainly due to the rigid core assumption.@xcite ours makes no attempt to explicitly overcome this simplification . however , comparisons with full - field calculations , where the core structure is not imposed from the start , but emerges as the result of solving the evolution equation for the displacement field , shows that this rigid - core assumption is acceptable on a quantitative basis as far as inertia is concerned , at least for velocities high , but not too close to @xmath128 . 99 e. hornbogen , acta metall . * 10 * , 978 ( 1962 ) clifton and x. markenscoff , j. mech . solids . * 29 * , 227 ( 1981 ) . d. tanguy , m. mareschal , p.s . lomdahl , t.c . germann , b.l . holian and r. ravelo , phys . b * 68 * , 144111 ( 2003 ) . hirth , h.m . zbib and j.p . lothe , modelling simul.mater . * 6 * , 165 ( 1998 ) . p. gumbsch and h. gao , science * 283 * , 965 ( 1999 ) . j. marian , w. cai and v.v . bulatov , nature materials * 3 * , 158 ( 2004 ) . c. denoual , phys . b * 70 * , 024 106 ( 2004 ) . l. pillon , c. denoual , r. madec and y .- p . pellegrini , j. phys . iv ( france ) * 134 * , 49 ( 2006 ) . c. denoual , comput . methods appl . engrg . * 196 * , 1915 ( 2007 ) . j. weertman in _ mathematical theory of dislocations _ , edited by t. mura ( asme , new york , 1969 ) , p. 178 . p. rosakis , phys . * 86 * , 95 ( 2001 ) . eshelby , phys . * 90 * , 248 ( 1953 ) . b. devincre and l.p . kubin , mater . a * 234 * -236 , 8 ( 1997 ) . v.v . bulatov , l.l . hsiung , m. tang , a. arsenlis , m.c . bartelt , w. cai , j.n . florando , m. hiratani , m. rhee , g. hommes , t.g . pierce and t. diaz de la rubia , nature * 440 * , 1174 ( 2006 ) . z.q . wang , i.j . beyerlein and r. lesar , los alamos preprint la - ur-06 - 3602 ( unpublished ) . e. bitzek and p. gumbsch mater . a * 400 * , 40 ( 2005 ) . eshelby , proc . r. soc . london , ser . a * 266 * , 222 ( 1962 ) . alshitz , v.l . indenbom and a.a . shtolberg , zh . eksp . fiz . * 60 * , 2308 ( 1971 ) [ sov . jetp * 33 * , 1240 ( 1971 ) ] . nabarro , proc . r. soc . london , ser . a * 209 * , 278 ( 1951 ) . alshits and v.l . indenbom , in _ dislocations in solids _ vol . 7 , edited by f.r.n . nabarro ( north - holland , amsterdam , 1986 ) . s. ishioka , j. phys . soc . japan * 34 * , 462 ( 1973 ) . r. peierls , proc . soc . * 52 * , 34 ( 1940 ) . nabarro , proc . soc . * 59 * , 256 ( 1947 ) . wu , acta mechanica * 158 * , 85 ( 2002 ) . hirth and j.p . lothe , _ theory of crystal dislocations _ ( krieger , new york , 1982 ) . j. weertman , in _ response of materials to high velocity deformations _ , edited by p.g . shewmon and v.f . zackay ( interscience , new york,1961 ) , p. 205 . eshelby , proc . a * 62 * , 307 ( 1949 ) .	an approximate equation of motion is proposed for screw and edge dislocations , which accounts for retardation and for relativistic effects in the subsonic range . good quantitative agreement is found , in accelerated or in decelerated regimes , with numerical results of a more fundamental nature .	9,109	66
in the field of disordered systems , the interest has been first on self - averaging quantities , like the free - energy per degree of freedom , or other thermodynamic observables that determine the phase diagram . however , it has become clear over the years that a true understanding of random systems has to include the sample - to - sample fluctuations of global observables , in particular in disorder - dominated phases where interesting universal critical exponents show up . besides these typical sample - to - sample fluctuations , it is natural to characterize also the large deviations properties , since rare anomalous regions are known to play a major role in various properties of random systems . among the various global observables that are interesting , the simplest one is probably the ground - state energy @xmath0 of a disordered sample . since it is the minimal value among the energies of all possible configurations , the study of its distribution belongs to the field of extreme value statistics . whereas the case of independent random variables is well classified in three universality classes @xcite , the problem for the correlated energies within a disordered sample remains open and has been the subject of many recent studies ( see for instance @xcite and references therein ) . for many - body models with @xmath1 degrees of freedom ( @xmath1 spins for disordered spin models , @xmath1 monomers for disordered polymers models ) , the interest lies \(i ) in the scaling behavior of the average @xmath2 and the standard deviation @xmath14 with @xmath1 . following the definitions of ref . @xcite , the ` shift exponent ' @xmath15 governs the correction to extensivity of the averaged value @xmath16 whereas the ` fluctuation exponent ' @xmath17 governs the growth of the standard deviation @xmath18 \(ii ) in the asymptotic distribution @xmath19 of the rescaled variable @xmath20 in the limit @xmath21 @xmath22 this scaling function @xmath5 describes the typical events where the variable @xmath23 is finite . \(iii ) in the large deviations properties . in the standard large deviation formalism ( see for instance the recent review @xcite and references therein ) , one is interested in the exponentially rare events giving rise to a _ finite difference _ @xmath24 between the intensive observable @xmath25 and its averaged value @xmath26 @xmath27 in disordered systems , the probability distributions of these rare events is not necessarily exponentially small in @xmath1 but can sometimes involve other exponents @xmath28 ( see examples below in the text ) @xmath29 in this paper , we discuss these properties for two types of disordered models : for the directed polymer of length @xmath1 in a two - dimensional medium , where many exact results exist , and for the sherrington - kirkpatrick ( sk ) spin - glass model of @xmath1 spins , where various possibilities have been proposed from numerical results or theoretical arguments . the main conclusions we draw from these two cases are the following : \(a ) it is very instructive to study _ the tails _ of the full probability distribution @xmath5 of eq . [ scalinge0 ] : these tails are usually described by the following form @xmath30 where the two tails exponents @xmath7 are usually different and in the range @xmath31 . in particular , the very common fits based on generalized gumbel distributions are very restrictive and very misleading since they correspond to the unique values @xmath32 and @xmath33 . we also discuss the consequences of eq . [ defetamu ] for the moments @xmath10 of order @xmath34 ( either positive or negative ) of the partition function @xmath35 at very low temperature . \(b ) simple rare events arguments can usually be found to obtain explicit relations between @xmath7 and @xmath4 . the probability distributions of these rare events usually correspond to anomalous large deviation properties of the generalized forms @xmath36 the paper is organized as follows . in section [ sec_dp ] , we recall the exact results concerning the directed polymer in a two - dimensional random medium , and discuss their meaning for the above points ( a ) and ( b ) . in section [ sec_sk ] , we discuss the case of the sherrington - kirkpatrick spin - glass model , and we present numerical results obtained for small sizes but with high statistics . our conclusions are summarized in section [ sec_conclusion ] . the directed polymer model in a two - dimensional random medium ( see the review @xcite ) is an exactly soluble model that has the following properties : \(i ) a single exponent @xcite @xmath37 governs both the correction to extensivity of the average @xmath2 ( eq . [ e0av ] ) and the width @xmath38 ( eq . [ deltae0 ] ) . \(ii ) the rescaled distribution @xmath5 of eq . [ scalinge0 ] is the tracy - widom distribution of the largest eigenvalue of random matrices ensembles @xcite . in particular , the two tails exponents of eq . [ defetamu ] read @xmath39 \(iii ) the exponents of the large deviations forms of eq . [ largedeve0 ] are respectively @xcite @xmath40 after this brief reminder of known results , we now turn to their physical interpretation . as explained in detail in @xcite , the large deviation exponents of eq . [ etapmdp ] can be understood as follows ( - ) to obtain a ground state energy which is extensively lower than the typical , it is sufficient to draw @xmath1 anomalously good on - site energies along the ground state path . this will happen with a probability @xmath41 corresponding to @xmath42 of eq . [ gammapmdp ] . ( + ) to obtain a ground state energy which is extensively higher than the typical , one needs to draw @xmath43 bad on - site energies ( i.e. in the whole sample ) . this will happen with a probability @xmath44 corresponding to @xmath45 of eq . [ gammapmdp ] . note that in the asymmetric exclusion process language , the interpretation is that to slow down the traffic , it is sufficient to slow down a single particle , whereas to speed up the traffic , one needs to speed up all particles @xcite . in the random matrix language , the interpretation is that to push the maximal eigenvalue inside the wigner sea , one needs to reorganize everything , whereas to pull the maximal eigenvalue outside the wigner sea , one may leave the wigner sea unchanged for the other eigenvalues @xcite . the fact that these large deviation exponents @xmath28 can be guessed via simple physical arguments is an important lesson that is very useful in other disordered models which are not exactly solvable : in particular , these arguments can be easily extended to the directed polymer in a random medium of higher dimensionality @xcite , or to other observables in various models @xcite . for an arbitrary probability distribution , the typical fluctuations in the bulk and the rare fluctuations in the far tails are in general different questions . however , for the probability distribution of the ground state energy @xmath46 ( or more generally the probability distributions of other global observables ) in disordered statistical physics models , it is very natural , from a physical point of view , to expect some matching between the typical fluctuation scaling regime where @xmath47 and the large deviations scaling regime where @xmath48 . more precisely , the tails in the regions @xmath49 of the rescaled distribution @xmath5 of typical fluctuations should match smoothly the large deviation region regime where the variable @xmath24 of eq . [ defv ] is finite , which corresponds to the regime where the variable @xmath23 of eq . [ defu ] is of order @xmath50 . if one plugs this scaling into the asymptotic form of eq . [ defetamu ] , and if one insists that one should then recover the large deviations exponents of eq . [ largedeve0 ] , one obtains the very simple relations between exponents @xcite @xmath51 for the directed polymer in a two - dimensional random medium , these relations are satisfied by the values quoted in eqs [ omegadp],[etapmdp ] and [ gammapmdp ] . this smooth matching has also been discussed in the equivalent problems concerning the current in the asymmetric exclusion process @xcite and the largest eigenvalue of gaussian random matrices @xcite . this matching property between typical fluctuation and large deviations is again an important lesson that can be used in other disordered models which are not exactly solvable . these relations have been checked in detail for the directed polymer in dimension @xmath52 @xcite as well as on hierarchical lattices @xcite . this matching property has also been used recently for the distribution of the dynamical barriers @xcite . it is also interesting from a physical point of view , because the asymmetry @xmath53 seen in the distribution of typical events can be seen as a consequence of the asymmetry of rare events @xmath54 . since a direct calculation of the probability distribution of the ground state of a disordered model is usually very difficult , analytical calculations usually focus on the moments @xmath55 of the partition function @xmath35 . then one can use two types of arguments to relate the distribution of @xmath56 to the distribution @xmath57 of the ground state energy @xmath46 : ( 1 ) at very low temperature @xmath58 , the partition function will be dominated by the ground state @xmath59 ( 2 ) moreover in some models , where the disorder - dominated phase @xmath60 is governed by a zero - temperature fixed point , one expects that in the whole region of temperatures @xmath60 , the probability distribution of the free - energy @xmath61 will actually have the same properties as the distribution of @xmath0 . since ( 2 ) is valid for the directed polymer model , but can not be taken for granted in all disordered models , we will restrict here to the point of view ( 1 ) of very low temperature @xmath58 . there exists a simple argument that has been proposed by zhang @xcite on the specific case on the directed polymer , that relates the scaling behaviors of the moments @xmath55 with the size @xmath1 and with the replica index @xmath34 to the properties of @xmath57 . the idea is to evaluate the moments by using the rescaled distribution of eq . [ scalinge0 ] @xmath62 for the case @xmath63 considered by zhang @xcite , the integral can be then evaluated by a saddle point method in the region @xmath64 , where one may use the asymptotic behavior of eq . [ defetamu ] with the exponent @xmath65 : the saddle point is of order @xmath66 that should be large @xmath67 , and one obtains @xmath68 for the case @xmath69 , the equivalent calculation yields in term of the other tail exponent @xmath70 : @xmath71 for the directed polymer in a two - dimensional random medium , one obtains , using @xmath72 with the explicit values of eqs [ omegadp],[etapmdp ] @xmath73 where one recognizes the combination @xmath74 that appears in the bethe ansatz replica calculation of ref . moreover , in zhang s argument @xcite , one actually imposes that the non - trivial term of eq . [ zhang2 ] should be extensive in @xmath1 ( because for positive integer @xmath34 , the moments of the partition function can be formulated in terms of the iteration of some transfer matrix , and thus they have to diverge exponentially in @xmath1 with some lyapunov exponent ) to obtain the relation @xmath75 ( which is equivalent here to the relation of eq . [ matchingtyplarge ] obtained previously by the rare event interpretation ) . for @xmath69 , the obtained behavior @xmath76 is rather different : the only extensive contribution of order @xmath1 in the exponential comes from @xmath2 . the leading contribution due to fluctuations is only of subleading order @xmath77 , and it involves a non - integer power of the replica index @xmath78 . to the best of our knowledge , the behavior of these negative moments has not been much discussed in the literature , in contrast to the case @xmath63 . these saddle - point calculations based on the facts that the tails exponents satisfy @xmath79 can be very useful in other non - exactly soluble models , for instance in the sherrington - kirkpatrick spin - glass model that we now consider . for short - ranged spin - glasses in any finite dimension @xmath80 , it has been proven that the fluctuation exponent of eq . [ deltae0 ] is exactly @xmath81 @xcite . accordingly , the rescaled distribution @xmath82 of eq . ( [ scalinge0 ] ) was numerically found to be gaussian in @xmath83 and @xmath84 @xcite , suggesting some central limit theorem . on the contrary , in mean - field spin - glasses , the width does not grows as @xmath85 and the distribution is not gaussian , as will be discussed in more details in this section . studies on long - ranged one - dimensional spin - glasses @xcite have confirmed that non - mean - field models are characterized by gaussian distributions , whereas mean - field models are not . the statistics of the ground state energy of the sherrington - kirkpatrick spin - glass model @xcite @xmath86 where the couplings @xmath87 are random quenched variables of zero mean @xmath88 and of variance @xmath89 , has been much studied recently with the following outputs : \(i ) there seems to be a consensus ( see for instance @xcite and references therein ) on the shift exponent of eq . [ e0av ] @xmath90 whereas the ` fluctuation exponent ' @xmath17 is still under debate between the value ( see @xcite and references therein ) @xmath91 and the value ( see @xcite and references therein ) @xmath92 \(ii ) the asymptotic distribution @xmath5 of eq . [ scalinge0 ] has been measured numerically by various authors ( see @xcite and references therein ) , but unfortunately it has almost always been fitted by generalized gumbel distributions of the form @xmath93 containing a single free - parameter @xmath94 for the shape . however these fits are very restrictive and very misleading since the tails exponents are fixed to be @xmath95 for any value of the parameter @xmath94 . in this paper , we propose instead that these exponents are in the range @xmath96 . \(iii ) the large deviation properties have been also very controversial . in @xcite , numerical results have been interpreted with the following values for the exponents @xmath28 of eq . [ largedeve0 ] @xmath97 : } \ \ \gamma_- & & \simeq 1.2 \nonumber \\ \gamma_+ & & \simeq 1.5 \label{ldandeanov}\end{aligned}\ ] ] other proposals are ( see @xcite and references therein ) @xmath98 : } \ \ \gamma_- & & = 1 \nonumber \\ \gamma_+ & & = 2 \label{ldrizzo}\end{aligned}\ ] ] after this brief summary of conflicting proposals , we now turn to the analysis along the same line as in the previous section concerning the directed polymer model . the simplest rare events one may consider for the sk model are the following : ( - ) to obtain a ground state energy which is much lower than the typical , it is natural to consider the anomalous ferromagnetic samples @xcite that appear with a small probability of order @xmath44 ( one needs to draw @xmath43 positive couplings in eq . [ defsk ] ) , and that will corresponds to anomalously low energy of order @xmath99 . these events corresponds to the very large deviation of the generalized form of eq . [ genelargedeve0 ] with the values @xcite @xmath100 this form has been checked numerically in @xcite . ( + ) to obtain a ground state energy which is much higher than the typical , one could consider the anomalous antiferromagnetic samples that appear with a small probability of order @xmath44 ( one needs to draw @xmath43 negative couplings ) and that will give an energy extensively higher . in the large deviation form of eq . [ largedeve0 ] , this would corresponds to @xmath101 this value corresponds to the proposal of eq . [ ldrizzo ] from ref . @xcite , but disagrees with the numerical proposal of eq . [ ldandeanov ] from ref . the question is whether to obtain an extensively higher energy , it is sufficient to draw anomalously only a number of order @xmath102 random couplings instead of @xmath43 . we are presently not aware of any simple argument in favor of this smaller power @xmath102 . in the ( + ) region , the matching between typical fluctuation and rare events leads to the same relation as in eq . [ matchingtyplarge ] @xmath103 in particular , the possible values of @xmath104 and @xmath4 leads to the following values for the tail exponent @xmath105 : @xmath106 or @xmath107 in the ( - ) region , the matching between typical fluctuation and the very large deviations of eq . [ ldandeanov ] leads to the relation @xmath108 using the values of eq . [ ferro ] one obtains the two possible values for @xmath109 @xmath110 if this matching works , the region of large deviation of eq . [ largedeve0 ] which is between the typical region and the very large deviation region , is constrained by consistency to involve the exponent @xmath111 the two possible values read @xmath112 both are close to the numerical value of eq . [ ldandeanov ] proposed by ref . both disagree with the value @xmath113 of eq . [ ldrizzo ] used in replica calculations of @xcite . as explained in detail in section [ moments ] , the moments of the partition function @xmath35 are then expected to follow eqs [ zhang2 ] and [ zhang3 ] @xmath114 for positive @xmath63 : the two possible values of @xmath4 and of the associated tail exponent @xmath109 ( see eq . [ etamoins ] ) correspond to the behaviors @xmath115 we note that in both cases , the non - trivial part is sub - extensive in @xmath1 , in contrast to the replica calculations of @xcite , but in agreement with the replica calculations of @xcite . it is also clear that the non - trivial part @xmath116 for the case @xmath117 , is simpler than the term @xmath118 for the case @xmath119 . in both cases , the powers of @xmath34 that appear are different from the value @xmath120 of perturbative replica calculations @xcite . for negative @xmath69 : for the case @xmath121 of eq . [ etaplusantiferro ] , the possible behaviors are @xmath122 for the case @xmath123 of eq . [ ldandeanov ] proposed in ref . @xcite , the behavior of the moments can be similarly evaluated using eq . [ etaplus3/2 ] . again in all cases , the non - trivial part is sub - extensive in @xmath1 , as already proposed in @xcite . concerning the powers of @xmath78 , the exponent @xmath124 for the case @xmath125 is in agreement with the replica calculations of @xcite . in the sk model : ( a ) the histograms @xmath126 of the rescaled variable @xmath23 of eq . [ defu ] measured for even sizes in the range @xmath127 almost coincide : this shows that the convergence in @xmath1 towards the asymptotic form is very rapid . ( b ) same data in log - scale to see the tails : one sees that the left tail does not change , whereas finite - size effects are visible on the right tail . , title="fig:",height=226 ] in the sk model : ( a ) the histograms @xmath126 of the rescaled variable @xmath23 of eq . [ defu ] measured for even sizes in the range @xmath127 almost coincide : this shows that the convergence in @xmath1 towards the asymptotic form is very rapid . ( b ) same data in log - scale to see the tails : one sees that the left tail does not change , whereas finite - size effects are visible on the right tail . , title="fig:",height=226 ] : examples of fits of our numerical rescaled histogram ( step function ) corresponding to the size @xmath128 ( a ) the smooth curve corresponds to the best three - parameter fit in the range @xmath129 by the form @xmath130 : the left tail exponent is of order @xmath131 . ( b ) the smooth curve corresponds to the best three - parameter fit in the range @xmath132 by the form @xmath133 : the right tail exponent is of order @xmath134 . , title="fig:",height=226 ] : examples of fits of our numerical rescaled histogram ( step function ) corresponding to the size @xmath128 ( a ) the smooth curve corresponds to the best three - parameter fit in the range @xmath129 by the form @xmath130 : the left tail exponent is of order @xmath131 . ( b ) the smooth curve corresponds to the best three - parameter fit in the range @xmath132 by the form @xmath133 : the right tail exponent is of order @xmath134 . , title="fig:",height=226 ] most numerical works on the distribution of the ground state energy in the sk model have followed the strategy to study the biggest sizes @xmath1 as possible , to measure the averaged value and the variance ( see @xcite and references therein ) . an opposite strategy has been followed in ref . @xcite where an exact enumeration of the disordered samples with the binomial distribution @xmath135 was performed for small sizes . as mentioned in @xcite , the results for the rescaled distribution @xmath5 at @xmath136 are already very good when compared to the results for larger @xmath1 . in other cases , we have also found that the distribution of rescaled variables converge much more rapidly than other observables @xcite . in the following , we thus follow the same strategy : we study the distribution of @xmath46 for the small sizes with a high statistics of disordered samples . on fig . [ fighisto ] ( a ) , we show the measured histograms @xmath126 of the the rescaled variable @xmath23 of eq . [ defu ] for even sizes in the range @xmath127 with a statistics of @xmath137 disordered samples : one clearly sees that all these histograms almost coincide . our conclusion is thus that the convergence in @xmath1 towards the asymptotic form is very rapid , so that these small - size data should provide a reliable measure of the asymptotic @xmath5 . as explained before , we are mainly interested into the tails exponents @xmath7 of eq . [ defetamu ] : as shown on fig . [ fighisto ] ( b ) the convergence of the left tail is extremely good , whereas the convergence of the right tail presents much stronger finite - size effects . let us first consider the left tail . the three - parameter fit of @xmath5 in the range @xmath129 by the form @xmath130 yields the value ( see fig [ figfit ] ( a ) ) @xmath138 that corresponds exactly to the value associated to @xmath139 ( see eq . [ etamoins ] ) . of course , it is probably not far enough from the alternative value @xmath140 corresponding to @xmath141 ( see eq . [ etamoins ] ) to really rule out the value @xmath141 . let us now turn to the right tail . the three - parameter fit of @xmath5 in the range @xmath132 by the form @xmath133 yields values for @xmath142 that are less precise , as a consequence of the finite - size effects visible on fig . [ fighisto ] we have already found in other studies that the right tail is usually more difficult to measure than the left tail @xcite . nevertheless our non - precise values of @xmath105 in the range @xmath143 $ ] ( see fig [ figfit ] ( b ) ) seem more compatible with the value @xmath144 than with the value @xmath145 ( see eq . [ etaplusantiferro ] and [ etaplus3/2 ] ) . in summary , even if a definitive agreement on the precise value of the fluctuation exponent @xmath4 remains difficult to reach ( see @xcite and references therein ) , our conclusions concerning the sk model are the following : \(i ) the numerical measure of the left tail exponent @xmath109 is in agreement with the matching argument based on rare ferromagnetic samples described by the very - large deviation form of eq . [ genelargedeve0 ] with the values of eq . [ ferro ] from ref . @xcite . then the large deviation form of eq . [ largedeve0 ] is constrained to involve an exponent @xmath146 given by eq . [ matchingskmoins ] @xmath147 this explains the numerical value of eq . [ ldandeanov ] proposed in ref . @xcite , and excludes the value @xmath113 of eq . [ ldrizzo ] used in the replica calculations of @xcite . we note moreover that this usual large deviation value @xmath113 would be satisfied only for the value @xmath148 , i.e. only if the fluctuation exponent @xmath4 would coincide with the shift exponent @xmath149 of eq . [ shiftsk ] . \(ii ) although less precise , the numerical measure of the right tail exponent @xmath105 is more in favor of the large deviation exponent @xmath144 , that can be justified with a simple rare events argument ( see eq . [ antiferro ] ) . \(iii ) finally , the facts that the tails exponents satisfy @xmath150 induces non - trivial behavior for the moments of the partition function ( see eqs . [ zhangsk ] ) when @xmath151 becomes large . in particular , from eqs [ zhangskpos ] and [ zhangskneg ] , our conclusion is that the only extensive term in @xmath1 comes from the trivial term @xmath2 both for negative and positive @xmath34 . moreover , the non - trivial sub - extensive terms can a priori involve non - integer powers of the replica index @xmath34 . in this paper , we have discussed the statistics of the ground state energy @xmath46 in two types of disordered models : ( i ) for the directed polymer of length @xmath1 in a two - dimensional medium , where many exact results exist ( ii ) for the sherrington - kirkpatrick spin - glass model , where various possibilities are still under debate both numerically and theoretically . our main conclusions are the following . besides the behavior of the disorder - average @xmath2 and of the standard deviation @xmath3 , it is very instructive to study the full probability distribution @xmath5 of the rescaled variable @xmath6 : \(a ) numerically , the convergence towards @xmath5 is usually very rapid , so that data on rather small sizes but with high statistics allow to measure the tails exponents @xmath7 that satisfy generically @xmath9 ( whereas the very common fits based on generalized gumbel distributions correspond to the unique values @xmath32 and @xmath33 ) . moreover , if one wishes to measure tails beyond the region probed via simple sampling , one may uses a monte - carlo procedure in the disorder , as done in @xcite for the sk model , and in @xcite for the directed polymer model . \(b ) simple rare events arguments can usually be found to obtain explicit relations between @xmath7 and @xmath4 . these rare events usually correspond to anomalous large deviation properties of the generalized form @xmath152 ( the usual large deviations formalism corresponding to @xmath13 is too restrictive for disordered models , as shown on explicit examples in the text ) . \(c ) we have also discussed the consequences of @xmath9 for the moments @xmath10 of order @xmath34 ( either positive or negative ) of the partition function @xmath35 . in the regime where @xmath11 $ ] becomes large , a saddle - point calculation leads to explicit non - trivial terms in the asymptotic behaviors of the moments @xmath10 of the partition function . we have shown in detail how this analysis for the directed polymer is in agreement with all known exact results . for the sk model , we have explained how this analysis agrees or disagrees with various possibilities debated in the literature . in conclusion , we believe that this type of analysis based on the matching between typical fluctuations and rare events is very useful to study disordered systems . here we have focused on the statistics of the ground state energy , but it can also be used for other global observables such as the maximal dynamical barrier of a disordered sample @xcite , or for the statistics of large excitations in ferromagnets and spin - glasses @xcite .	for the statistics of global observables in disordered systems , we discuss the matching between typical fluctuations and large deviations . we focus on the statistics of the ground state energy @xmath0 in two types of disordered models : ( i ) for the directed polymer of length @xmath1 in a two - dimensional medium , where many exact results exist ( ii ) for the sherrington - kirkpatrick spin - glass model of @xmath1 spins , where various possibilities have been proposed . here we stress that , besides the behavior of the disorder - average @xmath2 and of the standard deviation @xmath3 that defines the fluctuation exponent @xmath4 , it is very instructive to study the full probability distribution @xmath5 of the rescaled variable @xmath6 : ( a ) numerically , the convergence towards @xmath5 is usually very rapid , so that data on rather small sizes but with high statistics allow to measure the two tails exponents @xmath7 defined as @xmath8 . in the generic case @xmath9 , this leads to explicit non - trivial terms in the asymptotic behaviors of the moments @xmath10 of the partition function when the combination @xmath11 $ ] becomes large ( b ) simple rare events arguments can usually be found to obtain explicit relations between @xmath7 and @xmath4 . these rare events usually correspond to anomalous large deviation properties of the generalized form @xmath12 ( the usual large deviations formalism corresponds to @xmath13 ) . # 1#2 # 1#2	7,916	395
supermassive central black holes ( bh ) have now been discovered in more than a dozen nearby galaxies ( e.g. , kormendy & richstone 1995 ; ford et al . 1998 ; ho 1998 ; richstone 1998 , and van der marel 1999a for recent reviews ) . bhs in quiescent galaxies were mainly found using stellar kinematics while the bhs in active galaxies were detected through the kinematics of central gas disks . other techniques deployed are vlbi observations of water masers ( e.g. , miyoshi et al . 1995 ) and the measurement of stellar proper motions in our own galaxy ( genzel et al . 1997 ; ghez et al . the bh masses measured in active galaxies are all larger than a few times @xmath8 , while the bh masses in quiescent galaxies are often smaller . the number of accurately measured bhs is expected to increase rapidly in the near future , especially through the use of stis on board hst . this will establish the bh ` demography ' in nearby galaxies , yielding bh masses as function of host galaxy properties . in this respect two correlations in particular have been suggested in recent years . first , a correlation between bh mass and host galaxy ( spheroid ) optical luminosity ( or mass ) was noted ( e.g. , kormendy & richstone 1995 ; magorrian et al . 1998 ; van der marel 1999b ) . however , this correlation shows considerable scatter ( a factor @xmath9 in bh mass at fixed luminosity ) . the scatter might be influenced by selection effects ( e.g. , it is difficult to detect a low mass bh in a luminous galaxy ) and differences in the dynamical modeling . second , a correlation between bh mass and either core or total radio power of the host galaxy was proposed ( franceschini , vercellone , & fabian 1998 ) . however , the available sample is still small and incomplete . establishing the bh masses for a large range of optical and radio luminosities is crucial to determine the nature of galactic nuclei . an accurate knowledge of bh demography will put constraints on the connection between bh and host galaxy formation and evolution and the frequency and duration of activity in galaxies harboring bhs . in this paper we measure the bh mass of ic 1459 . ic 1459 is an e3 giant elliptical galaxy and member of a loose group of otherwise spiral galaxies . it is at a distance of @xmath10 with @xmath11 ( faber et al . williams & schwarzschild ( 1979 ) noted twists in the outer optical stellar isophotes . stellar spiral ` arms ' outside the luminous stellar part of the galaxy were detected in deep photographs ( malin 1985 ) . several stellar shells at tens of kpc from the center were discovered by forbes & reitzel ( 1995 ) . a remarkable feature is the counter - rotating stellar core ( franx & illingworth 1988 ) with a maximum rotation of @xmath12 . ic 1459 also has an extended emission gas disk ( diameter @xmath13 ) with spiral arms ( forbes et al . 1990 , goudfrooij et al . 1990 ) aligned with the galaxy major axis . the disk rotates in the same direction as the outer part of the galaxy ( franx & illingworth 1988 ) . the nuclear region of ic 1459 has line ratios typical of the liner class ( see e.g. , heckman 1980 , osterbrock 1989 for the definition of liners ) . a warped dust lane is also present . it is misaligned by @xmath14 from the galaxy major axis and some dust patches are observed at a radius of @xmath15 ( carollo et al . ic 1459 has a blue nuclear optical source with @xmath16 ( carollo et al . 1997 ; forbes et al . 1995 ) which is unresolved by hst . it also has a variable compact radio core ( slee et al . there is no evidence for a radio - jet down to a scale of @xmath17 ( sadler et al . ic 1459 has a hard x - ray component , with properties typical of low - luminosity agns ( matsumoto et al . 1997 ) . given the abovementioned properties , ic 1459 might best be described as a galaxy in between the classes of active and quiescent galaxies . this makes it an interesting object for extending our knowledge of bh demography , in particular since there are only few other galaxies similar to ic 1459 for which an accurate bh mass determination is available . we therefore made ic 1459 , and in particular its central gas disk , the subject of a detailed study with the hubble space telescope ( hst ) . we observed the emission gas of ic 1459 with the second wide field and planetary camera ( wfpc2 ) through a narrow - band filter around h@xmath2+[nii ] and took spectra with the faint object spectrograph ( fos ) at six locations in the inner @xmath17 of the disk . in section [ s : wfpc2 ] we discuss the wfpc2 observations and data reduction . in section [ s : spec ] we describe the fos observations and data reduction , and we present the inferred gas kinematics . to interpret the data we construct detailed dynamical models for the kinematics of the h@xmath3 and h@xmath2+[nii ] emission lines in section [ s : modelh ] , which imply the presence of a central bh with mass in the range @xmath18@xmath5 . in section [ s : species ] we discuss how the kinematics of other emission line species differ from those for h@xmath3 and h@xmath2+[nii ] , and what this tells us about the central structure of ic 1459 . in section [ s : starkin ] we present dynamical models for ground - based stellar kinematical data of ic 1459 , for comparison to the results inferred from the hst data . we summarize and discuss our findings in section [ s : discon ] . we adopt @xmath19 throughout this paper . this does not directly influence the data - model comparison for any of our models , but does set the length , mass and luminosity scales of the models in physical units . specifically , distances , lengths and masses scale as @xmath20 , while mass - to - light ratios scale as @xmath21 . we observed ic 1459 in the context of hst program go-6537 . we used the wfpc2 instrument ( described in , e.g. , biretta et al . 1996 ) on september 20 , 1996 to obtain images in two narrow - band filters . the observing log is presented in table [ t : wfpc2 ] . the ` linear ramp filters ' ( lrfs ) of the wfpc2 are filters with a central wavelength that varies as a function of position on the detector . the lrf fr680p15 was used as ` on band ' filter , with the galaxy position chosen so as to center the filter transmission on the h@xmath2+[nii ] emission lines . the narrow - band filter f631n was chosen as ` off - band ' filter , and covers primarily stellar continuum6300 emission , but the equivalent width of this line is small enough to have negligible influence on the off - band subtraction . ] . the position of the galaxy on the chip in the off - band observations was chosen to be the same as in the on - band observations . in all images the galaxy center was positioned on the pc chip , yielding a scale of @xmath22/pixel . the images were calibrated with the standard wfpc2 ` pipeline ' , using the most up to date calibration files . this reduction includes bias subtraction , dark current subtraction and flat - fielding . a flatfield was not available for the lrf filter , so we used the flatfield of f658n , a narrow - band filter with similar central wavelength ( 6590 ) . three back to back exposures were taken through each filter . in each case , the third exposure was offset by ( 2,2 ) pc pixels to facilitate bad pixel removal . the alignment of the exposures ( after correction for intentional offsets ) was measured using both foreground stars and the galaxy itself , and was found to be adequate . for each filter we combined the three available images without additional shifts , but with removal of cosmic rays , bad pixels and hot pixels . construction of a h@xmath2+[nii ] emission image requires subtraction of the stellar continuum from the on - band image . to this end we first fitted isophotes to estimate the ratio of the stellar continuum flux in the on - band and off - band image . this ratio could be fitted as a slowly varying linear function of radius in regions with no emission flux . the off - band image was multiplied by this ratio and subtracted from the on - band image . the resulting h@xmath2+[nii ] emission image was calibrated to units of @xmath23 using calculations with the stsdas / synphot package in iraf . the resulting flux scale was found to be in agreement with that inferred from our fos spectra ( see section [ s : spec ] ) . figure [ f : images ] shows both the f631n stellar continuum image of the central region of ic1459 , as well as the h@xmath2+[nii ] image . the continuum image shows a weakly obscuring warped dust lane across the center which is barely visible in the emission image . this dust lane is evident even more clearly in a @xmath24 image of ic 1459 ( carollo et al . the lane makes an angle of @xmath25 with the stellar major axis . the h@xmath2+[nii ] emission image shows the presence of a gas disk . the existence of this disk was already known from ground - based imaging , which showed that it has a total linear extent of @xmath13 ( goudfrooij et al . 1990 ; forbes et al . the outer parts of the disk show weak spiral structure and dust patches . inside the central @xmath26 the disk has a somewhat irregular non - elliptical distribution , with filaments extending in various directions . our hst image shows that the distribution becomes more regular again in the central @xmath27 . throughout its radial extent , the position angle ( pa ) of the disk coincides with the pa of the stellar distribution . in the case of the central @xmath28 , we derive from isophotal fits @xmath29 for the gas disk . this agrees roughly with the pa of the major axis of the stellar continuum in the same region , for which the f631n image yields @xmath30 . assuming an intrinsically circular disk , forbes & reitzel ( 1995 ) infer from the ellipticity @xmath31 of the gas disk at several arcseconds an inclination of @xmath32 . we performed a fit to the contour levels of the extended gas emission published by goudfrooij et al . ( 1994 ) , which also yields an inclination of @xmath32 . by contrast , the gas distribution in the central @xmath28 of the hst image is rounder than that at large radii . the ellipticity increases from @xmath33 at @xmath34 to @xmath35 at @xmath36 ( approximately the smallest radius at which the ellipticity is not appreciably influenced by the hst point spread function ( psf ) ) . while this could possibly indicate a change in the inclination angle of the disk , it appears more likely that the gas disk becomes thicker towards the center . this latter interpretation receives support from an analysis of the gas kinematics , as we will discuss below ( see section [ ss : ctiocomp ] ) . in the following we therefore assume an inclination angle of @xmath32 for ic 1459 , as suggested by the ellipticity of the gas disk at large radii . for the purpose of dynamical modeling we need a model for the stellar mass density of ic 1459 . carollo et al . ( 1997 ) obtained a hst / wfpc2 f814w ( i.e. , @xmath37-band ) image of ic 1459 , and from isophotal fits they determined the surface brightness profile reproduced in figure [ f : sbprof ] . carollo et al . corrected their data approximately for the effects of dust obscuration through use of the observed @xmath24 color distribution , so dust is not an important factor in the following analysis . to fit the observed surface brightness profile we adopt a parameterization for the three - dimensional stellar luminosity density @xmath38 . we assume that @xmath38 is oblate axisymmetric , that the isoluminosity spheroids have constant flattening @xmath39 as a function of radius , and that @xmath38 can be parameterized as @xmath40^{\beta } , \quad m^2 \equiv r^2 + z^2 q^{-2 } . \label{e : lumdendef}\ ] ] here @xmath41 are the usual cylindrical coordinates , and @xmath2 , @xmath3 , @xmath42 and @xmath43 are free parameters . when viewed at inclination angle @xmath44 , the projected intensity contours are aligned concentric ellipses with axial ratio @xmath45 , with @xmath46 . the projected intensity for the luminosity density @xmath38 is evaluated numerically . in the following we adopt @xmath47 , based on the discussion in section [ ss : wfpc2_ana ] . we take @xmath48 based on the isophotal shape analysis of carollo et al . ( their figure 1o ) , which shows an ellipticity @xmath49 of @xmath50 in the inner @xmath26 with variations @xmath51 . the isophotal pa is almost constant , with a monotonic increase of @xmath52 between @xmath17 and @xmath26 . the carollo et al . results show larger variations in @xmath49 and pa in the inner @xmath17 , but these are probably due to the residual effects of dust obscuration . our model with constant @xmath49 and pa is therefore expected to be adequate in the present context . the projected intensity of the model was fit to the observed surface brightness profile between @xmath53 and @xmath54 . the best fit model has @xmath55 , @xmath56 , @xmath57 and @xmath58 . its predictions are shown by the solid curve in figure [ f : sbprof ] . the fit was restricted to the range @xmath59 . as a result , the fit is somewhat poor at larger radii . this can of course be improved by extending the fit range , but with the simple parametrization of equation ( [ e : lumdendef ] ) this would have led to a poorer fit in the region @xmath60 . since this is the region of primary interest in the context of our spectroscopic hst data ( described below ) , we chose to accept the fit shown in the figure . the central @xmath53 were excluded from the fit because ic 1459 has a nuclear point source . this point source has a blue @xmath24 color . it is most likely of non - thermal origin ( similar to the point source in m87 ; kormendy 1992 ; van der marel 1994 ) and associated with the core radio emission in ic 1459 . if so , the point source does not contribute to the mass density of the galaxy ( which is what we are interested in here ) , and it is therefore appropriate to exclude it from consideration . in section [ s : discon ] we briefly discuss the implications of the alternative possibility that the point source is a cluster of young stars . we used the red side detector of the fos ( described in , e.g. , keyes et al . 1995 ) on november 30 , 1996 to obtain spectra of ic 1459 . the costar optics corrected the spherical aberration of the hst primary mirror . the observations started with a ` peak - up ' target acquisition on the galaxy nucleus . the sequence of peak - up stages was similar to that described in van der marel , de zeeuw & rix ( 1997 ) and van der marel & van den bosch ( 1998 ; hereafter vdmb98 ) . we then obtained six spectra , three with the 0.1-pair square aperture ( nominal size , @xmath61 ) and three with the 0.25-pair square aperture ( nominal size , @xmath62 ) . the g570h grating was used in ` quarter - stepping ' mode , yielding spectra with 2064 pixels covering the wavelength range from 4569 to 6819 . periods of earth occultation were used to obtain wavelength calibration spectra of the internal arc lamp . at the end of the observations fos was used in a special mode to obtain an image of the central part of ic 1459 , to verify the telescope pointing . galaxy spectra were obtained on the nucleus and along the major axis . a log of the observations is provided in table [ t : fossetup ] . target acquisition uncertainties and other possible systematic effects caused the aperture positions on the galaxy to differ slightly from those commanded to the telescope . we determined the actual aperture positions from the data themselves , using the independent constraints provided by the target acquisition data , the fos image , and the ratios of the continuum and emission - line fluxes observed through different apertures . this analysis was similar to that described in appendix a of vdmb98 . the inferred aperture positions are listed in table [ t : fossetup ] , and are accurate to @xmath63 in each coordinate . the roll angle of the telescope during the observations was such that the sides of the apertures made angles of @xmath64 and @xmath65 with respect to the galaxy major axis . figure [ f : aperpos ] shows a schematic drawing of the aperture positions . henceforth we use the labels ` s1'`s3 ' for the small aperture observations , and ` l1'`l3 ' for the large aperture observations . most of the necessary data reduction steps were performed by the hst calibration pipeline , including flat - fielding and absolute sensitivity calibration . we did our own wavelength calibration using the arc lamp spectra obtained in each orbit , following the procedure described in van der marel ( 1997 ) . the relative accuracy ( between different observations ) of the resulting wavelength scale is @xmath66 ( @xmath67 ) . uncertainties in the absolute wavelength scale are larger , @xmath68 ( @xmath69 ) , but influence only the systemic velocity of ic 1459 , not the inferred bh mass . the spectra show several emission lines , of which the following have a sufficiently high signal - to - noise ratio ( @xmath70 ) for a kinematical analysis : h@xmath3 at 4861 ; the [ oiii ] doublet at 4959 , 5007 ; the [ oi ] doublet at 6300 , 6364 ; the h@xmath2+[nii ] complex at 6548 , 6563 , 6583 ; and the [ sii ] doublet at 6716 , 6731 . to quantify the gas kinematics we fitted the spectra under the assumption that each emission line is a gaussian . this yields for each line the total flux , the mean velocity @xmath71 and the velocity dispersion @xmath72 . for doublets we fitted both lines simultaneously under the assumption that the individual lines have the same @xmath71 and @xmath72 . the h@xmath2+[nii ] complex is influenced by blending of the lines , and for this complex we made the additional assumptions that h@xmath2 and the [ nii ] doublet have the same kinematics , and for the [ nii ] doublet that the ratio of the fluxes of the individual lines equals the ratio of their transition probabilities ( i.e. , 3 ) . figure [ f : emlines ] shows the observed spectra for each of the five line complexes listed above , with the gaussian fits overplotted . the figure shows that the observed emission lines are not generally perfectly fit by gaussians ; they often have a narrower core and broader wings . it was shown in vdmb98 that this arises naturally in dynamical models such as those constructed below . in the present paper we will not revisit the issue of line shapes , but restrict ourselves to gaussian fits ( both for the data and for our models ) . the mean and dispersion of the best - fitting gaussian are well - defined and meaningful kinematical quantities , even if the lines themselves are not gaussians . the gaussian fit parameters for each of the line complexes are listed in table [ t : fosgaskin ] . the listed velocities are measured with respect to the systemic velocity of ic 1459 . the systemic velocity was estimated from the hst data themselves , by including it as a free parameter in the dynamical models described below ( see section [ ss : dyn_mod ] ) . this yields @xmath73 ( but with the possibility of an additional systematic error due to uncertainties in the fos absolute wavelength calibration ) . this result is a bit higher than values previously reported in the literature ( e.g. , @xmath74 by sadler 1984 ; @xmath75 by franx & illingworth 1988 ; @xmath76 by da costa et al . in fact , systemic velocities that are up to @xmath77 smaller than our value have been reported as well ( e.g. , davies et al . 1987 ; drinkwater et al . 1997 ) . figure [ f : gaskin ] shows the inferred kinematical quantities for the five line complexes as function of major axis distance . the observational setup provides only sparse sampling along the major axis and with apertures of different sizes , but nonetheless , two items are clear . first , for all apertures and line species there is a steep positive mean velocity gradient across the nucleus ( i.e. , between observations s1 and s2 , or l1 and l2 ) . second , the velocity dispersion tends to be highest for the smallest aperture closest to the nucleus ( observation s1 ) ; this is true for all line species with the exception of [ sii ] , for which the dispersion peaks for observation s2 . the steep central velocity gradient and centrally peaked velocity dispersion profile are similar to what has been found for other galaxies with nuclear gas disks ( e.g. , ferrarese , ford & jaffe , 1996 ; macchetto et al . 1997 ; bower et al . 1998 ; vdmb98 ) . the kinematical properties of the different emission line species show both significant similarities and differences . for example , the kinematics of h@xmath3 and h@xmath2+[nii ] are in excellent quantitative agreement . by contrast , [ oiii ] shows a significantly steeper central mean velocity gradient , and both [ oiii ] and [ oi ] have a higher velocity dispersion for several apertures ; the central velocity dispersion for [ oiii ] exceeds that for h@xmath3 and h@xmath2+[nii ] by more than a factor two . the kinematics of the [ sii ] emission lines deviates somewhat from that for h@xmath3 and h@xmath2+[nii ] , but only for the small apertures . there is no a priori reason to expect identical flux distributions , and hence identical kinematics for the different species , because they differ in their atomic structure , ionization potential , critical density , etc . differences of similar magnitude have been detected in the kinematics of other gas disks as well ( e.g. , harms et al . 1994 ; ferrarese , ford & jaffe 1996 ) . the former authors studied the gas disk in m87 , and also found that the [ oiii ] line indicates a larger mean velocity gradient and higher dispersion than h@xmath3 and h@xmath2+[nii ] . we discuss the differences in the kinematics of the different line species in section [ s : species ] , after first having analyzed in detail the kinematics of h@xmath3 and h@xmath2+[nii ] in section [ s : modelh ] . in our modeling it proved useful to complement the fos spectroscopy with ground - based data that extends to larger radii . we therefore reanalyzed a major axis long - slit spectrum of ic 1459 obtained at the ctio 4 m telescope . the data were taken with a @xmath78-wide slit using a ccd with @xmath79 pixels , in seeing conditions with fwhm @xmath80 . the spectra have a smaller spectral range than the fos spectra , but do cover the emission lines of h@xmath3 and [ oiii ] . fluxes and kinematics for these lines were derived using single gaussian fits , as for the fos spectra . the inferred gas kinematics are listed in table [ t : ctio ] . the stellar kinematics implied by the _ absorption _ lines in the same spectrum ( presented previously by van der marel & franx 1993 ) are used in section [ s : starkin ] for the construction of stellar dynamical models . the fos spectra of the h@xmath2+[nii ] and h@xmath3 lines yield similar relative fluxes ( cf . figure [ f : fluxfit ] below ) and similar mean velocities and dispersions ( cf . figure [ f : gaskin ] ) . we therefore assume that these emission lines have the same intrinsic flux distributions and kinematics . we start in sections [ ss : flux_gas][ss : ctiocomp ] with the construction of models for the h@xmath3 and h@xmath2+[nii ] gas kinematics in which the gas disk is assumed to be an infinitesimally thin structure in the equatorial plane of the galaxy . however , as discussed in section [ ss : wfpc2_ana ] , this assumption may not be entirely appropriate at small radii , where the projected isophotes of the gas disk become rounder . in section [ ss : asymdrift ] we therefore discuss models in which the gas distribution is extended vertically . to model the h@xmath3 and h@xmath2+[nii ] gas kinematics we need a description of the intrinsic ( i.e. , the deconvolved and de - inclined ) flux profile for these emission lines . we model the ( face - on ) intrinsic flux distribution as a triple exponential , @xmath81 and assume that the disk is infinitesimally thin and viewed at an inclination @xmath82 , ( cf . section [ ss : wfpc2_ana ] ) . the total flux contributed by each of the three exponential components is @xmath83 ( @xmath84 ) , and the overall total flux is @xmath85 . the best - fitting parameters of the model flux distribution were determined by comparison to the available data . flux data are available for h@xmath2+[nii ] from both the wfpc2 imaging and fos spectra . for h@xmath3 they are available from the fos and the ctio spectra . for the spectra we determined the fluxes in the relevant lines ( and their formal errors ) using single gaussian fits . for the wfpc2 image data we included for simplicity not the full two - dimensional brightness distribution in the fit , but only image cuts along the major and minor axes . image fluxes outside @xmath86 are dominated by read - noise , and were excluded . the errors for each image data - point were computed taking into account the poisson - noise and the detector read - noise . the combined flux data from all sources are shown in figure [ f : fluxfit ] . we performed an iterative fit of the triple exponential to all the available flux data , taking into account the necessary convolutions with the appropriate psf , pixel size and aperture size for each setup . the hst and ctio fluxes constrain the flux distribution predominantly for @xmath87 and @xmath88 , respectively , due to their relatively narrow and broad psf . the wfpc2 data have a pixel area that is @xmath89 and @xmath90 times smaller than the respective fos apertures , and therefore provide the strongest constraints on the flux distribution close to the center . the solid line in figure [ f : fluxfit ] shows the predictions of the model that best fits all available data ( which we will refer to as ` the standard flux model ' ) . this model has parameters @xmath91 , @xmath92 , @xmath93 , @xmath94 , @xmath95 and @xmath96 . the absolute calibration gives @xmath97 for h@xmath2+[nii ] and @xmath98 for h@xmath3 . the total h@xmath2+[nii ] flux inferred from our model agrees to within 25% with that inferred from a previous ground - based observation of ic 1459 ( macchetto et al . figure [ f : fluxprof ] shows the intrinsic flux distribution as function of radius for the standard flux model . approximately one quarter of the total flux is contained in a component that is essentially unresolved at the spatial resolution of hst . the standard flux model provides an adequate fit to the observed fluxes , but the fit is not perfect . the model predicts too little flux in the central wfpc2 pixel , while at the same time predicting too much flux in the small fos aperture closest to the galaxy center . so the different data sets are not fully mutually consistent under the assumptions of our model . this is presumably a result of uncertainties in the psfs and aperture sizes for the different observations . to explore the influence of this on the inferred flux distribution we performed fits to two subsets of the flux data the first subset consists only of the fos and ctio data , while the second subset consists only of the wfpc2 and ctio data . the flux distribution models that best fit these subsets of the data are also shown in figure [ f : fluxprof ] . the results show that the standard flux model represents a compromise between the fos and the wfpc2 data . at small radii the fos data by themselves would imply a broader profile , while the wfpc2 data by themselves would imply a narrower profile . at larger radii the situation reverses . as will be discussed in section [ ss : bestfit ] , the uncertainties in the intrinsic flux distribution have only a very small effect on the inferred bh mass . our thin - disk models for the gas kinematics are similar to those employed in vdmb98 . the galaxy model is axisymmetric , with the stellar luminosity density @xmath99 chosen as in section [ ss : mass_stars ] to fit the available surface photometry . the stellar mass density @xmath100 follows from the luminosity density upon the assumption of a constant mass - to - light ratio @xmath101 . the mass - to - light ratio can be reasonably accurately determined from the ground - based stellar kinematics for ic 1459 . this yields @xmath102 in solar @xmath37-band units , cf . section [ s : starkin ] below . we keep the mass - to - light ratio fixed to this value in our modeling of the gas kinematics . we assume that the gas is in circular motion in an infinitesimally thin disk in the equatorial plane of the galaxy , and has the circularly symmetric flux distribution @xmath103 given in section [ ss : flux_gas ] . we take the inclination of the galaxy and the gas disk to be @xmath82 , as discussed in section [ ss : wfpc2_ana ] . the circular velocity @xmath104 is calculated from the combined gravitational potential of the stars and a central bh of mass @xmath105 . the line - of - sight velocity profile ( vp ) of the gas at position @xmath106 on the sky is a gaussian with mean @xmath107 and dispersion @xmath108 , where @xmath109 is the radius in the disk . the velocity dispersion of the gas is assumed to be isotropic , with contributions from thermal and non - thermal motions : @xmath110 . we refer to the non - thermal contribution as ` turbulent ' , although we make no attempt to describe the underlying physical processes that cause this dispersion . it suffices here to parameterize @xmath111 through : @xmath112 . \label{eq : turbdef}\ ] ] the parameter @xmath113 was kept fixed to @xmath114 , as suggested by the ctio data for h@xmath3 with @xmath115 ( see figure [ f : modelctio ] below ) . the predicted vp for any given observation is obtained through flux weighted convolution of the intrinsic vps with the psf of the observation and the size of the aperture . the convolutions are described by the semi - analytical kernels given in appendix a of van der marel et al . ( 1997 ) , and were performed numerically using gauss - legendre integration . a gaussian is fit to each predicted vp for comparison to the observed @xmath71 and @xmath72 . the model was fit to the fos gas kinematics for h@xmath2+[nii ] and h@xmath3 . rotation velocity and velocity dispersion measurements were both included , yielding a total of 24 data points . three free parameters are available to optimize the fit : @xmath105 , and the parameters @xmath116 and @xmath117 that describe the radial dependence of the turbulent dispersion . the temperature of the gas is not an important parameter : the thermal dispersion for @xmath118 is @xmath119 , and is negligible with respect to @xmath111 for all plausible models . we define a @xmath120 quantity that measures the quality of the fit to the kinematical data , and the best - fitting model was found by minimizing @xmath120 using a ` downhill simplex ' minimization routine ( press et al . 1992 ) . the curves in figure [ f : modelfos ] show the predictions of the model that provides the overall best fit to the h@xmath2+[nii ] and h@xmath3 kinematics , using the standard flux model of section [ ss : flux_gas ] . its parameters are : @xmath121 , @xmath122 and @xmath123 . this model ( which we will refer to as ` the standard kinematical model ' ) adequately reproduces the important features of the hst kinematics , including the central rotation gradient and the nuclear velocity dispersion . to determine the range of bh masses that provides an acceptable fit to the data we compared the predictions of models with different fixed values of @xmath105 , while at each @xmath105 varying the remaining parameters to optimize the fit . the radial dependence of the intrinsic velocity dispersion of the gas is essentially a free function in our models , so the observed velocity dispersion measurements can be fit equally well for all plausible values of @xmath105 . thus only the predictions for the hst rotation velocity measurements depend substantially on the adopted @xmath105 . to illustrate the dependence on @xmath105 , figure [ f : rotfits ] compares the predictions for the hst rotation measurements for three different models . the solid curves show the predictions of the standard kinematical model defined above . the dotted and dashed curves are the predictions of models in which @xmath105 was fixed a priori to @xmath124 and @xmath125 , respectively . the model without a bh predicts a rotation curve slope which is too shallow and the model with @xmath126 predicts a rotation curve slope which is too steep . both these bh masses are ruled out by the data at more than the @xmath127 confidence level ( see discussion below ) . to assess the quality of the fit to the hst rotation velocity measurements we define a new @xmath120 quantity , @xmath128 , that measures the fit to these data only . at each @xmath105 , the parameters @xmath116 and @xmath117 are fixed almost entirely by the velocity dispersion measurements . these parameters can therefore not be varied independently to improve the fit to the hst rotation velocity measurements . as a result , @xmath128 is expected to follow approximately a @xmath120 probability distribution with @xmath129 degrees of freedom ( there are 12 hst measurements , and there is one free parameter , @xmath105 ) . the expectation value for this distribution is @xmath130 . however , for the standard kinematical model we find @xmath131 . to determine the cause of this statistically poor fit we inspected the goodness of fit as function of bh mass for each line species separately . figure [ f : mbhrange ] shows @xmath128 as function of @xmath105 for both h@xmath2+[nii ] and h@xmath3 . the kinematics of h@xmath2+[nii ] are formally poorly fitted , despite the apparently good qualitative agreement in figure [ f : modelfos ] . in particular , the observed h@xmath2+[nii ] velocity gradient between the fos-0.1 apertures s1 and s2 is steeper than predicted by the best - fit model with @xmath121 , which suggests that the bh mass may actually be twice as high ( since @xmath132 in our models ) . the poor formal fit may not be too surprising , given that our modeling of the gas as a flat circular disk in bulk circular rotation with an additional turbulent component is almost certainly an oversimplification of what in reality must be a complicated hydrodynamical system . the fits to the kinematics of h@xmath3 are statistically acceptable , but this may be in part because the formal errors on the h@xmath3 kinematics are twice as large as for h@xmath2+[nii ] . this would cause any shortcomings in the models to be less apparent for this emission line . nonetheless , an important result in figure [ f : mbhrange ] is that the bh masses implied by the h@xmath2+[nii ] and h@xmath3 kinematics are virtually identical . formal errors on the bh mass can be inferred using the @xmath133 statistic , as illustrated in figure [ f : mbhrange ] . for h@xmath3 this yields @xmath134 at @xmath135% confidence ( i.e. , 1-@xmath72 ) , and @xmath136 at 99% confidence . the formal @xmath133 confidence intervals inferred from the h@xmath2+[nii ] lines are smaller , but this is not necessarily meaningful since the @xmath120 itself is not acceptable for these lines . in section [ ss : flux_gas ] we showed that there is some uncertainty in the flux distributions of h@xmath2+[nii ] and h@xmath3 . the mean velocities and velocity dispersions predicted by the dynamical model are flux - weighted quantities , and therefore depend on the adopted flux distribution . to assess the influence on the inferred bh mass we repeated the analysis using the two non - standard flux distributions shown in figure [ f : fluxprof ] . with these distributions we found fits to the kinematical data of similar quality as for the standard kinematical model . the inferred values of @xmath105 agree with those for the standard kinematical model to within 10% . this shows that the uncertainties in the flux distribution have negligible impact on the inferred bh mass . the model parameters in section [ ss : bestfit ] were chosen to best fit the fos data . here we investigate what this model predicts for the setup of the ctio data . figure [ f : modelctio ] shows the resulting data - model comparison ( without any further changes to the model parameters ) . at radii @xmath137 the standard kinematical model fits the data acceptably well . the agreement in the velocity dispersion is trivial since it is the direct result of our choice of the model parameter @xmath113 . however , the agreement for the rotation velocities is quite important . it shows that outside the very center of the galaxy , the observations are consistent with the assumed scenario of gas rotating at the circular velocity in an infinitesimally thin disk . moreover , it suggests that the value of mass - to - light ratio @xmath101 used in the models ( derived from ground - based stellar kinematics ) is accurate . by contrast , the fit to the ctio data is less good at radii @xmath138 . in particular , the predicted rotation curve is too steep , and the central peak in the predicted velocity dispersion is too high . these discrepancies can not be attributed to possible errors in the assumed value for the seeing fwhm for the ctio observations . the latter was calibrated from the spectra themselves . due to their superior spatial resolution , the wfpc2 data set the inner intrinsic flux profile . we could therefore determine the ctio psf by optimizing the agreement between the predicted and observed central three ctio fluxes using this intrinsic inner flux profile . the resulting fwhm determination ( @xmath139 ) is quite accurate , and is inconsistent with the large fwhm values needed to make the standard kinematical model fit the ctio data . the discrepancies must therefore be due to an inaccuracy or oversimplification in the modeling . the models discussed so far assume that the observed velocity dispersion is due to local turbulence in gas that has bulk motion along circular orbits . however , an alternative interpretation could be that the gas resides in individual clouds , and that the observed dispersion of the gas is due to a spread in the velocities of individual clouds . in this case the gas would behave as a collisionless fluid obeying the boltzmann equation . an important consequence would be that the velocity dispersion @xmath72 of the gas clouds would provide some of the pressure responsible for hydrostatic support , so that the mean rotation velocity @xmath140 would be less than the circular velocity @xmath141 by a certain amount @xmath142 . this effect is know as asymmetric drift ( for historical reasons having to do with the stellar dynamics of the milky way ; see e.g. , binney & merrifield 1998 ) . the presence of a certain amount of asymmetric drift would simultaneously explain several observations . first , if close to the center the gas receives a certain amount of pressure support from bulk velocity dispersion , and if this dispersion would be near - isotropic , it would induce a thickening of the disk . this would cause the isophotes of the gas close to the center to be rounder than those at larger radii , which is exactly what is observed ( cf . sections [ ss : wfpc2_ana ] and [ s : discon ] ) . second , asymmetric drift would cause the gas to have a mean velocity less than the circular velocity , which would explain why the models of section [ ss : bestfit ] overpredict the observed rotation velocities ( cf . figure [ f : modelctio ] ) . third , the central peak in the velocity dispersion seen in the ground - based data is due in part to rotational broadening ( spatial convolution of the steep rotation gradient near the center ) . so if the mean velocity of the gas in the models were lowered due to asymmetric drift , then the predicted central velocity dispersion would go down as well . this would tend to improve the agreement between the predicted and the observed velocity dispersions in figure [ f : modelctio ] . if the gas in ic 1459 indeed has a non - zero asymmetric drift at small radii , then the models of section [ ss : dyn_mod ] would have _ under_-estimated the enclosed mass within any given radius ( and models without a bh would be ruled out at even higher confidence than already indicated by figure [ f : mbhrange ] ) . in the following we address how much of an effect this would have on the inferred bh mass . in the limit @xmath143 one has that @xmath144 ^ 2)$ ] ( e.g. , binney & tremaine 1987 ) , and any asymmetric drift correction to @xmath105 would be fairly small . however , at the resolution of hst we find that @xmath145 , so the approximate formulae that exist for the limit @xmath143 can not be used . for a proper analysis the gas kinematics would have to be modeled as a ` hot ' system of point masses , using any of the techniques that have been developed in the context of stellar dynamical modeling of elliptical galaxies ( e.g. , merritt 1999 ) . while fully general collisionless modeling of the gas in ic 1459 is beyond the scope of the present paper , it is important to establish whether such an analysis would yield a very different bh mass . to address this issue we constructed spherical isotropic models for the gas kinematics using the jeans equations , as in van der marel ( 1994 ) . the three - dimensional density of the gas was chosen so as to reproduce the major axis profile given by equation ( [ e : fluxparam ] ) after projection . as before , the gravitational potential of the system was characterized by a variable @xmath105 and a fixed @xmath102 . any turbulent velocity dispersion component of the gas was assumed to be zero . we then calculated the rms projected line - of - sight velocity @xmath146 predicted for the smallest fos aperture positioned on the galaxy center . the h@xmath2+[nii ] and h@xmath3 observations yield @xmath147^{1/2 } \approx 600 { \>{\rm km}\,{\rm s}^{-1}}$ ] ( cf . table [ t : fosgaskin ] ) . we found that the spherical isotropic jeans models require @xmath148 to reproduce this value . larger bh masses would be ruled out because they predict more rms motion than observed . in more general collisionless models the required bh mass will depend on the details of the model , but not very strongly . velocity anisotropy and axisymmetry influence the projected dispersion of a population of test particles in a kepler potential only at the level of factors of order unity ( de bruijne , van der marel & de zeeuw 1996 ) . so as expected , if the velocity dispersion of the gas is interpreted as gravitational motion of individual clouds , then the bh mass must be larger than inferred in section [ ss : bestfit ] . however , the increase would only be a factor of @xmath894 . models without a bh would remain firmly ruled out . we emphasize that both types of model that we have studied are fairly extreme . in one case we assume that the gas resides in an infinitesimally thin disk , and has a large turbulent ( or otherwise non - thermal ) velocity dispersion and no bulk velocity dispersion or asymmetric drift . in the other case we assume the opposite , that the gas is in a spherical distribution , and has no turbulent velocity dispersion but instead a large bulk velocity dispersion . the truth is likely to be found somewhere between these extremes , and we therefore conclude that ic 1459 has a bh with mass in the range @xmath149@xmath5 . spectroscopic information is not only available for h@xmath2+[nii ] and h@xmath3 , but also for three other line species : [ oi ] , [ oiii ] and [ sii ] . interestingly , the flux distributions and kinematics of these lines show some notable differences from those of h@xmath3 and h@xmath2+[nii ] . we have no narrow - band imaging for [ oi ] , [ oiii ] and [ sii ] , so information on the flux distributions of these lines is available only from the six apertures for which we obtained fos spectra . figure [ f : speciesflux ] shows the relative surface brightness @xmath150 for the different apertures and line species , using the definition @xmath151 \ > \big / \ > [ f { \rm ( l2,species ) } / a { \rm ( l2 ) } ] , \label{eq : relfluxdef}\ ] ] where @xmath152 is the flux observed through an aperture for a given species , and @xmath153 is the area of the aperture ; the aperture l2 is the observation with the large 0.25-pair aperture on the nucleus ( cf . figure [ f : aperpos ] ) . the relative surface brightnesses for the different species as seen through the large aperture are all quite similar . by contrast , those for the small 0.1-pair aperture differ considerably . the profiles for [ oi ] and especially [ oiii ] are more centrally peaked than that for h@xmath2+[nii ] , while the profile for [ sii ] is less centrally peaked than that for h@xmath2+[nii ] . to quantify this , we have defined a measure of the peakedness of the surface brightness profile as @xmath154 . \label{eq : peakedness}\ ] ] the values of this quantity for the different species are @xmath155 , @xmath156 , @xmath157 , @xmath158 and @xmath159 for h@xmath3 , [ oiii ] , [ oi ] , h@xmath2+[nii ] and [ sii ] , respectively . the kinematics for the different species also show differences , as already pointed out in section [ ss : spec_ana ] . figure [ f : gaskin ] shows that the most pronounced differences are seen in the value of the velocity dispersion as observed through the smallest aperture on the nucleus ( observation s1 ) . this quantity varies from @xmath160 for [ sii ] to @xmath161 for [ oiii ] , cf . table [ t : fosgaskin ] . such differences in the line width for different species have previously been found in the central regions of other liner galaxies and seyferts , and have been extensively modeled ( e.g. , whittle 1985 ; simpson & ward 1996 ; simpson et al . 1996 ; and references therein ) . one result from these studies has been that there is generally a correlation between velocity dispersion and critical density of the lines . we find a similar result for the nucleus of ic 1459 . figure [ f : critical]a shows the velocity dispersion for the different lines observed through aperture s1 , versus the critical density ( the balmer lines are not plotted because for them interpretation of the critical density is complicated by the effects of radiative transfer for permitted lines ; see e.g. filippenko & halpern 1984 ) . there is indeed a rough correlation . the fact that [ oiii ] has a larger dispersion than [ oi ] is somewhat surprising in view of this , but this has also been found for other galaxies ( whittle 1985 ) . it has been hypothesized that at a basic level the approximate correlation between velocity dispersion and critical density can be understood as the result of differences in the spatial distribution of the line flux for different species ( osterbrock 1989 , p. 366 ) . lines with a high critical density tend to be more strongly concentrated towards the ionizing source in the galaxy nucleus than species with a low critical density . so for a line with a high critical density , the observed flux will on average come from smaller radii . either the presence of a central bh or increased turbulence ( cf.equation [ eq : turbdef ] ) would naturally cause the gas at smaller radii to move faster , which qualitatively explains the correlation . to make detailed quantitative predictions one would need to model the complete ionization structure ( ionizing flux , electron density , temperature , etc . as function of radius ) and kinematics of the gas , which can generally be done only with simplifying assumptions . however , even without a detailed model we can test the basic interpretation in an observational sense . figure [ f : speciesflux ] shows that we are observing different flux distributions for the different lines , which implies that we are actually resolving the region from which the emission arises . thus we can test directly whether the lines for which the velocity dispersion is high have a flux that is strongly concentrated towards the nucleus . figure [ f : critical]b confirms this . it shows the velocity dispersion for the different lines observed through aperture s1 versus the flux - peakedness parameter @xmath162 . the strong correlation provides direct support for the proposed interpretation . in our dynamical models , the gas motions are a function of position in the disk only ; they do not depend on the physical properties of the line species . hence , differences in the observed kinematics of the lines must be due entirely to differences in their flux distributions . for accurate modeling it is therefore essential that the intrinsic flux distributions are well - constrained by the observations . this is true for h@xmath2+[nii ] , for which a narrow - band image is available at the wfpc / pc resolution of @xmath22/pixel . however , this is not true for [ oi ] , [ oiii ] and [ sii ] , for which only the six fos spectral flux measurements are available , with resolutions no better than @xmath61/aperture . this is insufficient for accurate modeling . hence , we can not test in detail whether the different line species all independently indicate the same bh mass . however , a very simple argument can be used to set an upper limit on how much the bh masses implied by the [ oi ] , [ oiii ] and [ sii ] data could differ from that inferred in section [ ss : bestfit ] from the h@xmath2+[nii ] and h@xmath3 data . none of the line species have either a central rotation velocity gradient @xmath163 or a central velocity dispersion @xmath164 that exceeds the value for h@xmath2+[nii ] by more than a factor of @xmath165 ( cf . table [ t : fosgaskin ] ) . in the models of section [ ss : dyn_mod ] one has approximately @xmath166 , while in the models of section [ ss : asymdrift ] one has approximately @xmath167 . so if we would assume ( incorrectly ) that all species have the same flux distribution , then we would infer bh masses that exceed that inferred from the h@xmath2+[nii ] and h@xmath3 data by at most a factor @xmath168 . however , this is a very conservative upper limit since the flux distributions for e.g. [ oi ] and [ oiii ] _ are _ actually more peaked than for h@xmath2+[nii ] and h@xmath3 , which would tend to reduce the inferred bh mass . so the data provide no compelling reason to believe that the [ oi ] , [ oiii ] and [ sii ] observations imply a very different bh mass than inferred from h@xmath2+[nii ] and h@xmath3 , but we can not test this in detail . for [ sii ] , both the mean velocity and the velocity dispersion profiles are quite irregular . the [ sii ] lines have a relatively low equivalent width and form a blended doublet , so this could be due to systematic problems in the extraction of the kinematics from the data . on the other hand , irregularities in the dispersion profiles are also seen for [ oiii ] and [ oi ] as observed through the small apertures . although such irregularities are not seen for h@xmath3 and h@xmath2+[nii ] , this may indicate that our modeling of the gas flux distribution ( eq . [ [ e : fluxparam ] ] ) and the turbulent velocity dispersion ( eq . [ [ eq : turbdef ] ] ) as smooth functions is somewhat oversimplified . we compared the observed line ratios for ic 1459 with modeled values for shocks ( allen et al . 1998 ; dopita et al . 1997 ) and found them to be consistent . shocks could naturally explain the turbulence that we invoke in our models of ic 1459 . at the same time , it would suggest that the gas properties could easily possess more small - scale structure than the smooth functions that we have adopted . ground - based stellar kinematical data are available from the same major axis ctio spectrum for which the emission lines were discussed in section [ ss : spec_ground ] . the stellar rotation velocities @xmath71 and velocity dispersions @xmath72 inferred from this spectrum were presented previously in van der marel & franx ( 1993 ) . to interpret these data we constructed a set of axisymmetric stellar - dynamical two - integral models for ic 1459 in which the phase - space distribution function @xmath169 depends only on the two classical integrals of motion . as before , the mass density was taken to be @xmath170 , with the luminosity density @xmath99 given by equation ( [ e : lumdendef ] ) . the gravitational potential is the sum of the stellar potential and the kepler potential of a possible central bh . predictions for the root - mean - square ( rms ) stellar line - of - sight velocities @xmath147^{1/2}$ ] were calculated by solving the jeans equations for hydrostatic equilibrium , projecting the results onto the sky , and convolving them with the observational setup . this modeling procedure is equivalent to that applied to large samples by , e.g. , van der marel ( 1991 ) and magorrian et al . the calculations presented here were done with the software developed by van der marel et al . ( 1994 ) . figure [ f : starkin ] shows the data for @xmath146 as function of major axis distance . the value of @xmath101 in the models was chosen so as to best fit the data outside the central region , yielding @xmath171 . this leaves @xmath105 as the only remaining free parameter . the curves in figure [ f : starkin ] show the model predictions for various values of @xmath105 . a model without a bh predicts a central dip in @xmath146 , which is not observed . the models therefore clearly require a bh . none of the models fits particularly well , but models with @xmath172@xmath173 provide the best fit . this exceeds the bh mass inferred from the hst gas kinematics by a factor of 10 or more . this suggests that the assumptions underlying the stellar kinematical analysis may not be correct . the velocity dispersion anisotropy of a stellar system can have any arbitrary value , and the sense of the anisotropy can have a large effect on inferences about the nuclear mass distribution . two - integral @xmath169 models can be viewed as axisymmetric generalizations of spherical isotropic models . such models will overestimate the bh mass if galaxies are actually radially anisotropic ( binney & mamon 1982 ) . van der marel ( 1999a ) showed that @xmath169 models can easily overestimate the bh mass by a factor of 10 when applied to ground - based data of similar quality as that available here , even for galaxies that are only mildly radially anisotropic . support that this may be happening comes from various directions . first , several detailed studies of bright galaxies similar to ic 1459 have concluded that these galaxies are radially anisotropic ( e.g. , rix et al . 1997 ; gerhard et al . 1998 ; matthias & gerhard 1999 ; saglia et al . 1999 ; cretton , rix & de zeeuw 2000 ) . second , detailed three - integral distribution function modeling of stellar kinematical hst data for several galaxies ( gebhardt et al . 2000 ) does indeed yield bh masses that are many times smaller than inferred by magorrian et al . ( 1998 ) from @xmath169 models for ground - based data for the same galaxies . and third , models of adiabatic bh growth for hst photometry also suggest that @xmath169 models for ground - based data yield bh masses that are too large ( van der marel 1999b ) . so in summary , the fact that the bh mass inferred from @xmath169 models for ground - based ic 1459 data does not agree with that inferred from the hst gas kinematics provides little reason to be worried . it simply shows that one can not generally place very meaningful constraints on bh masses from ground - based stellar kinematics of @xmath174 spatial resolution . while the bh mass inferred from ground - based stellar kinematical data is generally very sensitive to assumptions about the structure and dynamics of the galaxy , this is not true for the inferred mass - to - light ratio . models with different inclination ( van der marel 1991 ) or anisotropy ( van der marel 1994 ; 1999a ) yield the same value of @xmath101 to within @xmath90% . additional support for the accuracy of the inferred @xmath102 comes from the fact that this value yields a good fit to the observed rotation velocities of the gas outside the central few arcsec ( cf . figure [ f : modelctio ] ) . we have presented the results from a detailed hst study of the central structure of ic 1459 . the kinematics of the gas disk in ic 1459 was probed with fos observations through six apertures along the major axis . in our modeling of the observed kinematics we took into account the stellar mass density in the central region by fitting wfpc2 broad - band imaging , and we determined the flux distribution of the emission - gas from wfpc2 narrow - band imaging . from the models we have determined that ic1459 harbors a black hole with a mass in the range @xmath175 @xmath176 , with the exact value depending somewhat on whether we model the gas as rotating on circular orbits , or as an ensemble of collisionless cloudlets . while the dynamical models that we have constructed provide good fits to the observations , the true structure of ic 1459 could of course be more complex than our models . below we discuss several aspects of this . ground - based observations ( goudfrooij et al . 1994 ; forbes & reitzel 1995 ) indicate an ellipticity @xmath31 for the gas disk at radii larger than a few arcseconds , implying an inclination angle @xmath82 . by contrast , from our hst emission - line image we found a monotonic increase in ellipticity from @xmath35 to @xmath177 between @xmath36 to @xmath178 . in our modeling we have assumed that these rounder inner isophotes are due to a thickening of the gas disk caused by asymmetric drift , and we estimated the effect of this on the inferred @xmath105 ( see sections [ ss : ctiocomp ] and [ ss : asymdrift ] ) . however , an alternative interpretation would be to assume that the disk is warped . the presence of a dust lane slightly misaligned with the gas disk , a counter - rotating stellar core , and stellar shells and ripples in the outer galaxy make it plausible that the central gas and dust were accreted from outside , displaying warps as it settles down . in this interpretation we infer an increase in inclination angle from @xmath179 to @xmath180 between @xmath36 to @xmath178 . the bh mass in the models of section [ ss : dyn_mod ] scales as @xmath181 due to the projection of the rotational velocities . hence the inferred differences in inclination angle would amount to an increase in @xmath105 by only a factor @xmath182 , which would not significantly change our results . ic 1459 hosts an unresolved blue nuclear point source . carollo et al . ( 1997 ) estimated a luminosity of @xmath183 . so far , we have assumed that this light is non - stellar radiation from the active nucleus . however , one could assume alternatively that the blue light is emitted by a cluster of young stars . if the cluster were to have a mass equal to the mass @xmath105 that we have inferred from our models , this would require @xmath184 . depending on age and metallicity , stellar evolution models typically predict @xmath185 for a young cluster ( e.g. , worthey 1994 ) . thus the assumption that the point source is stellar in origin can not lift the need for a central concentration of non - luminous matter , most likely a bh . the data show differences between the fluxes and kinematics for the various line species . the main characteristics of the observed kinematics are similar for all species , but we see that for the species with higher critical densities the flux distribution is generally more concentrated towards the nucleus and the observed velocities are higher . this can be well understood qualitatively in the framework of a single @xmath105 . however , to actually verify quantitatively whether each species implies the same value for @xmath105 would require information on the flux distributions for each species at high spatial resolution as well as detailed knowledge on the ionization structure of the gas , neither of which is available . an extremely conservative upper limit on the differences in the @xmath105 implied by the different line species is obtained by assuming that all species have the same flux distribution ( not actually correct ) , in which case @xmath105 values are obtained that are up to @xmath168 times larger than inferred from h@xmath3 and h@xmath2+[nii ] . irregularities in the velocity dispersion profiles of [ oiii ] , [ oi ] and [ sii ] suggest localized turbulent motions . we incorporated turbulent motion in our models in a very simple manner ( cf . equation [ eq : turbdef ] ) , using a parametrization that fits the main trend of an increase of the velocity dispersion toward the nucleus . nevertheless , going to the extreme , one could assume that all observed motions have a non - gravitational origin . the overall kinematics would then be due to in- or outflows . there are several objections to this interpretation . a spherical in- or outflow could not produce any net mean velocity . a bi - directional flow is unlikely since no hint of this is seen in the hst h@xmath2+[nii ] emission image . next we consider the location of ic 1459 with respect to the correlations between @xmath105 and host total optical and radio luminosity , mentioned in the introduction . the ratio of @xmath105 and galaxy mass is in the range @xmath6@xmath7 . this is somewhat lower than the average value of @xmath186 seen for other galaxies ( kormendy & richstone 1995 ) , but still comfortably within the observed scatter . as discussed in the introduction , ic 1459 is probably the end - product of a merger between two galaxies . apparently , this merger history has not moved ic 1459 to an atypical spot in the @xmath105 vs. galaxy mass scatter diagram . the fact that ic 1459 has a liner type spectrum and core radio emission , but no jets , makes it interesting to see where it is located on a @xmath187 plot . ic 1459 has a radio luminosity of @xmath188 at 5 ghz ( wright et al . interestingly , this puts ic 1459 within the scatter observed around the correlation between @xmath105 and total radio luminosity inferred for a small sample of galaxies with available bh mass determinations , but quite off the correlation with core radio luminosity ( see figs . 3 and 4 of franceschini , vercellone , & fabian 1998 ) . however , one should be weary of beaming and variability of core radio sources , and possible resolution differences among the observations . our dynamical modeling has used the gas disk in ic 1459 as a diagnostic tool to constrain @xmath105 . a logical next step to improve on our work would be to obtain better two - dimensional coverage of the gaseous and stellar kinematics . a definite advance in our understanding of nuclear gas disks would also be obtained if we had a better knowledge of properties of the gas disk such as the electron density , metallicity , and the ionization structure and ionization mechanism . one could then try to simultaneously understand these properties and the gas dynamics . we now know that quite likely most , or even all , bright ellipticals host a massive central bh . more detailed knowledge of the chemical properties and kinematics of the dust and gas surrounding the bh could tell us about the origin of this material : accretion of small satellites , stripping of companions , or internal mass loss from stars . this would immediately constrain the frequency and probability with which any bright elliptical hosts this material . a better understanding of the kinematics , such as the importance of dissipative shocks generated by turbulence , could then help to determine the accretion rate of the black hole . these pieces of information together could tell us what fraction of the observed @xmath105 could have come from this process over the life time of a galaxy . knowledge on the ratio of black hole and stellar mass as function of time would be valuable for understanding the formation and evolution of early - type galaxies in general . support for this work was provided by nasa through grant number # go-06537.01 - 95a , and through c.m.c.s hubble fellowship # hf-01079.01 - 96a , awarded by the space telescope science institute which is operated by the association of universities for research in astronomy , incorporated , under nasa contract nas5 - 26555 . the authors would like to thank marijn franx for helpful comments on an earlier version of the manuscript . s1 & h@xmath3 & -22 & 60 & 599 & 63 + & [ oiii ] & -146 & 57 & 1014 & 47 + & [ oi ] & -55 & 45 & 748 & 46 + & h@xmath2+[nii ] & -108 & 26 & 545 & 23 + & [ sii ] & 34 & 32 & 211 & 25 + + s2 & h@xmath3 & 185 & 73 & 473 & 76 + & [ oiii ] & 328 & 40 & 491 & 40 + & [ oi ] & 241 & 50 & 414 & 50 + & h@xmath2+[nii ] & 247 & 21 & 422 & 19 + & [ sii ] & 531 & 226 & 409 & 133 + + s3 & h@xmath3 & 162 & 56 & 363 & 56 + & [ oiii ] & 277 & 68 & 562 & 68 + & [ oi ] & 243 & 91 & 630 & 98 + & h@xmath2+[nii ] & 116 & 17 & 280 & 13 + & [ sii ] & -35 & 42 & 241 & 35 + + l1 & h@xmath3 & -158 & 39 & 381 & 40 + & [ oiii ] & -395 & 44 & 848 & 36 + & [ oi ] & -99 & 43 & 526 & 48 + & h@xmath2+[nii ] & -121 & 10 & 321 & 8 + & [ sii ] & -113 & 22 & 216 & 15 + + l2 & h@xmath3 & 92 & 31 & 408 & 30 + & [ oiii ] & 39 & 45 & 920 & 38 + & [ oi ] & 62 & 38 & 564 & 41 + & h@xmath2+[nii ] & 72 & 15 & 473 & 13 + & [ sii ] & 75 & 43 & 311 & 32 + + l3 & h@xmath3 & 55 & 54 & 202 & 53 + & [ oiii ] & 252 & 35 & 243 & 36 + & [ oi ] & 43 & 35 & 158 & 34 + & h@xmath2+[nii ] & 106 & 17 & 280 & 13 + & [ sii ] & 91 & 26 & 188 & 19 + 4.65 & 2 & 335 & 21 & 122 & 22 & 343 & 10 & 125 & 10 + 3.55 & 1 & 284 & 26 & 125 & 27 & 313 & 15 & 154 & 15 + 2.82 & 1 & 249 & 30 & 110 & 31 & 301 & 12 & 132 & 12 + 2.09 & 1 & 245 & 25 & 144 & 26 & 265 & 17 & 185 & 17 + 1.36 & 1 & 104 & 20 & 195 & 21 & 104 & 15 & 261 & 16 + 0.63 & 1 & 31 & 11 & 212 & 12 & 47 & 9 & 296 & 9 + -0.10 & 1 & 0 & 10 & 218 & 10 & 0 & 7 & 320 & 8 + -0.83 & 1 & -50 & 10 & 172 & 10 & -63 & 9 & 285 & 9 + -1.56 & 1 & -81 & 13 & 157 & 14 & -130 & 10 & 183 & 10 + -2.29 & 1 & 164 & 23 & 138 & 24 & -159 & 13 & 170 & 13 + -3.02 & 1 & 191 & 21 & 107 & 22 & -223 & 15 & 155 & 15 + -3.75 & 1 & 212 & 18 & 66 & 17 & -267 & 19 & 133 & 19 + -4.85 & 2 & 346 & 53 & 138 & 55 & -315 & 21 & 151 & 22 +	the peculiar elliptical galaxy ic 1459 ( @xmath0 , @xmath1 ) has a fast counterrotating stellar core , stellar shells and ripples , a blue nuclear point source and strong radio core emission . we present results of a detailed hst study of ic 1459 , and in particular its central gas disk , aimed a constraining the central mass distribution . we obtained wfpc2 narrow - band imaging centered on the h@xmath2+[nii ] emission lines to determine the flux distribution of the gas emission at small radii , and we obtained fos spectra at six aperture positions along the major axis to sample the gas kinematics . we construct dynamical models for the h@xmath2+[nii ] and h@xmath3 kinematics that include a supermassive black hole , and in which the stellar mass distribution is constrained by the observed surface brightness distribution and ground - based stellar kinematics . in one set of models we assume that the gas rotates on circular orbits in an infinitesimally thin disk . such models adequately reproduce the observed gas fluxes and kinematics . the steepness of the observed rotation velocity gradient implies that a black hole must be present . there are some differences between the fluxes and kinematics for the various line species that we observe in the wavelength range 4569 to 6819 . species with higher critical densities generally have a flux distribution that is more concentrated towards the nucleus , and have observed velocities that are higher . this can be attributed qualitatively to the presence of the black hole . there is some evidence that the gas in the central few arcsec has a certain amount of asymmetric drift , and we therefore construct alternative models in which the gas resides in collisionless cloudlets that move isotropically . all models are consistent with a black hole mass in the range @xmath4@xmath5 , and models without a black hole are always ruled out at high confidence . the implied ratio of black holes mass to galaxy mass is in the range @xmath6@xmath7 , which is not inconsistent with results obtained for other galaxies . these results for the peculiar galaxy ic 1459 and its black hole add an interesting data point for studies on the nature of galactic nuclei .	20,165	571
the black - hole binary grs 1915 + 105 is highly variable in x - rays ( belloni et al . 2000 , and references therein ) . still , even its hardest spectra are relatively soft , consisting of a blackbody - like component and a high - energy tail ( vilhu et al . they are softer than those of other black - hole binaries in the hard state , which @xmath1 spectra peak at @xmath2 kev ( e.g. , cyg x-1 , gierliski et al . 1997 ) , and are similar to their soft state ( e.g. , cyg x-1 , gierliski et al . 1999 , hereafter g99 ; lmc x-1 , lmc x-3 , wilms et al . 2001 ) . the blackbody component arises , most likely , in an optically - thick accretion disk . on the other hand , there is no consensus at present regarding the origin of the tail . all three main models proposed so far involve comptonization of the blackbody photons by high - energy electrons . they differ , however , in the distribution ( and location ) of the electrons , which are assumed to be either thermal ( maxwellian ) , non - thermal ( close to a power law ) , or in a free fall onto the black hole . a discussion of these models is given in zdziarski ( 2000 ) , who shows that the thermal and free - fall models of the soft state of black hole binaries can be ruled out , mostly by the marked absence of a high - energy cutoff around 100 kev in the _ cgro _ data ( grove et al . 1998 ; g99 ; tomsick et al . 1999 ; mcconnell et al . the present best soft - state model appears to involve electron acceleration out of a maxwellian distribution ( i.e. , a non - thermal process ) , which leads to a hybrid electron distribution consisting of both thermal and non - thermal parts ( zdziarski , lightman & macioek - niedwiecki 1993 ; poutanen & coppi 1998 ; g99 ; coppi 1999 ) . in this _ letter _ , we present all osse observations of grs 1915 + 105 . we then choose two osse spectra corresponding to the lowest and highest x - ray flux and fit them together with spectra from simultaneous _ rxte _ pointed observations . the spectra , showing extended power laws without any cutoff up to at least 600 kev , provide strong evidence for the presence of non - thermal comptonization . more extensive presentation of the combined x - ray / osse data will be given elsewhere . table 1 gives the log of the 9 osse observations , together with results of power - law fits and basic data about the corresponding x - ray and radio states . the osse instrument accumulated spectra in a sequence of 2-min . measurements of the source field alternated with 2-min . , offset - pointed measurements of background . the background spectrum for each source field was derived bin - by - bin with a quadratic interpolation in time of the nearest background fields ( see johnson et al . figure [ fig : osse ] shows the osse spectra ( including standard energy - dependent systematic errors ) , which were fitted up to energies at which the source signal was still detected . the uncertainty for a fitted parameter corresponds hereafter to 90% confidence ( @xmath3 ) . we see that the source went through wide ranges of radio and x - ray fluxes and types of x - ray variability during those observations . in spite of that variety , 8 out of 9 osse spectra are best - fitted by a power law with a photon index of @xmath4 and the flux varying within a factor of 2 . the only exception is the osse spectrum corresponding to the highest x - ray flux measured by the asm ( 1999 april 2127 ) , which is much harder , @xmath5 , and has a much lower flux . we then consider the osse spectra corresponding to the extreme x - ray fluxes measured by the _ rxte_/asm , i.e. , from 1997 may 1420 ( vp 619 ) and 1999 april2127 ( vp 813 ) . we fit them together with spectra from the pointed _ rxte _ observations of 1997 may 15 and 1999 april 23 ( the observation ids are 20187 - 02 - 02 - 00 , 40403 - 01 - 07 - 00 ; 1% systematic error is added to the pca data with the responses of 2001 february ) . these pca data correspond to the variability classes ( belloni et al . 2000 ) of @xmath6 and @xmath7 , in which the variability is moderate and the source spends most of the time in two basic low ( @xmath8 ) and high ( @xmath9 ) x - ray flux state , respectively . we fit the data with the xspec ( arnaud 1996 ) model eqpair ( coppi 1999 ; g99 ) , which calculates self - consistently microscopic processes in a hot plasma with electron acceleration at a power law rate with an index , @xmath10 , in a background thermal plasma with a thomson optical depth of ionization electrons , @xmath11 . the electron temperature , @xmath12 , is calculated from the balance of compton and coulomb energy exchange , as well as @xmath13 pair production ( yielding the total optical depth of @xmath14 ) is taken into account . the last two processes depend on the plasma compactness , @xmath15 , where @xmath16 is a power supplied to the hot plasma , @xmath17 is its characteristic size , and @xmath18 is the thomson cross section . we then define a hard compactness , @xmath19 , corresponding to the power supplied to the electrons , and a soft compactness , @xmath20 , corresponding to the power in soft seed photons irradiating the plasma ( which are assumed to be emitted by a blackbody disk with the maximum temperature , @xmath21 ) . the compactnesses corresponding to the electron acceleration and to a direct heating ( i.e. , in addition to coulomb energy exchange with non - thermal @xmath13 and compton heating ) of the thermal @xmath13 are denoted as @xmath22 and @xmath23 , respectively , and @xmath24 . details of the model are given in g99 . = 8.7 cm we also take into account compton reflection with a solid angle of @xmath25 ( magdziarz & zdziarski 1995 ) and an fe k@xmath26 emission from an accretion disk assumed to extend down to @xmath27 ( which results in a relativistic smearing ) . the equivalent width , @xmath28 , with respect to the _ scattered _ spectrum only is tied to @xmath25 via @xmath29 ev ( george & fabian 1991 ) . the elemental abundances of anders & ebihara ( 1982 ) , an absorbing column of @xmath30 @xmath31 ( dickey & lockman 1990 ; vilhu et al . 2001 ) , and an inclination of @xmath32 are assumed . as discussed in g99 , @xmath33 depends weakly on @xmath20 in a wide range of this parameter . an increase of @xmath20 leads to increasing @xmath13 pair production , which then leads to an annihilation feature around 511 kev . the presence of such a feature is compatible with the osse data ( fig . [ fig : osse ] ) , but only very weakly constrained . g99 found that @xmath34 provides a good fit to cyg x-1 data . here , we find that a good fit is provided with @xmath35 , compatible with the high luminosity of grs 1915 + 105 . for example , for 1/2 of the eddington luminosity , @xmath36 , and spherical geometry , the size of the plasma corresponds then to @xmath37 . this model provides very good description of our two broad - band spectra ( as well as of other _ rxte_-osse spectra , s. v. vadawale et al . , in preparation ) . for vp 619 , we assume a free relative normalization of the hexte and osse spectra with respect to those from the pca . on the other hand , the hexte spectrum for vp 813 has relatively few counts at its highest energies , and thus we use the actual osse normalization in that fit . table 2 gives the fit results , and figure [ f : data ] shows the spectra . = 8.2 cm our model predicts the power law emission extending with no cutoff well above 1 mev and a weak annihilation feature ( with the plasma allowed to be pair - dominated , i.e. , with @xmath38 , for vp 619 ) . those predictions can be tested by future soft @xmath7-ray detectors more sensitive than the osse . we note that the comptel has already detected a power law tail up to @xmath39510 mev in the soft state of cyg x-1 ( mcconnell et al . 2000 ) . figure [ f : model]a shows the spectral components of the fit to the vp-619 spectrum . compton reflection with @xmath40 is detected at a very high significance ( @xmath41 gives @xmath42 , corresponding to the chance appearance of reflection of @xmath43 from the @xmath44-test ) , and it is responsible for the convex curvature in the @xmath3910100 kev spectrum . figure [ f : model]a also shows that the scattered component has a spectral break at @xmath2 kev but continues as a power law ( with addition of the broad annihilation feature ) above it due to the domination of non - thermal scattering at those energies . comptonization by the thermal electrons dominates at energies close to the blackbody component and thus pca data of grs 1915 + 105 can be reasonably modeled up to 60 kev by thermal comptonization of a disk blackbody ( vilhu et al . when the osse data are included , the probability that non - thermal electrons are not present ( i.e. , @xmath45 ) is only @xmath46 . the best - fit thermal compton model shown by the long dashes in figure [ f : model]a strongly underestimates the flux above 100 kev . the statistical significance of the presence of non - thermal electrons can be further increased by fitting the _ rxte _ spectrum together with the average spectrum from osse , which has virtually identical shape to that of vp 619 , but much better statistics . then , allowing for @xmath47 reduces @xmath48 from @xmath49 to @xmath50 , which corresponds to the chance probability of @xmath51 . thus , we strongly rule out the pure - thermal comptonization model . during the vp 813 , compton reflection is statistically not required , as indeed expected at the large @xmath52 of the scattering medium covering the disk ( which would completely smear out any disk reflection and fluorescence features ) . thus , we set @xmath41 is the fit . the presence of non - thermal electrons is now required at an extremely high significance ( @xmath53 ) due to the presence of the very distinct hard high - energy tail above the thermally cut - off spectrum , see figures [ f : data ] , [ f : model]b . = 8.4 cm on the other hand , shrader & titarchuk ( 1998 ) have fitted a model of bulk - motion comptonization of blackbody photons to _ rxte _ and batse data from a hard state of grs 1915 + 105 similar to that of vp 619 . we fit their model ( bmc in xspec ) at a free @xmath54 to our broad - band spectra , and find it is completely unacceptable statistically , with @xmath55 and @xmath56 for vp 619 and 813 , respectively . however , the _ specific _ feature of bulk - motion comptonization is a high - energy cutoff at @xmath57 kev due to the effects of compton recoil and gravitational redshift close to the black - hole horizon ( e.g. laurent & titarchuk 1999 ) . such a cutoff is _ not _ included in the bmc model ( and its inclusion would further worsen the fits above ) . thus , we use monte carlo results of laurent & titarchuk ( 1999 ) , which include the cutoff , to test whether the osse data ( regardless of the data at lower energies ) are compatible with its presence . we find that their theoretical spectrum for the accretion rate of @xmath58 matches well the slope of the average osse spectrum at low energies ( the histogram in fig . [ f : bmc ] ) . the monte - carlo spectrum can then be very well reproduced by a power law times a step function convolved with a gaussian ( model plabs(step ) in xspec , with the cutoff energy of 150 kev and the gaussian width of 35 kev , the solid curve in fig . [ f : bmc ] ) . we see that the osse average spectrum lies well above that model at @xmath57 kev . quantitatively , the bulk - motion compton model yields @xmath59 . in comparison , the power - law and eqpair models yield @xmath60 and @xmath61 , respectively . thus , the bulk - compton model is completely ruled out . further problems with that model are discussed in zdziarski ( 2000 ) . = 7.0 cm also , the commonly used phenomenological models of disk blackbody and either a power law or an e - folded power law give very bad fits to our data . the latter yields @xmath62 , @xmath63 for vp 619 , 813 , respectively . in fact , even the thermal part of the vp-813 spectrum is very poorly described by a disk blackbody , with @xmath64 for a fit to the 3.520 kev pca data , whereas the same data are well modeled by thermal comptonization , @xmath65 ( with neither model including a high - energy tail ) . thus , we find physical models of the spectra of grs 1915 + 105 in terms of thermal and non - thermal comptonization and ( in some cases ) compton reflection to be vastly superior to any other model proposed so far . we have found that broad - band spectra of grs 1915 + 105 in its two main spectral states ( @xmath9 , @xmath8 ) are very well fitted by comptonization of disk blackbody photons in a ( hybrid ) plasma with both electron heating and acceleration . the presence of strong reflection indicates the plasma is located in coronal regions ( possibly magnetic flares ) above the disk . this physical model is the same as that fitted to the soft state of cyg x-1 by g99 . differences in the variability properties of cyg x-1 and grs 1915 + 105 are likely to be due to the disk being stable in the former case and unstable in the latter , most likely due to its much higher @xmath66 . the corona is unstable in both cases . the comptonizing medium in the high-@xmath67 state ( @xmath9 ) is thomson - thick , and it can represent the surface layer of an overheated disk accreting at a super - eddington rate ( beloborodov 1998 ) . an issue to be addressed by future research is the origin of the hardening of the tail in this state as compared to other ones ( fig . [ fig : osse ] ) . our model predicts broad annihilation features , although their strength depends on the unknown size of the plasma . grs 1915 + 105 was in the power - law @xmath7-ray state in the classification of grove et al . ( 1998 ) during the 9 osse observations . this state usually corresponds to the high / soft x - ray state . indeed , x - ray spectra observed so far from grs 1915 + 105(vilhu et al . 2001 ) are substantially softer than those with @xmath68 and a sharp thermal cutoff at @xmath69 kev , characteristic to the hard state of other black - hole binaries . we thank p. coppi and m. gierliski for their work on the eqpair model , w. n. johnson for help with the osse data reduction , and ph . laurent for supplying his monte carlo results . this research has been supported by grants from kbn ( 2p03d00614 and 2p03c00619p1,2 ) , the foundation for polish science ( aaz ) , and the swedish natural science research council and the anna - greta and holger crafoord fund ( jp ) . jp and aaz acknowledge support from the royal swedish academy of sciences , the polish academy of sciences and the indian national science academy through exchange programs . zdziarski , a. a. 2000 , in iau symp . 195 , highly energetic physical processes and mechanisms for emission from astrophysical plasmas , ed . martens , s. tsuruta & m. a. weber ( san francisco : asp ) , 153 ( astro - ph/0001078 )	grs 1915 + 105 was observed by the _ cgro_/osse 9 times in 1995 - 2000 , and 8 of those observations were simultaneous with those by _ rxte_. we present an analysis of all of the osse data and of two _ rxte_-osse spectra with the lowest and highest x - ray fluxes . the osse data show a power - law like spectrum extending up to @xmath0 kev without any break . we interpret this emission as strong evidence for the presence of non - thermal electrons in the source . the broad - band spectra can not be described by either thermal or bulk - motion comptonization , whereas they are well described by comptonization in hybrid thermal / non - thermal plasmas .	4,741	191
axial killing fields play an important role in classical general relativity when defining the angular momentum of a black hole or a similar compact , gravitating body . the simplest example is the komar angular momentum , which applies to regions of spacetime that admit a global axial killing field @xmath1 . the formula is @xmath2 where @xmath3 is a spacelike 2-sphere , @xmath4 is the area bivector normal to @xmath3 , @xmath5 is the extrinsic curvature of a cauchy surface @xmath6 containing @xmath3 , @xmath7 is the spacelike normal to @xmath3 within @xmath6 , and @xmath8 is the intrinsic area element on @xmath3 . similar integrals such as the quasi - local formulae due to brown and york @xcite and for dynamical horizons @xcite apply more generally , when only the _ intrinsic , two - dimensional _ metric on @xmath3 is symmetric . but the definition of each does rely on the existence of such an intrinsic symmetry . in order to justify an integral like ( [ komar ] ) as a physical angular momentum , @xmath1 should have some standard , global characteristics of an axial killing field . in particular , for a general riemannian geometry @xmath9 , these characteristics include that 1 . @xmath1 vanishes at at least one fixed point @xmath10 , 2 . every orbit of the flow @xmath11 generated by @xmath1 is either a fixed point or a circle , and 3 . all of the circular orbits close only at integer multiples of a common period @xmath12 . one may then scale @xmath1 by a constant as needed so that @xmath13 , the standard period for a group of rigid rotations . the angular momenta mentioned above take unique values only when one restricts to this preferred normalization . if @xmath14 happens to be a ( topological ) 2-sphere , then condition ( a ) is usually strengthened to 1 . @xmath1 vanishes at exactly two fixed points @xmath15 and @xmath16 in @xmath14 . these conditions distill the intuitive , geometric features of axial symmetries in standard geometries ( _ i.e. _ , euclidean , gaussian , or lobachevskian ) . the purpose of this note is to show that the conditions above are redundant . the _ global _ conditions ( b ) and ( c ) as well as condition ( a ) when @xmath14 derive from condition ( a ) and one other _ local _ condition at @xmath15 , which is that @xmath17 that is , any killing field @xmath1 with at least one fixed point @xmath15 , whose derivative satisfies the local condition ( [ fcon ] ) , is necessarily an axial killing field in the global sense described above . furthermore , ( [ fcon ] ) _ always _ holds if @xmath18 is 2- or 3-dimensional , so in those cases any killing field with a fixed point is an axial killing field . finally , every vector field on a 2-sphere must vanish at at least one point , so every killing field on a 2-sphere is axial . condition ( a ) then follows by a topological argument . one can also show that conditions ( a ) , ( b ) , and ( c ) hold whenever @xmath14 is a 2-sphere ( * ? ? ? * see 3 ) using the classical uniformization theorem : every metric @xmath19 on a 2-sphere is conformally related to a round metric @xmath20 . any killing field of @xmath19 must also be a killing field of @xmath20 , and therefore must satisfy conditions ( a ) , ( b ) , and ( c ) . the approach we take here is more direct , and actually lays the foundation @xcite for a proof of the uniformization theorem via ricci flow @xcite . here we extend this direct approach to arbitrary riemannian geometries and highlight its utility in physical applications . the global flow @xmath11 generated by a killing field @xmath1 on a riemannian manifold @xmath9 is closely related to the corresponding inifnitesimal flow @xmath21 induced on the tangent space @xmath22 at any point @xmath15 where @xmath1 vanishes . the key point is that the latter form a subgroup of the symmetry group @xmath23 for the euclidean geometry @xmath24 . the natural flow @xmath21 in the tangent space at @xmath15 consists of the differentials @xmath25 of the diffeomorphisms @xmath11 in the global flow on @xmath18 . the differential of a diffeomorphism @xmath26 is just the natural push - forward map @xmath27 on vectors . if @xmath28 is fixed , then @xmath29 becomes a linear map from the tangent space @xmath30 to itself . furthermore , if @xmath31 is a global isometry of the riemannian geometry @xmath9 that fixes @xmath32 , then @xmath29 is a linear isometry of the euclidean geometry @xmath33 . it follows that @xmath34 is a one - parameter group of linear isometries of a eucliean vector space . they must be rotational isometries , not translational , because they fix the origin of @xmath22 . thus , each orbit of @xmath21 must either be a fixed point or a circle . this is the tangent - space analogue ( b@xmath35 ) of condition ( b ) above . next , recall that the lie derivative along @xmath1 of a vector field @xmath36 is defined in terms of the push - forward under the flow @xmath11 generated by @xmath1 : @xmath37_{t = 0}.\ ] ] when @xmath1 vanishes at @xmath38 , this determines the generator , in the ordinary sense of matrices , of the one - parameter group of linear isometries @xmath21 via @xmath39_{t = 0 } = - \lie_\varphi\ , v^b \bigr\|_{p_0 } = v^a\ , \grad\!_a\ , \varphi^b \bigr\|_{p_0}.\ ] ] thus , the tensor @xmath40 from ( [ fcon ] ) generates the rotations @xmath21 in @xmath22 in the sense that @xmath41 one can always find an orthonormal basis for @xmath24 that puts an anti - symmetric tensor like @xmath40 in canonical , block - diagonal form with one or more @xmath42 anti - symmetric blocks along the diagonal , possibly followed by one or more zeroes . there can be at most one non - zero block if @xmath18 is 2- or 3-dimensional , but in higher dimensions there may be several . in principle , each block could have a different pair of values @xmath43 for its off - diagonal elements . but the period of @xmath44 within the 2-dimensional subspace of @xmath22 corresponding to one of these blocks is @xmath45 . these periods are equal if ( [ fcon ] ) holds because , working in the preferred basis , we have @xmath46 for each @xmath47 . thus , ( [ fcon ] ) yields the tangent - space analogue ( c@xmath35 ) of condition ( c ) . to summarize , the rigid , euclidean geometry of @xmath24 suffices to prove the tangent - space analogues of our claims above . more precisely , we have shown that 1 . the 1-parameter group of linear isometries @xmath21 fixes the origin in @xmath22 , _ whence _ 2 . every other orbit of @xmath21 in @xmath22 is either a fixed point or a circle , and 3 . all of those circular orbits close for a common period @xmath12 as long as ( [ fcon ] ) holds . in addition , ( [ fcon ] ) holds identically if @xmath18 is 2- or 3-dimensional . finally , note that the natural normalization of @xmath40 sets each @xmath48 , which is equivalent to @xmath49 this one normalization condition implies that _ all _ orbits of @xmath21 close at @xmath13 . the results in the tangent space @xmath22 extend back to the whole manifold @xmath18 essentially because the global isometries @xmath11 map geodesics of @xmath9 to other geodesics . recall the exponential map @xmath50 where @xmath51 denotes the unique geodesic of @xmath9 starting from @xmath52 , having initial tangent @xmath53 , and affinely parameterized so that @xmath54 for all @xmath55 . each global isometry @xmath11 maps this geodesic to another one , @xmath56 , which of course has its initial tangent rotated in @xmath22 . in short , @xmath57 for all @xmath58 and all @xmath59 . if @xmath18 is connected , compact , and without boundary , then there exists at least one geodesic between @xmath15 and any of its other points . it therefore follows from ( [ pcomm ] ) that every orbit of @xmath11 in @xmath18 must be the image under the exponential map of at least one orbit of @xmath21 in @xmath22 . the exponential generally is not invertible , of course , so the orbit in @xmath22 generally is not unique . but the orbits of @xmath11 can not intersect one another , or themselves , non - trivially . thus , although the exponential mapping of orbits may be many - to - one , and may map circular orbits of @xmath21 to fixed - point orbits of @xmath11 , it nonetheless preserves orbits . it follows in particular that each orbit of @xmath11 must either be a fixed point or a closed circle . in other words , condition ( b ) for the orbits of @xmath11 follows from condition ( b@xmath35 ) for the orbits of @xmath21 . it is also clear that every circular orbit of @xmath11 must close with the same period as one of the orbits of @xmath21 that is mapped to it by the exponential . but all orbits of @xmath21 have the same period as long as ( [ fcon ] ) holds , and condition ( c ) for the orbits of @xmath11 then follows from condition ( c@xmath35 ) for the orbits of @xmath21 . let us summarize the chain of arguments here . condition ( a ) on a killing field on a riemannian geometry @xmath9 implies condition ( a@xmath35 ) for the linear isometries @xmath21 of the euclidean geometry @xmath24 . this in turn implies conditions ( b@xmath35 ) and ( c@xmath35 ) , as long as ( [ fcon ] ) holds . finally , these imply conditions ( b ) and ( c ) globally on @xmath18 via the exponential map . note that the _ local _ normalization condition ( [ fnorm ] ) at @xmath15 therefore dictates the _ global _ scaling of the killing field @xmath1 such that all of its orbits have the desired period @xmath13 . every vector field , and thus every killing field , on a 2-sphere @xmath3 vanishes at at least one point , and ( [ fcon ] ) holds identically in 2 dimensions . every killing field on a 2-sphere therefore satisfies conditions ( a ) , ( b ) , and ( c ) . but the intuitive picture of an axial killing field on a 2-sphere is that it should vanish at exactly two poles , as in condition ( a ) , not just at a single point . we now show that this is necessarily the case . suppose that a 2-sphere geometry @xmath60 admits an axial killing field @xmath1 , which we take to be normalized according to ( [ fnorm ] ) . each point @xmath61 in @xmath3 lies on a unique orbit of @xmath1 , which divides the sphere into two regions . let @xmath62 denote the area of the region bounded by that orbit and containing the fixed point @xmath15 . equivalently @xcite , we may set @xmath63 the smooth function @xmath62 takes both maximum and minimum values somewhere on the ( compact ) sphere , and each must occur at a fixed point where @xmath64 vanishes . the minimum value is zero , and occurs only at @xmath38 . the maximum value @xmath65 also occurs at a fixed point @xmath16 , which can not coincide with @xmath15 because any fixed point of @xmath1 must be _ one - sided_. that is , the local topology of @xmath66 is that of the 2-dimensional plane , so the fixed points of @xmath1 must be isolated from one another , and each must be surrounded by a _ single _ family of circular orbits . ( if two distinct families of circular orbits surrounded a given fixed point , then @xmath3 would have the local topology of a two - sided cone there . ) similarly , every circular orbit of @xmath1 must be _ two - sided _ in the sense that there are exactly two distinct families of circular orbits surrounding it . ( there can not be only one such family because @xmath3 has no boundary , and there can not be more than two because this would violate the assumed manifold structure of @xmath3 at each point of such an orbit . ) the manifold of orbits @xmath67 of @xmath1 therefore has the topology of a closed interval @xmath68 $ ] , with exactly two fixed points at the two ends and only circular orbits in the interior . thus , @xmath1 must have exactly two isolated fixed points . finally , we show that the exponential map maps a closed disk with _ finite _ radius @xmath69 in @xmath70 onto _ all _ of @xmath3 , with the entire circular boundary of that disk mapped to the conjugate fixed point @xmath71 . this mapping is invertible in the interior of the disk , but obviously not on the boundary . to see this , observe that @xmath72{\dot\gamma_v(s ) } = 0 \qquad\text{and}\qquad \grad\!_{\dot\gamma_v(s)}\ , \bigl ( \varphi \cdot \dot\gamma_v(s ) \bigr ) = 0\ ] ] because @xmath51 in ( [ expdef ] ) is an affinely parameterized geodesic and @xmath1 is a killing field . it follows that the tangent to @xmath51 has constant norm equal to @xmath73 for all @xmath55 and , since @xmath1 vanishes at @xmath15 , is everywhere orthogonal to the circular orbits of @xmath1 . the proper distance in @xmath3 along @xmath51 from @xmath15 to @xmath74 is therefore equal to @xmath73 in @xmath75 . ( this holds generally , not just on the 2-sphere . ) it also follows from ( [ gprop ] ) that the proper geodetic distance from @xmath15 to @xmath16 is @xmath76 where @xmath77 is the proper radius of the orbit of @xmath1 through any given point , and primes denote derivatives with respect to @xmath78 . this integral is irregular because @xmath79 vanishes at both endpoints . but the integral converges nonetheless because the conditions to avoid conical singularities in @xmath3 at @xmath15 and @xmath16 are @xmath80 thus , although @xmath81 diverges at both endpoints in ( [ sigint ] ) , the indefinite integral of @xmath82 vanishes like @xmath79 near @xmath83 . the difference @xmath84 likewise vanishes like @xmath79 near @xmath85 . there can be no other irregular points with @xmath86 , so the integral ( [ sigint ] ) is well - defined and @xmath69 is finite . we have seen that any killing field @xmath1 that vanishes at at least one point @xmath15 in a riemannian manifold @xmath9 , and satisfies ( [ fcon ] ) there , necessarily has only fixed - point and closed , circular orbits throughout @xmath18 , the latter all having the same period @xmath87 . this offers a convenient way to characterize axial killing fields in terms of local data at a fixed point . furthermore , scaling @xmath1 globally such that ( [ fnorm ] ) holds locally at @xmath15 guarantees that the period of every circular orbit is @xmath0 . the condition ( [ fcon ] ) is not needed in 2 or 3 dimensions , and in fact one might imagine abandoning it altogether for manifolds of arbitrary dimension . the main disadvantage of this would be that some orbits of @xmath21 may no longer be closed . rather , if some of the @xmath88 from ( [ fivals ] ) were incommensurate , some orbits @xmath21 would densely fill a torus in @xmath22 . the same would then apply to the orbits of @xmath11 by the same methods used above . although this would be rather less convenient , and less physical , such `` quasi - axial '' symmetries could potentially be interesting in certain applications . one additional feature of these results is that they suggest a natural and convenient way to normalize an _ approximate _ killing field on a riemannian geometry @xmath9 that has no actual symmetries @xcite . namely , suppose that a candidate approximate killing field @xmath89 can be found up to constant scaling on @xmath18 and vanishes at at least one point , which again is guaranteed if @xmath14 . then , one may define @xmath90}$ ] , and use ( [ fnorm ] ) to normalize @xmath89 throughout @xmath18 without having to compute the orbits of @xmath89 and ensure that they close , etc . we explore this proposal further in a companion paper @xcite .	this note describes a local scheme to characterize and normalize an axial killing field on a general riemannian geometry . no global assumptions are necessary , such as that the orbits of the killing field all have period @xmath0 . rather , any killing field that vanishes at at least one point necessarily has the expected global properties .	4,953	81
the extraction of the cosmic far infrared background ( cfirb ) , induced by the emission of light from distant galaxies ( partridge & peebles , 1967 ; bond et al . , 1986 and references therein ) , requires an accurate subtraction of the interstellar medium ( ism ) foreground emissions . the two instruments dirbe and firas on board the cobe satellite provide actually the best available data to study , on the whole sky , the distribution and properties of the ism far infrared ( far - ir ) emission . + boulanger et al . ( 1996 ) have extensively studied the emission of the dust associated with the hi component using the spatial correlation between the far - ir dust emission as measured by dirbe and firas and the 21 cm hi emission as measured by the leiden / dwingeloo survey of the northern hemisphere . the dust emission spectrum derived from this correlation ( for n@xmath10 4.5 10@xmath11 @xmath12 ) can be quite well represented by a single modified planck curve characterized by t=17.5 k and @xmath13 @xmath8 . this emissivity law is very close to the one predicted by the draine @xmath14 lee ( 1984 ) dust model . + dust emission associated with molecular clouds has been recently studied through far - ir and submillimeter ( submm ) observations with the dirbe , firas and spm / pronaos instruments . in a previous paper ( lagache et al . , 1998 ) , we have extensively studied the spatial distribution of the temperature of the dust at thermal equilibrium using the dirbe and firas experiment . we have found at large angular scale the presence of a cold dust component ( with a median temperature of 15 k ) , very well correlated with molecular complexes with low star forming activity such as taurus . the lowest values of the temperature found in the cold regions ( @xmath15 k ) are comparable with that obtained for dense clouds in star forming regions by the balloon - borne experiment spm / pronaos ( ristorcelli et al . , 1996 , 1998 , serra et al . , 1997 ) . the association between the cold dust component and molecular clouds is further demonstrated by the fact that all sky pixels with significant cold emission have an excess ir emission with respect to the high latitude ir / hi correlation . a threshold value of the column density , n@xmath16=2.5 @xmath17 h @xmath12 , below which cold dust is not detected within the firas beam of @xmath18 has been deduced . this knowledge on the spatial distribution of the dust associated with cold molecular clouds is important for the search of the cfirb since it allows to select parts of the sky for which cold dust is not detected . + on the other hand , the knowledge of the dust emission associated with the h@xmath0 component is very poor . observations of h@xmath19 emission at high galactic latitudes and of dispersion measures in the direction of pulsars at high @xmath20 indicate that the low - density ionised gas ( the warm interstellar medium , wim ) accounts for some 30@xmath2 of the gas in the solar neighborhood ( reynolds , 1989 ) . there is also evidence that part of the wim is spatially correlated with the hi gas ( reynolds et al . , 1995 ) . consequently , a significant fraction of the far - ir emission associated with the wim may contribute to the spectrum of the dust associated with the hi gas . however , the scale height of the h@xmath0 medium is much larger than the hi one , so a significant part of the h@xmath0 is completely uncorrelated with the hi . since most of the grain destruction is expected to occur in the low - density component of the ism ( mc kee 1989 ) , the wim could also be dust poor . depletion studies of elements that form the grains show that grains are indeed partly destroyed in the low density phases of the ism ( review by savage & sembach , 1996 ) . measuring the dust emission from the wim could allow to understand the evolution of the dust in the low - density gas . however , this measure is difficult because one can not easily separate the contribution of the h@xmath0 gas from that of the hi . boulanger & perault ( 1988 ) unsuccessfully searched in the 100 @xmath4 iras all - sky map for such a contribution . the unfruitful analysis may be due to the spatial correlation between the hi and h@xmath0 emissions . boulanger et al . ( 1996 ) have searched for such a component in the residual firas emission after the removal of the hi component . they found that the residual emission is consistent with an emission spectrum like that of the hi gas for n@xmath21 4 10@xmath22 @xmath12 . however , they consider this as an upper limit for the contribution of the h@xmath0 component since they could have measured emission from low contrasted molecular clouds . arendt et al . ( 1998 ) have also investigated potential ir wim dust emission . they conclude that they were unable to detect any ir emission associated with low density ionised gas at high galactic latitude ( the fraction of the sky used is less than 0.3@xmath2 ) . however , very recently , howk & savage ( 1999 ) have pointed out , for the first time , the existence of al- and fe - bearing dust grains towards two high - z stars . they have shown that the degree of grain destruction in the ionised medium , through these two stars , is not much higher than in the warm neutral medium . if dust is present in the wim , one should detect its infrared emission . + the cfirb is expected to have two components : direct starlight redshifted in the far - ir and submm , and the stellar radiation absorbed by dust . we concentrate here on the submm part of this background . its detection is very difficult because of the strong and fluctuating galactic foregrounds . first , upper limits have been reported : hauser et al . ( 1995 ) from dirbe data and mather et al . ( 1994 ) from firas data . lower limits on the cfirb have been obtained from the deepest iras and k galaxy counts ( hauser et al . , 1994 and references therein ) . the first direct detection of the cfirb has been reported by puget et al . all the galactic foregrounds were modeled and removed using independant dataset in addition to the firas data . its spectrum indicates the presence of sources at large redshift . the main uncertainty on the cfirb comes from galactic foregrounds . therefore , we stress that the puget et al . ( 1996 ) results were confirmed in the cleanest parts ( n@xmath23 @xmath12 in a 7@xmath24 beam ) of the sky ( guiderdoni et al . , 1997 ) . more recently , fixsen et al . ( 1998 ) and hauser et al . ( 1998 ) have also confirmed the detection of the cfirb using firas and dirbe data . + a general problem with all these determinations of the cfirb comes from a potential far - ir emission from the wim which has never been determined . the goal of this paper is to push our knowledge of the galactic emission one step forward by deriving the far - ir spectrum of the wim dust emission . then , we use our understanding of the interstellar dust emissions associated with the hi and h@xmath0 components to give a more accurate estimate of the cfirb spectrum . the paper is organised as follow : in sect . [ sect_data_prep ] , we present the data we have used . the variations of the dust emission spectrum associated with different hi gas column densities are studied in sect . [ sect_hi_var ] . in sect . [ sect_effect_hii ] , we show that the spatial variations of the dust emission spectrum in the low hi column density regions can be due to the presence of the non - correlated h@xmath0 component . after the removal of the dust hi component , we detect a residual galactic emission which is attributed to the wim ( sect . [ sect_hii_spec ] ) . this is the first detection of the wim dust emission . we show ( sect . [ sect_cfirb_lh ] ) that the firas spectra in the very low hi column density regions exhibit a large excess over the emission of dust associated with hi and h@xmath0 components . in these regions , the cfibr dominates the firas emission . we test its isotropy in sect . [ sect_cfirb_iso ] . all the results are summarised in sect . [ sect_cl ] . the firas instrument is a polarising michelson interferometer with @xmath25 resolution and two separate bands which have a fixed spectral resolution of 0.57 @xmath26 ( fixsen et al . 1994 ) . the low frequency band ( 2.2 to 20 @xmath27 was designed to study the cmb ( cosmic microwave background ) and the high frequency band ( 20 to 96 @xmath26 ) to measure the dust emission spectrum in the galaxy . we use the so - called llss ( left low short slow ) and rhss ( right high short slow ) data from the `` pass 3 '' release which cover the low and high frequency bands respectively ( see the firas explanatory supplement ) . + dirbe is a photometer with ten bands covering the range from 1.25 to 240@xmath28 with 40 arcmin resolution ( silverberg et al . we choose to use annual averaged maps because they have a higher signal to noise ratio than maps interpolated at the solar elongation of @xmath29 ( see the dirbe explanatory supplement ) . in our analysis , we only use the 140 and 240 @xmath4 maps . the present study is based on `` pass 2 '' data . + since in our analysis we combine the firas and dirbe data at 140 and 240 @xmath4 , we convolve the dirbe maps with the firas point spread function ( psf ) . the psf is not precisely known for all wavelengths , so we use the approximation suggested by mather ( private communication ) of a @xmath25 diameter circle convolved with a line of @xmath30 length perpendicular to the ecliptic plane ( mather et al . + before studying the far - ir emission , we have subtracted the cmb and its dipole emission from the firas data using the parameters given by mather et al . ( 1994 ) and fixsen et al . ( 1994 ) . to remove the interplanetary ( ip ) dust emission , we first consider the @xmath31 map as a spatial template for the ip dust . then , we compute the ip dust emission at 100 @xmath4 using the zodiacal emission ratio given by boulanger et al . ( 1996 ) : @xmath32 . we remove the ip dust emission at @xmath33 using the ip emission template at 100@xmath4 and considering a zodiacal spectrum @xmath34 ( reach et al . , 1995 ) . + the hi data we used are those of the leiden / dwingeloo survey , which covers the entire sky down to @xmath35 with a grid spacing of 30 in both l and b ( hartmann @xmath14 burton , 1997 ) . the 36 half power beam width ( hpbw ) of the dwingeloo 25-m telescope provides 21-cm maps at an angular resolution which closely matches that of the dirbe maps . these data represent an improvement over earlier large scale surveys by an order of magnitude or more in at least one of the principal parameters of sensitivity , spatial coverage , or spectral resolution . details of the observationnal and correction procedures are given by hartmann ( 1994 ) and by hartmann @xmath14 burton ( 1997 ) . the 21cm - hi data are convolved with the firas psf . we derive the hi column densities with 1 k km s@xmath36=1.82 @xmath37 h @xmath12 ( optically thin emission ) . + throughout this paper , diffuse parts of the sky are selected following lagache et al . ( 1998 ) . to remove molecular clouds and hii regions , we use the dirbe map of the 240 @xmath4 excess with respect to the 60@xmath4 emission : @xmath38s = s@xmath39(240)- c@xmath40s@xmath39(60 ) with c=4@xmath60.7 . this map shows as positive flux regions , the cold component of the dust emission , and as negative flux regions , regions where the dust is locally heated by nearby stars ( like the hii regions ) . therefore , diffuse emission pixels are selected with @xmath41 , @xmath3 being evaluated from the width of the histogram of @xmath38s and @xmath42 being chosen for our different purposes . for example , in sect . [ sect_hii_spec ] , we take @xmath43 , which is very restrictive , to ensure that the selected pixels are mostly coming from the diffuse medium ; in sect . [ sect_cfirb_iso ] , we take @xmath44 , since we need a fraction of the sky as large as possible . + we also use the 240 @xmath4/hi map excess of reach et al . regions for which this excess is greater than 3@xmath3 are systematically discarded . + from lagache et al . ( 1998 ) , we have for each line of sight the spectrum of the cold component of the dust emission ( if cold dust is detected ) and the cirrus spectrum . these spectra are used to compute the contribution of the cold dust emission in sect . [ sect_hi_var ] ( table [ tbl-1 ] ) . in this part , we first concentrate on the spatial variations of the dust emission spectrum with the hi gas column densities . we deduce a column density threshold above which the contribution of the cold dust component induces a significant submm excess with respect to the @xmath45 emissivity law . then , we investigate the spectrum of the dust associated with regions containing hi column densities below this threshold . + [ cols="^,^,^,^,^ , < " , ] + @xmath46 hauser et al . , 1998 @xmath47 schlegel et al . , 1998 + = 8.cm = 7.cm + the cfirb can also be determined at 240 and 140 @xmath4 using dirbe data for the same part of the sky . results are presented in table [ cfirb_dirbe ] together with previous determinations . in this table , uncertainties on total emissions are the dispersions measured for the selected pixels , uncertainties on the hi dust emissions are given by eq . [ emiss_hi ] , and uncertainties on the wim dust emissions correspond to the 21@xmath2 dispersion observed in the four bins of galactic longitude ( table [ tbl - var - hii - long ] ) . our residual emissions ( res2 in table [ cfirb_dirbe ] ) are significantly lower than the ones reported in hauser et al . ( 1998 ) and schlegel et al . ( 1997 ) since they have neglected the contribution of the dust associated with the wim . we clearly see that it is essential to take into account the dust emission associated to the wim below 240 @xmath4 . in firas data at 240 @xmath4 , the cfirb emission ( fig . [ firback_lh_fir ] ) is very close to our dirbe value . at 140 @xmath4 , the comparison between dirbe and firas data is not possible due to the considerable increase of the firas data noise ( that is why we have prefered to cut the spectrum at 200 @xmath4 in fig . [ firback_lh_fir ] ) . we test in this part the isotropy of the cfirb on large scales since we do not know the spatial distribution of the wim dust emission at small angular scales . this test at large scales is important to detect potential systematic effects caused by an inacurate subtraction of ism dust emissions . as expected from our detection of the wim dust emission , we show that the cfirb is isotropic only if we consider the emission of dust associated with the h@xmath48 component . + = 8.cm = 7.cm + thus , we have to compute the cfirb spectrum on a fraction of the sky as large as possible . for that , we prefer to use a galactic template based on far - ir emissions rather than hi gas , ( 1 ) to take into account variations in the dust temperature from place to place and ( 2 ) to avoid the large hole in the southern hemisphere of the leiden / dwingeloo survey . we combine the dirbe 140 and 240 @xmath4 data with firas spectra . first , we extract from dirbe data ( at 7@xmath24 resolution ) a galactic emission template . then , the dirbe galactic emission is extrapolated to longer wavelengths and subtracted from each individual selected firas spectrum to derive the cfirb emission and test its isotropy . + the two ( 140 and 240 @xmath4 ) galactic dirbe templates , @xmath49 , are computed by removing the cfirb ( table [ cfirb_dirbe ] ) from the dirbe emissions . assuming a @xmath45 emissivity law , temperature and optical depth of each pixel are estimated using the two dirbe galactic templates . the residual firas emission is computed in the following way . first , we discard pixels located in known molecular clouds or hii regions . we keep pixels with @xmath50 and work at @xmath51 ( 51@xmath2 of the sky ) . then , for each selected pixel with its associated temperature , we derive the ratio @xmath52 between the modified planck curve computed at each firas wavelength and the modified planck curve computed at 240 @xmath4 . finally , the residual emission is computed in the following way : @xmath53 the mean residual emission spectrum is shown fig . [ 51_cfirb ] together with the spectrum obtained on 0.54@xmath2 of the sky and its analytical determination ( eq . [ analy_cfirb ] ) . the two spectra agree very well . = 8.cm = 7.cm + = 8.cm = 7.cm = 8.cm = 7.cm + the isotropy of our residual emission is addressed by looking at its variation with the galactic longitude or latitude . the variation of our residual with the galactic longitude is derived by computing the mean residual emission ( in the [ 300 , 609 ] @xmath4 band ) in 8 different longitude bins ( each bin representing around 3.5@xmath2 of the sky ) . the profile , shown in fig . [ cfirb_long ] , does not show any particular galactic structure . to test the isotropy with the galactic latitude , we compute the emission profile versus the latitude ( cosecant variation ) of our residual emission ( fig . [ cfirb_lat]a ) . we clearly see a residual galactic component . the slope of the fitted cosecant law is @xmath54 7 10@xmath55 w m@xmath56 sr@xmath36 which is 4 times smaller than the slope of the h@xmath57 cosecant component . + the pixels selected before represent only the diffuse medium but contain both the neutral and ionised emission and thus can not be very well represented by a single temperature component . therefore , our residual firas emission could be affected by an inaccurate subtraction of the interstellar medium emissions . to test the effect of the presence of the wim dust emission on our residual emission , we apply the same method as previously but remove from the dirbe data , before determining the temperatures , the h@xmath58 dust emission defined as @xmath59 @xmath16@xmath60 with @xmath61 ( see sect . [ sect_effect_hii ] ) . @xmath62 represents the non - correlated @xmath63 spectrum derived in sect . [ sect_hii_spec ] . the firas residual emission is computed in the following way : @xmath64 where @xmath65 and @xmath66 represent the new ratio and dirbe 240 @xmath4 neutral galactic template respectively , obtained after removing the contribution of the dust emission associated with the non - correlated hii component . the mean residual emission is in very good agreement with the previous estimates ( fig . [ 51_cfirb ] ) . moreover , we see no more residual cosecant variations ( fig . [ cfirb_lat]b ) : the residual component of fig . [ cfirb_lat]a was due to the non - correlated @xmath67 emission . + in conclusion , we clearly show that the main uncertainty to the cfirb determination on individual pixels of the sky comes from dust emission associated with the non - correlated h@xmath48 component . a new analysis of the correlation between gas and dust at high galactic latitude has been presented . for the first time , we are able to present a decomposition of the far - ir emission over the hi , h@xmath48 and h@xmath1 gas with the determination of the dust emission spectrum for each of these components . this decomposition is important to study the evolution of interstellar dust as well as to give new constraints on the cfirb ( spectrum and test of isotropy ) . the data used in this analysis are the cobe far - ir data and the leiden - dwingeloo hi survey . our decomposition is based on the ir / hi correlation and dirbe and firas ir colors . the main quantitative results of this work follow : + 1 ) dust emission spectrum associated with the hi gas : + we confirm that the emission spectrum of dust in hi gas is well fitted by a modified planck curve with a @xmath45 emissivity law and a temperature of 17.5 k. however , the emissivity normalised per h atom varies by about 30@xmath2 depending on the pieces of the sky used to compute the spectrum . we show that this variation may be explained by the emission of the dust associated with the h@xmath48 component . taking into account this contribution , we derive a new value of the dust emissivity nomalised per hydrogen atom : @xmath68= 8.7@xmath6 0.9 @xmath69 @xmath8 with a temperature of 17.5 k , slightly lower than the previous determination of boulanger et al . + 2 ) dust emission spectrum associated with the h@xmath48 gas : + the existence of far - ir emission from the wim is demonstrated by the latitude dependance of the residuals of the ir / hi correlation . this cosecant variation gives a spectrum which is clearly different from the hi dust spectrum . this component is detected at a 10@xmath3 level in the [ 200 - 350 ] @xmath4 band . dust associated with the wim has an emissivity @xmath70 3.8@xmath6 0.8 @xmath7 @xmath8 with a temperature of 29.1 k. with a spectral index equal to 2 , the emissivity law becomes @xmath5 1.0 @xmath6 0.2 @xmath9 @xmath8 with a temperature of 20 k. the variation in dust spectrum from the hi to the wim dust component can be explained by only changing the upper cutoff of the big grain size distribution from 0.1 @xmath4 to 30 nm . + 3 ) cosmic far - ir background ( cfirb ) : + firas spectra in low hi column density regions ( 0.54@xmath2 of the sky ) clearly show the presence of a component which is not associated with the hi and h@xmath0 gas and which is interpreted , following puget et al . ( 1996 ) , as the cfirb due to the integrated light of distant galaxies . the determination of this component on 51@xmath2 of the sky confirms its isotropy at large scale on a suitable fraction of the sky . with our determination of the wim contribution to the far - ir sky emission , we find for the cfirb values at 140 and 240 @xmath4 , @xmath71 10@xmath72 and @xmath73 10@xmath72 w m@xmath56 sr@xmath36 respectively , which are significantly lower than the hauser et al . ( 1998 ) determinations . the contribution of the ir dust emission from the wim relative to the cfirb is negligible at longer wavelengths . arendt , r.g . et al . , 1998 , apj 508 , 74 bohlin , r.c . , 1978 , apj 224 , 132 bond , r.j . 1986 , apj 306 , 428 boulanger , f. , & prault , m. 1988 , apj 330 , 964 boulanger , f. et al . 1996 , a&a 312 , 256 dsert , f.x . , boulanger , f. , puget , j.l . 1990 , a&a 237 , 215 draine , b.t . & lee , h.m . 1984 , apj 285 , 89 fixsen , d.j . 1994 , apj 420 , 457 fixsen , d.j . et al . 1998 , apj 508 , 123 gispert , r. , lagache , g. & puget , j.l . in preparation guiderdoni , b. et al . , 1997 , nature 390 , 257 hartmann , d. 1994 , ph.d . thesis , university of leiden hartmann , d. & burton , w.b . , `` atlas of galactic neutral hydrogen '' , cambridge university press , 1997 hauser , m.g . et al . , in `` examining the big bang and diffuse background radiations , iau symp . 168 '' , the hague , 1994 hauser , m.g . et al . , in `` unveiling the cosmic infrared background '' ed . e. dwek , aip conf . 1995 hauser , m.g . et al . , 1998 , apj 508 , 25 howk , j.c . & savage , b.d . , apj , in press jahoda , k. , lockman , f.j . , mccammon , d. 1990 , apj 354 , 184 jones , a.p . et al . , 1996 , apj 469 , 740 lagache , g. , abergel , a. , boulanger , f. , puget , j.l . 1998 , a&a 333 , 709 mather , j.c . 1986 , app opt 25 , 16 mather , j.c . 1994 , apj 420 , 439 mckee , c.f . 1989 , in `` interstellar dust '' , eds l.j . allamandola and a.g.g.m . tielens , p431 , kluwer osterbrock , d.e , `` astrophysics of gaseous nebulae and active galactic nuclei '' , university science book , 1989 partridge , r.b . & peebble , p.j.e . 1967 , apj , 148 , 377 puget , j.l . 1996 , a&a 308 , l5 reach , w.t . 1998 , apj 507 , 507 reach , w.t . 1995 , in `` unveiling the cosmic infrared background '' ed . e. dwek , aip conf . reynolds , r.j . , 1984 , 282 , 191 reynolds , r.j . 1989 , apj 339 , l29 reynolds , r.j . 1991 , apj 372 , l17 reynolds , r.j . 1992 , apj 392 , l35 reynolds , r.j . et al . , 1995 , apj 448 , 715 reynolds , r.j . , 1998 , pasa 15 , 14 ristorcelli , i. , et al . 1998 , to be submitted ristorcelli , i. et al . 1996 , in `` diffuse infrared radiation and the irts '' , ed . h. okuda , t. matsumoto , t. roelling tufte , s.l . , et al . , 1996 , baas , 28 , 890 savage , b.d & sembach , k.r . 1996 , araa , 34 , 279 schlegel , d.j . , finkbeiner , d.p . , davis , m. 1998 , apj 500 , 525 serra , g. et al . 1997 , in `` the far infrared and submillimetre universe '' , esa sp-401 silverberg , r.f . 1993 , in spie conference proc . 2019 on infrared spaceborne remote sensing , san diego	we present a new analysis of the far - ir emission at high galactic latitude based on cobe and hi data . a decomposition of the far - ir emission over the hi , h@xmath0 and h@xmath1 galactic gas components and the cosmic far infrared background ( cfirb ) is described . + for the first time the far - ir emission of dust associated with the warm ionised medium ( wim ) is evidenced . this component determined on about 25@xmath2 of the sky is detected at a 10@xmath3 level in the [ 200 , 350 ] @xmath4 band . the best representation of the wim dust spectrum is obtained for a temperature of 29.1 k and an emissivity law @xmath5 3.8 @xmath6 0.8 @xmath7 @xmath8 . with a spectral index equal to 2 , the emissivity law becomes @xmath5 1.0 @xmath6 0.2 @xmath9 @xmath8 , with a temperature of 20 k , which is significantly higher than the temperature of dust associated with hi gas . the variation in the dust spectrum from the hi to the wim component can be explained by only changing the upper cutoff of the big grain size distribution from 0.1 @xmath4 to 30 nm . + the detection of ir emission of dust in the wim significantly decreases the intensity of the cfirb , especially around 200 @xmath4 which corresponds to the peak of energy . +	7,988	382
after the first successful space flight use of the x - ray charge coupled device ( ccd ) of the sis ( @xcite ) on board asca , the ccd has been playing a major role in imaging spectroscopy in the field of x - ray astronomy . however , the charge transfer inefficiency ( cti ) of x - ray ccds increases in orbit due to the radiation damage ; the cti is defined as the fraction of electrons that are not successfully moved from one ccd pixel to the next during the readout . since the amount of charge loss depends on the number of the transfers , the energy scale of x - ray ccds depends on the location of an x - ray event . furthermore , there is a fluctuation in the amount of the lost charge . therefore , without any correction , the energy resolution of x - ray ccds in orbit gradually degrades . in the case of the x - ray imaging spectrometer ( xis ) @xcite on board the suzaku satellite @xcite launched on july 10 , 2005 , the energy resolution in full width at half maximum ( fwhm ) at 5.9 kev was @xmath0140 ev in august 2005 , but had degraded to @xmath0230 ev in december 2006 . the increase of the cti is due to an increase in the number of charge traps at defects in the lattice structure of silicon made by the radiation . since the trap distribution is not uniform , it would be best if we could measure the cti of each pixel as chandra acis @xcite . in the case of the xis , however , it is impossible to measure the cti values of all the pixels , mainly because the onboard calibration sources do not cover the entire field of view of the xis . therefore , we use the cti of each column to correct the positional dependence of the energy scale . the xis is equipped with a charge injection structure @xcite which can inject an arbitrary amount of charge in arbitrary positions . using this capability , we can precisely measure the cti of each column @xcite . by applying the column - to - column cti correction , the positional dependence of the cti corrected energy scale is greatly reduced , and the over - all energy resolution is also improved @xcite . in @xcite , the results of the cti correction was mainly based on the ground - based charge injection experiments . in - orbit measurements were limited within one year after the launch . this paper reports more comprehensive and extended in - orbit experiments up to two years after the launch . the results are based on the data with the normal full window mode @xcite without a spaced - row charge injection @xcite , and have been implemented to the suzaku calibration database and applied to all the data obtained with the same mode . all the errors are at the 1@xmath1 confidence level throughout this paper unless mentioned . the xis is the set of four x - ray ccd camera systems . three sensors ( xis 0 , 2 , and 3 ) contain front - illuminated ( fi ) ccds and the other ( xis 1 ) contains back illuminated ( bi ) ccd . the xis 2 sensor became unusable on november 9 , 2006 . therefore there are no data for xis 2 after that day . the detailed structure of the ccd has been provided in @xcite . in this paper , we define a `` row '' and a `` column '' as a ccd line along the @xmath2 and @xmath3 axes , respectively ( see figure 3 in @xcite ) . in the imaging area , the _ actx _ value runs 0 to 1023 from the segment a to d , while the _ acty _ value runs from 0 to 1023 from the readout node to the charge injection structure . the charge injection structure lies adjacent to the top row ( _ acty _ = 1023 ) in the imaging area . we can inject charges from @xmath0 50 e@xmath4 to @xmath0 4000 e@xmath4 per pixel ; the equivalent x - ray energy ranges from @xmath00.2 kev to @xmath015 kev . a charge packet generated by an incident x - ray is transferred to the readout node , then is converted to a pulse - height value . we define @xmath5 to be the original pulse height generated by the x - ray . in the real case , the readout pulse height of the packet ( @xmath6 ) is smaller than @xmath5 , because some amount of charges is lost during the transfer . to measure the charge loss , we have to know both @xmath5 and @xmath7 . however , we can usually measure only @xmath6 , and hence it is difficult to obtain @xmath5 . @xcite and @xcite reported a technique to solve this problem by the charge injection method , and @xcite applied this technique to the xis . we briefly repeat by referring figure 3 in @xcite . first , we inject a `` test '' charge packet into the top ccd row ( _ acty _ then , after the gap of a few rows , five continuous packets are injected with the same amount of charge of the test packet . the former four packets are called `` sacrificial '' charge packets , while the last one is called a `` reference '' charge packet . the test packet loses its charge by the charge traps . on the other hand , the reference packet does not suffer from the charge loss , because the traps are already filled by the preceding sacrificial packets . thus we can measure the charge loss by comparing the pulse - height values of the reference charge ( @xmath8 ) and the test charge ( @xmath9 ) . the relation between sacrificial charge packets and reference charge packets is described in gendreau ( 1995 ) . we can obtain a checker flag pattern by these injected packets in the x - ray image ( right panel of figure 3 in @xcite ) because of the onboard event - detection algorithm @xcite . therefore in this paper , we call this technique a `` checker flag charge injection ( cfci ) . '' a charge packet in the xis loses its charge during ( a ) the fast transfer ( 24 @xmath10s pixel@xmath11 ) along the _ acty _ axis in the imaging area , ( b ) the fast transfer along the _ acty _ axis in the frame - store region , ( c ) the slow transfer ( 6.7 ms pixel@xmath11 ) along the _ acty _ axis in the frame - store region , ( d ) the fast transfer to the readout node along the _ actx _ axis . the cti depends on many parameters such as the transfer speed and the number density of the charge traps @xcite . the frame - store region is covered by the shield and is not exposed to the radiation directly . furthermore , the pixel size of the frame - store region ( 21 @xmath10m@xmath1213.5 @xmath10 m ) is different from that of the imaging area ( 24 @xmath10m@xmath1224 @xmath10 m ) . thus the number of traps per pixel may be different between the imaging area and the frame - store region . then we assumed that the four transfers have different cti values . we examined the transfer ( d ) by using the calibration source data taken in april 2007 , and found no significant decrease of the pulse height along the _ actx _ axis . we , therefore , ignore the charge loss in the transfer ( d ) . we define that @xmath13 is the transfer number in the imaging area ( @xmath13=@xmath3 + 1 ; here , @xmath3 is a coordinate value where an incident x - ray generates a charge packet ) . then the relation between @xmath6 and @xmath5 is expressed as , @xmath14 where @xmath15 , @xmath16 and @xmath17 are the cti values in the transfers ( a ) , ( b ) , and ( c ) , respectively . here we used the fact that the cti values are much smaller than 1 . thus we can separate the charge loss into @xmath13-dependent component ( the second term in the right - hand side of equation [ eq : cti_def_0 ] ) and constant component ( the third term ) . we therefore substitute the cti with cti1 ( the former component ) and cti2 ( the latter component ) , which have the cti values of @xmath18 and @xmath19 , respectively . then equation [ eq : cti_def_0 ] can be written as @xmath20 since the cti values depend on the amount of transfer charge which is proportional to the pulse height , we assume the cti is described by a power function of the pulse height ( prigozhin et al . 2004 ) and expressed as @xmath21 where @xmath22 and @xmath23 are scale factors for the cti1 and cti2 , and the index @xmath24 is common to the cti1 and cti2 . we have conducted the cfci experiments six times in orbit . effective exposure time for each experiment ranges from a few to @xmath020 ks . the equivalent x - ray energy of the injected charge packets ranges from @xmath00.3 kev to @xmath08 kev . since june 2006 , we injected various amounts of charge in one experiment . the log is summarized in table 1 . in the cfci experiments , the test charge is injected to the row at @xmath25 ( @xmath26 ) , and hence @xmath27 . the reference charge should be equal to the original charge which does not suffer from the charge loss , and hence @xmath28 . then equation [ eq : cti_def ] can be written as @xmath29 we determined @xmath30 by measuring the ratio @xmath31 for each column . from equation 3 , we can obtain the relation in the cfci experiments as , @xmath32 the index @xmath24 and @xmath33 were derived by fitting equation [ eq : cfci_cti2 ] to the values of @xmath30 obtained with the cfci experiments with multiple amount of charge injections ( multiple @xmath8s ; log number 36 in table 1 ) . the mean and standard deviation of the best - fit @xmath24 of equation [ eq : cfci_cti2 ] averaged over each sensor are shown in figure [ fig : ctipow ] . the mean value of @xmath24 shows no time variation , and the time averaged values of xis 0 , 1 , and 3 are 0.31 , 0.22 , 0.15 for xis 0 , 1 , and 3 , respectively . as for xis 2 , there was only one data point , and we obtained @xmath34 . if a charge packet has a volume proportional to the number of electrons and is spherically symmetric , the probability that one electron encounters a charge trap is proportional to the cross section of the charge packet . in this case , we can expect @xmath35 . from equation [ eq : cfci_cti ] , the cti value is proportional to @xmath36 , and hence @xmath37 . thus the simple model is roughly consistent with the observed @xmath24 values . using the results of all the cfci experiments ( log number 16 in table 1 ) and above determined @xmath24 values , we then re - estimated @xmath22 and @xmath23 separately . in this process , we assumed @xmath38 is equal to @xmath39 which was estimated by the 6.4 kev line from the sgr c region to be 0.67 and 1.5 for the fi and bi ccd , respectively ( @xcite ) . from this @xmath22 , we can obtain the final value of @xmath40 . figure [ fig : baratuki ] shows an example of the distribution of @xmath40 in july 2006 . we can see significant column - to - column dispersion . figure [ fig : ctihenka ] shows the change of @xmath40 from july 2006 to september 2007 . we can see that the cti values of all columns increased , but the increasing rate was different from column to column . the results of figures 2 and 3 indicate that the cti correction at the column level is strongly required . in figure [ fig : ctijikanhatten ] , we show the column - averaged @xmath40 value as a function of time . since the cfci experiments were only sparsely conducted ( see table 1 ) , we interpolate the @xmath40 and @xmath41 values for the observations of inter - cfci epochs . as for the determination of the cti values before the first cfci experiment , see appendix . a cti correction , which is the conversion of @xmath42 to @xmath5 , is made with equation [ eq : cti_def ] , where @xmath40 and @xmath41 are calculated from equation 3 by using the @xmath22 , @xmath23 , and @xmath24 values determined in section 3 . we used the emission lines from the onboard calibration sources , the perseus cluster of galaxies , and the supernova remnant 1e0102.2@xmath437219 . we retrieved the data from the data archives and transmission system of isas / jaxa . all data were acquired with the normal full window mode and the 3@xmath123 or 5@xmath125 editing mode @xcite . we used the data with the asca grades of 0 , 2 , 3 , 4 , and 6 . as is mentioned in @xcite , a small fraction of the charge in a pixel is left behind ( trailed ) to the next pixel in the same column during the transfer . all data used in this paper were corrected for the trail phenomenon . the observations are summarized in table [ tab : objlog ] . the calibration source @xmath44fe produces the mn k@xmath45 line . the theoretical line center energy is 5895 ev @xcite . we used the data from august 2005 to april 2007 . this is one of the x - ray brightest clusters of galaxies in the sky . the x - ray spectrum is that of a thin thermal plasma with the strong k@xmath45 line of fe . the plasma temperature changes smoothly from @xmath46 4 kev to @xmath0 7 kev toward the outer region @xcite , and the center energy of the fe k@xmath45 triplet is almost constant ( @xmath06.56 kev at @xmath47 ) within this temperature range . its radius of @xmath48 can cover the entire field of view of the xis ( @xmath49 ) . thus this source is suitable for measuring the positional dependence of the energy scale . this is one of the brightest supernova remnants in the small magellanic cloud . with the spatial resolution of suzaku , it can be regarded as a point source . there are many bright emission lines originated from thermal plasma in the x - ray spectrum below 2 kev . these lines are resolved with the xmm - newton rgs , and the accurate energies of the line centroids are known @xcite . this object has been used by many instruments for the calibration in the low - energy band , and an empirical model to describe the spectrum has been established . we used this source as the energy - scale calibrator in the low - energy band . for the data of the perseus cluster of galaxies , we divided the imaging area into four regions along the _ acty _ axis , and extracted a spectrum from each region . then we fitted the spectra in the 57.3 kev band with a power - law model and a gaussian function , and obtained the center pulse height of the fe k@xmath45 line . figure [ fig : perseus ] shows the center pulse height as a function of @xmath13 . triangles and circles indicate the data before and after the cti correction , respectively . we can see no significant @xmath13 dependence after the cti correction , and this supports the validity of our correction . the goal is to determine a relation of @xmath5 and x - ray energy @xmath51 . from the ground experiments , we found that the @xmath5-@xmath51 relation can be expressed as a broken - linear function linked at the si - k edge energy of 1839 ev @xcite . we then determined the @xmath5-@xmath51 relation of each segment by using the lines of the calibration sources ( mn k@xmath45 line at 5895 ev ) and 1e@xmath50 ( k@xmath45 lines of o , ne and ne around 6501020 ev ) . we show the results after the cti correction and the @xmath5-@xmath51 conversion . figure [ fig : calpeak ] shows measured center energies of the mn k@xmath45 line as a function of time . each mark in the plot has an effective exposure of more than 60 ks . the mean values of the center energy are 5896.2 , 5895.4 , 5895.0 , and 5895.4 ev for xis 0 , 1 , 2 , and 3 , respectively . the deviation around the theoretical center energy ( 5895 ev ) is 7.8 , 4.4 , 6.6 , and 7.8 ev for xis 0 , 1 , 2 , and 3 , respectively . therefore , the time - averaged uncertainty of the absolute energy is @xmath52 0.1 % for the mn k@xmath45 line of the calibration sources . we also studied the time evolution of the deviation around the theoretical center energy , and the results are shown in figure [ fig : peakbunsanhenka ] . we can see that the deviation gradually increases with time . figure [ fig : sciofflowpeak ] shows the center energy of the o k@xmath45 line from the 1e@xmath50 data . the mean values of the center energy are 652.6 , 653.8 , 652.7 , and 652.8 ev for xis 0 , 1 , 2 , and 3 , respectively . the deviation around the center energy of the empirical model ( 653 ev ) is 1.4 , 1.4 , 2.3 , and 1.1 ev for xis 0 , 1 , 2 , and 3 . therefore , the uncertainty of the absolute energy is @xmath53 0.2% for the o k@xmath45 line of 1e@xmath50 . we examined the energy resolution in fwhm ( @xmath54 ) @xcite for each sensor ; @xmath54 is common to all segments . we expressed @xmath54 as @xmath55 where @xmath56 and @xmath57 are time dependent parameters and @xmath58 is the energy resolution determined by the ground experiments and obtained using equation 1 in @xcite . we determined @xmath56 and @xmath57 by using the time history of the calibration sources and 1e@xmath50 . the @xmath54 values obtained in this way is incorporated into the redistribution matrix file ( rmf ) . figure [ fig : segcolumn ] shows the energy resolution of the mn k@xmath45 line after the column - to - column cti correction . we also plot the results of the cti correction , where we used the cti values averaged over a segment ( the column - averaged cti correction ) . we can see that the energy resolution is greatly improved by the column - to - column cti correction . for example , the energy resolution in december 2006 was greatly improved from @xmath0230 ev to @xmath0 190 ev . on the other hand , with the column - averaged cti correction , the energy resolution is @xmath0230 ev and is not significantly improved . in figure [ fig : calsig ] , we compared the energy resolution of the mn k@xmath45 line with our rmf model . the deviation of the data points around our model is 5.6 , 4.9 , 3.4 , and 6.3 ev for xis 0 , 1 , 2 , and 3 , respectively . we have conducted the cfci experiments six times in orbit . the cti correction has been done with the cfci results . we calibrated the energy scale of the xis precisely using the onboard calibration sources and 1e@xmath50 . our calibration results have been applied to all the data obtained with the normal full window mode without the spaced - row charge injection . the results of the cfci experiments and the current calibration status are summarized as follows : 1 . we determined the cti1 and cti2 values of each column precisely based on the data of the cfci experiments . we also found that the pulse height dependence of the cti does not change with time . 2 . after the column - to - column cti correction , we determined the @xmath5-@xmath51 relation . we also modeled the time - dependent energy resolution . the uncertainty of the energy scale is @xmath59 0.2 % for the o k@xmath45 line ( @xmath0 0.65 kev ) of 1e@xmath50 , and @xmath59 0.1 % for the mn k@xmath45 line ( @xmath605.9 kev ) of the calibration sources . 4 . with the column - to - column cti correction , the energy resolution at 5.9 kev in december 2006 was greatly improved from @xmath0230 ev to @xmath0 190 ev . the authors thank all the xis members for their support and useful information . this work was supported by the grant - in - aid for the global coe program `` the next generation of physics , spun from universality and emergence '' from the ministry of education , culture , sports , science and technology ( mext ) of japan . m.o . , h.u . , and h.n . are financially supported by the japan society for the promotion of science . is also supported by the mext , grant - in - aid for young scientists ( b ) , 18740105 , 2008 , and by the sumitomo foundation , grant for basic science research projects , 071251 , 2007 . h.t . and k.h . were supported by the mext , grant - in - aid 16002004 . first , we determined the cti values of the segments a and d in august 2005 . we combined the data of the calibration sources from august 11 to 31 , 2005 , and obtained the @xmath6 of the mn k@xmath45 line . we also estimated @xmath5 at 5895 ev from the @xmath5-@xmath51 relation determined by the ground experiments , and obtained the ratio @xmath61 . from equation [ eq : cti_def ] , the ratio can be expressed approximately as @xmath62 , where @xmath63 is the mean transfer number of the calibration events ( typically @xmath0900 ) . we determined @xmath40 and @xmath41 at 5895 ev from @xmath61 with the @xmath39 ratio fixed to the values shown in section 3 . the @xmath40 and @xmath41 for other pulse - height values were calculated from equation [ eq : cti_ph ] , where we used the column - averaged and time - averaged @xmath24 values determined in section 3 . then for segments b and c , we took the average cti values of the segments a and d. we regard the cti values obtained in this procedure as those on august 11 , 2005 ( the day of the xis first light ) . note that these values are determined for each segment , not for each column .	the x - ray imaging spectrometer ( xis ) on board the suzaku satellite is an x - ray ccd camera system that has superior performance such as a low background , high quantum efficiency , and good energy resolution in the 0.212 kev band . because of the radiation damage in orbit , however , the charge transfer inefficiency ( cti ) has increased , and hence the energy scale and resolution of the xis has been degraded since the launch of july 2005 . the ccd has a charge injection structure , and the cti of each column and the pulse - height dependence of the cti are precisely measured by a checker flag charge injection ( cfci ) technique . our precise cti correction improved the energy resolution from 230 ev to 190 ev at 5.9 kev in december 2006 . this paper reports the cti measurements with the cfci experiments in orbit . using the cfci results , we have implemented the time - dependent energy scale and resolution to the suzaku calibration database .	6,363	273
oscillators providing stable sub-100 fs pulses in the near - infrared region around 1.5 @xmath0 m are of interest for a number of applications including infrared continuum generation @xcite and high - sensitivity gas spectroscopy @xcite . to date , the typical realization of such sources is based on a femtosecond er : fiber oscillator with an external pulse amplification . a promising alternative to such combination is a solid - state cr@xmath1:yag mode - locked oscillator @xcite . such an oscillator allows a direct diode pumping and possesses the gain band providing the few - optical cycle pulses . however , attempts to increase the pulse energy in a cr : yag oscillator is limited by its relatively small gain coefficient . because of the low gain , the oscillator has to operate with low output couplers and , thereby , the intra - resonator pulse energy has to be high . as a result , the instabilities appear @xcite . to suppress the instabilities in the negative dispersion regime ( ndr ) a fair amount of the group - delay - dispersion ( gdd ) is required . the resulting pulse is a relatively long soliton with reduced peak power . such a pulse is nearly transform - limited and is not compressible . a remedy is to use the positive dispersion regime ( pdr ) , when the pulse is stabilized due to substantial stretching ( up to few picoseconds ) caused by a large chirp @xcite . such a pulse is dispersion - compressible down to few tens of femtoseconds . for both ndr and pdr , the oscillator will eventually become unstable at high power . the main scenarios of the pulse destabilization have been identified with the multipulsing in the ndr @xcite and the cw - amplification in the pdr @xcite . it has been found , that the higher - order dispersions ( i.e. , the frequency dependent gdd ) and losses significantly modify the stability conditions @xcite . hence , the study of the stability conditions affected by both linear and nonlinear processes inherent in a mode - locked oscillator remains an important task . here , we present a study of the destabilization mechanisms of a cr : yag mode - locked oscillator , operating in both ndr and pdr . we put a special emphasis on the influence of the spectral dependence of gdd and losses on the oscillator stability . the cr : yag oscillator has been built on the basis of the scheme published in refs . the mode - locking and the dispersion control were provided by sesam and chirped - mirrors ( cms ) , respectively . the gdd of intra - resonator elements as well as the net - gdd are shown in fig . [ fig1 ] . as a result of the gdd variation of the 51-layer cms and the uncertainty of the sesam dispersion , the real net - gdd has some uncertainty , too ( gray region in fig . [ fig1 ] , _ a _ ) . a ) gdd of three sets of chirped mirrors cm ( as designed ) , the yag crystal , and the sesam . b ) the net dispersion of the resonator of cr@xmath1:yag oscillator . black line : as designed , grey area - uncertainty region due to the chirped mirrors.,height=302 ] selection of the different cm combinations allows over- and under - compensation of the dispersion . selecting a 2cm@xmath2 + 2cm@xmath2 allows stabilizing the oscillator at the 144.5 mhz pulse repetition rate and 150 mw average output power . the corresponding spectra shown in fig . [ fig2 ] have truncated profiles , that is typical for an oscillator operating in the pdr @xcite . a ) spectra of the cr : yag oscillator operating in the pdr at different values of intracavity pulse energy . b ) spectra of the cr : yag oscillator with different output couplers.,height=302 ] to study the stability limits of pdr , the numerical simulations based on the nonlinear cubic - quintic complex ginzburg - landau model @xcite have been realized . the evolution of the slowly varying field envelope @xmath3 can be described in the following way : @xmath4p - \kappa \varsigma p^2 \right\}a . \hfill\end{aligned}\ ] ] here @xmath5 is the propagation distance normalized to the cavity length @xmath6 ( i.e. , the cavity round - trip number ) , @xmath7 is the local time . the reference time frame moves with the pulse group - velocity defined at the reference frequency @xmath8 corresponding to the gain maximum at @xmath91.5 @xmath0 m . the term @xmath10 describes the action of the gain spectral profile in parabolic approximation . parameter @xmath110.028 is the saturated gain coefficient at @xmath8 , and it is close to the net - loss value at this frequency . parameter @xmath129.5 fs is the inverse gain bandwidth . the term @xmath13 describes the net - gdd action in the fourier domain . the term @xmath14 describes the action of the net loss spectral profile in the fourier domain . frequency @xmath15 corresponds to the transmission minimum of the output coupler at @xmath161.53 @xmath0 m . mw@xmath18 describes the self - phase modulation inside the active medium , @xmath190.05@xmath20 is the self - amplitude modulation parameter , @xmath210.6@xmath20 is the parameter defining saturation of the self - amplitude modulation with power @xcite . parameter @xmath22 is the difference between the saturated gain @xmath23 and the net loss at the reference frequency @xmath8 . it was assumed , that this parameter depends on the full pulse energy:@xmath24 , where @xmath25 corresponds to the full energy stored inside an oscillator in the cw regime @xcite . parameter @xmath26 equals to -0.03 . it was found , that the gdd decrease in the pdr results in the cw - amplification ( see the black curve in fig . [ fig3 ] ) . the cw - amplification appears in the vicinity of the spectral net - loss minimum , where the saturated net - gain becomes positive . the latter occurs because gain saturation decreases with gdd approaching to zero . simultaneously , the spectrum gets broader , which enhances the spectral losses and , thereby reduces the pulse energy . ( a ) - simulated spectra , ( b ) - intensity profiles in the positive dispersion regime . gdd corresponds to @xmath27 200 fs@xmath28 ( gray curves ) and 150 fs@xmath28 ( black curves).,height=302 ] when the gdd is spectrally dependent ( i.e. , there are the higher - order dispersions ) , the scenario of destabilization with an approaching of gdd to zero changes ( fig . [ fig4 ] , _ a _ ) . in this case , the chaotic oscillations of the peak power appear ( _ chaotic mode - locking _ the spectrum edges in such a regime become smoothed and the spectrum shifts to the local gdd - minimum . it was found @xcite , that the pdr stability is improved in the vicinity of the local minimum of the gdd ( i.e. , when the fourth - order dispersion is positive ) . [ fig4 ] , _ b _ shows the spectrum ( gray curve ) corresponding to the lower on gdd stability border of pdr . the spectrum is asymmetrical , `` m - shape '' , and the spectrum maximum is situated in the vicinity of the gdd local minimum . simulated spectra ( solid curves ) , with different net - gdd ( dashed curves ) . @xmath291.54 @xmath0 m , other parameters correspond to fig . [ fig3 ] , the case without spectrally - dependent losses.,height=302 ] in the ndr and in the absence of higher - order dispersions , the pulse can be stabilized only by a signifivant amount of the negative gdd ( @xmath30 - 9900 fs@xmath28 in our case ) . however , the contribution of higher - order dispersions can stabilize the pulse in the immediate vicinity of zero gdd ( black curves in fig . [ fig5 ] ) . the pulse is stable if even a part of spectrum is situated within the positive gdd range ( left picture in fig . [ fig5 ] ) . gdd spectral profiles ( dashed ) and corresponding simulated pulse spectra without ( black ) and with ( gray ) spectral dependence of the losses . the losses are assumed to have a local minimum at 1.53 @xmath0 m about -0.025 fs@xmath18 , height=226 ] however , the ndr stability in the vicinity of zero gdd is very sensitive to the spectral dependence of the losses . presence of such dependence leads to the multipulsing ( harmonic mode - locking , gray curves in fig . [ fig5 ] ) . as it was found @xcite , the source of multipulsing is the insufficiently saturated net - gain , which appears in the vicinity of zero gdd due to decrease of the pulse energy caused by the spectral losses . in analogy with the pdr , the spectral dependence of gdd can initiate the chaotic mode - locking , when the negative gdd approaches zero ( gray curve in fig . [ fig4 ] ) . the spectrum shifts from the region , where the local minimum of gdd is located . this behaviour is reverse in the comparison with that in the pdr . thus , the stability regions of pdr and ndr are disjointed by a region of chaotic mode - locking in the vicinity of zero gdd . just as the `` m - shaped '' spectra appear in the pdr , when the fourth - order dispersion leads to the gdd growth on the spectrum edges , the humps of the spectrum envelope exist in the ndr ( fig . [ fig6 ] ) . in contrast to the pdr , the local spectral maxima appear in the vicinity of local maxima of gdd . gdd spectral profile ( dashed ) and corresponding simulated pulse spectrum in the negative dispersion regime with modulated dispersion and no spectral dependence of the losses.,height=302 ] we have studied stability conditions of a cr : yag oscillator in both , positive and negative dispersion regimes . in the ndr , the oscillator exhibits strong tendency to harmonic mode - locking ( i.e. , multipulsing ) , which can be suppressed only by a significant amount of the negative gdd . the gdd value providing the multipulsing suppression depends on the gdd shape , so that the soliton - like regime can exist even if the part of the spectrum lies within the positive dispersion range . the spectrum maximum shifts in the region , where the local maximum of gdd is located . the presence of the spectrally - dependent losses enhances the multipulsing in the vicinity of zero gdd . when the gdd is frequency - dependent , the positive- and negative - dispersion single - pulse regimes are disjointed by the gdd ranges with chaotic and harmonic mode locking . in the pdr , approaching of the gdd to zero results in a cw - amplification or a chaotic mode - locking . the last regime appears if the gdd is frequency - dependent . the stabilizing factor of such frequency dependence is the presence of a local minimum ( for the pdr ) or maximum ( for the ndr ) of gdd in the vicinity of the pulse spectrum . e. sorokin , v. l. kalashnikov , s. naumov , j. teipel , f. warken , h. giessen and i. t. sorokina,``intra- and extra - cavity spectral broadening and continuum generation at 1.5 @xmath0 m using compact low - energy femtosecond cr : yag laser , '' _ appl . * b 77 * , pp . 197204 , 2003 . v. l. kalashnikov , e. sorokin , s. naumov , i. t. sorokina , v. v. ravi kanth kumar and k. george , `` low - threshold supercontinuum generation from an extruded sf6 pcf using a compact cr@xmath1:yag laser , '' _ appl . phys . _ * b 79 * , pp . 591596 , 2004 . j. mandon , g. guelachvili , n. picqu , f. druon and p. georges , `` femtosecond laser fourier transform absorption spectroscopy , '' _ opt . lett . _ * 32 * , pp . 16771679 , 2007 . s. naumov , e. sorokin , v. l. kalashnikov , g. tempea , i. t. sorokina , `` self - starting five optical pulse generation in cr@xmath1:yag laser , '' _ appl . b _ * 76 * , pp . 111 , 2003 . c. g. leburn , a. a. lagatsky , c. t. a. brown , and w. sibbett , `` femtosecond cr@xmath1:yag laser with 4 ghz pulse repetition rate , '' _ electron . _ * 40 * , pp . 805807 , 2004 . v. l. kalashnikov , e. sorokin , and i. t. sorokina , `` multipulse operation and limits of the kerr - lens mode - locking stability , '' _ ieee j. quant . _ * 39 * , pp . 323336 , 2003 . s. naumov , a. fernandez , r. graf , p. dombi , f. krausz , and a. apolonski , `` approaching the microjoule frontier with femtosecond laser oscillators , '' _ new journal of physics _ * 7 * , p. 216 v. l. kalashnikov , e. podivilov , a. chernykh , a. apolonski , `` chirped - pulse oscillators : theory and experiment , '' _ appl . _ b * 83 * , pp . 503510 , 2006 . v. l. kalashnikov , a. fernndez , a. apolonski , `` high - order dispersion in chirped - pulse oscillators , '' _ optics express _ , pp . 42064216 , 2008 . v. l. kalashnikov , e. sorokin , s. naumov , and i. t. sorokina , `` spectral properties of the kerr - lens mode - locked cr@xmath1:yag laser , '' _ j. opt _ * 20 * , pp . 20842092 , 2003 . s. naumov , e. sorokin , i. t. sorokina , _ osa trends in optics and photonics vol . 83 , advanced solid - state photonics _ * 83 * , p. 163 n. n. akhmediev , a. ankiewicz , _ solitons : nonlinear pulses and beams _ , chapman & hall , london , 1997 .	we analyze the influence of spectrally modulated dispersion and loss on the stability of mode - locked oscillators . in the negative dispersion regime , a soliton oscillator can be stabilized in a close proximity to zero - dispersion wavelength , when spectral modulation of dispersion and loss are strong and weak , respectively . if the dispersion is close to zero but positive , we observe _ chaotic _ mode - locking or a stable coexistence of the pulse with the cw signal . the results are confirmed by experiments with a cr : yag oscillator .	4,432	150
over the last decade several diagnostic diagrams have been proposed to quantify the contribution of star formation and agn activity to the infrared luminosity of infrared galaxies based on mid - infrared ( to far - infrared ) continuum slope , pah line - to - continuum ratio , pah to far - infrared luminosity ratio and the ratio of a high to a low ionization forbidden line such as [ nev]/[neii ] @xcite . however , none of these diagrams takes into account the effects of strong obscuration of the nuclear power source . with the advent of the infrared spectrograph ( _ irs _ ; * ? ? ? * ) on board the _ spitzer _ space telescope @xcite astronomers have been handed a powerful tool to study the 537@xmath0 m range for a wide range of galaxy types at an unprecedented sensitivity . this enables for the first time a systematic study of a large number of galaxies over the wavelength range in which amorphous silicate grains have strong opacity peaks due to the si o stretching and the o si o bending modes centered at 9.7 and 18@xmath0 m , respectively . here we will introduce the strength of the 9.7@xmath0 m silicate feature as a tool to distinguish between different dust geometries in the central regions of ( ultra-)(luminous ) infrared galaxies as part of a new diagnostic diagram and mid - infrared galaxy classification scheme . the results presented in this paper are based on both _ spitzer irs _ and _ iso - sws _ observations . the core sample is formed by the irs gto ulirg sample ( pid 105 ; j.r.houck pi ) , which comprises @xmath1100 ulirgs in the redshift range 0.02 @xmath2 0.93 . this sample is compared to samples of agn and starburst templates from the irs gto programs 14 @xcite and 96 and to selected agns from the gto programs 82 and 86 @xcite . additional irgs , lirgs and ulirgs were taken from program 159 ( ngc1377 ; * ? ? ? * ) and from the spitzer ddt program 1096 . iso - sws _ spectra of starburst nuclei were taken from @xcite . the _ spitzer _ observations were made with the short - low ( sl ) and long - low ( ll ) modules of the _ irs_. the spectra were extracted from droop level images provided by the spitzer science center ( 34 using pipeline version s11.0.2 , 131 using pipeline version s14.0 ) and background - subtracted by differencing the first and second order apertures . the spectra were calibrated using the _ irs _ standard stars hd173511 ( 5.219.5@xmath0 m ) and @xmath3dra ( 19.538.5@xmath0 m ) . after flux calibration the orders were stitched to ll order 1 , requiring order - to - order scaling adjustments of typically less than 10% . for all the spectra in our sample we have measured the equivalent width of the 6.2@xmath0 m pah emission feature as well as the strength of the 9.7@xmath0 m silicate feature and plotted the two quantities in a diagnostic diagram ( fig.[fig1 ] ) . the flux in the 6.2@xmath0 m pah emission band is measured by integrating the flux above a spline interpolated local continuum from 5.956.55@xmath0 m . the equivalent width ( ew ) of the pah feature is then obtained by dividing the integrated pah flux by the interpolated continuum flux density below the peak ( @xmath1 6.22@xmath0 m ) of the pah feature . the apparent strength of the 9.7@xmath0 m silicate feature is inferred by adopting a local mid - infrared continuum and evaluating the ratio of observed flux density ( f@xmath4 ) to continuum flux density ( f@xmath5 ) at 9.7@xmath0 m and defining @xmath6 for sources with a silicate absorption feature @xmath7 can be interpreted as the negative of the apparent silicate optical depth . given the great diversity among our mid - infrared galaxy spectra , there is no `` one - size - fits - all '' procedure to define the local continuum in all spectra . we therefore developed separate methods for three distinctly different types of mid - infrared galaxy spectra : continuum - dominated spectra , pah - dominated spectra and absorption - dominated spectra . these methods are illustrated in fig.[fig2 ] and described in its caption . we are forcing the local continuum to touch down at 14.014.5@xmath0 m , because the ism dust cross - section decreases between the two silicate peaks in this region . detailed radiative transfer calculations verify that our interpolation procedure properly reproduces the emission that would be generated by dust stripped of its silicate features ( sirocky et al . in preparation ) . the galaxy spectra in fig.[fig1 ] are classified into 9 different classes based on their 6.2@xmath0 m pah ew and 9.7@xmath0 m silicate strength . the parameter space covered by the various classes is indicated by shaded rectangles in fig.[fig1 ] . average mid - infrared spectra for the eight populated classes are shown in fig.[fig3 ] . the average spectra were constructed by normalizing all spectra to unity at 14.5@xmath0 m flux before the averaging process . in order to maximize the signal - to - noise ( s / n ) of the average spectra , low s / n spectra were discarded from the process . below we describe the 8 average spectra in the order ( bottom to top ) they are presented in fig.[fig3 ] . the ninth class , 3c , is not populated . the class 1a spectrum is characterized by a nearly featureless hot dust continuum with a very weak silicate absorption feature at 9.7@xmath0 m . the class 1b spectrum differs from the class 1a spectrum by clearly showing the family of pah emission features at 6.2 , 7.7 , 8.6 , 11.2 , 12.7 and 17.3@xmath0 m on top of a hot dust continuum . this hot dust continuum is nearly absent in the class 1c spectrum , allowing the pah emission features to dominate the mid - infrared spectral appearance . silicate absorption at 9.7@xmath0 m becomes noticable in the class 2c spectrum as an increased depth of the depression between the 69 and 1113@xmath0 m pah emission complexes . another marked difference with the class 1c spectrum is the steepening of the 2030@xmath0 m continuum and the appearance of an 18@xmath0 m silicate absorption feature . in the class 2b spectrum the pah features appear weaker than in the class 2c spectrum . in addition , the spectrum starts to show a 6@xmath0 m water ice and a 6.85@xmath0 m aliphatic hydrocarbon absorption band . in the class 3b spectrum these absorption features reach their maximum depths , while the 9.7 and 18@xmath0 m silicate features continue to increase in depth up to class 3a . absorption features of _ crystalline _ silicates @xcite appear in the spectra of the classes 3b and 3a at 16 , 19 and 23@xmath0 m . equivalent widths of pah emission features and emission lines decrease from class 2b to 3b and 3a . note especially the change in shape of the 7.7@xmath0 m pah feature as it first broadens and then disappears going from class 2b to 3b and 3a . finally , the class 2a spectrum differs from the class 3a spectrum mainly by a clearly lower apparent depth of the 9.7 and 18@xmath0 m silicate features . note that the spectral structure in individual spectra may differ substantially from the average properties of the classes . in fig.[fig3 ] this is represented by the 1-@xmath8 dispersion ranges around the individual average spectra . the galaxies shown in fig.[fig1 ] are color - coded according to their galaxy classification . however , their positions may differ from their original assignments , as in fig.[fig1 ] the 6.2@xmath0 m pah ew has been corrected for the effect of 6@xmath0 m water ice absorption on the 6.2@xmath0 m continuum through substitution of the observed 6.2@xmath0 m continuum by the 6.2@xmath0 m continuum that was defined to infer the 9.7@xmath0 m silicate strength . the ice - correction is justified assuming that the ice is part of the obscuring medium _ behind _ the pah emitting region . the correction is applied only to spectra clearly showing the imprint of water ice absorption , as found among the spectra of classes 2b , 2c , 3a and 3b . the extent of the correction for individual spectra is indicated in fig.[fig1 ] by horizontal dotted lines . the ulirgs , seyferts , quasars and starburst galaxies shown in fig.[fig1 ] are not distributed randomly through the diagram . instead , the galaxies appear to be distributed along two branches . one extending horizontally from the positions of the prototypical seyfert-1 nucleus ngc4151 @xcite to the prototypical starburst nucleus ngc7714 @xcite , the other branch extending diagonally from the prototypical deeply obscured nucleus of ngc4418 @xcite to the starburst nuclei of m82 @xcite and ngc7714 . very few sources are found above and to the right of the diagonal branch and in between the two branches ( the class 2a domain ) . the three extremes distinguish between agn - heated hot dust spectra ( class 1a ) , pah - dominated spectra ( classes 1c and 2c ) and absorption - dominated spectra ( class 3a ) . the galaxies lined up along the two branches in between the extremes show signatures of both extremes to varying proportions . the galaxies along the horizontal branch may be thought of as combinations of agn and starburst activity , while the galaxies along the diagonal branch may be thought of as intermediate stages in between a fully obscured galactic nucleus and an unobscured nuclear starburst . the latter is illustrated by the two dotted lines in fig.[fig1 ] , which represent mixing lines between the spectrum of ngc4418 on one extreme and either ngc7714 or m82 on the other . the large majority of galaxies on the diagonal branch fall in between these mixing lines . one notable exception is arp220 . relatively few galaxies are found in the section between the two branches : the domain of the class 2a sources . spectroscopically , there are several simple scenarios which will populate this class from adjacent classes . a class 3a galaxy can be turned into a class 2a galaxy by adding a featureless hot dust continuum to the class 3a absorbed continuum spectrum . the main effect of this continuum dilution is the filling in of the deep 9.7@xmath0 m silicate feature , resulting in a less pronounced feature . increasing the dilution not only further decreases the depth of the silicate feature but also increases the continuum at 6.2@xmath0 m , resulting in both a less negative silicate strength and a lower 6.2@xmath0 m pah ew . the effect of this type of continuum dilution is illustrated in fig.[fig1 ] by a curved arrow . alternatively , a class 1a galaxy may be turned into class 2a galaxy by passing its mid - infrared spectrum through a foreground cold dust screen . the path that a galaxy will take is straight up , since the pah ew is not affected by cold foreground extinction . the presence of foreground dust or dust mixed in within the pah emitting region may also explain the difference between class 1c and 2c starburst galaxies , and the difference between the classification of the prototypical starburst galaxies ngc7714 and m82 in particular . in the latter case this may be primarily an orientation effect , as the nuclear starburst in ngc7714 is observed face - on , whereas the starburst ring in m82 is seen under an inclination of @xmath180 degrees . finally , the complete absence of galaxy spectra classified as 3c must imply that strong , pure foreground , cold dust obscuration is not common in starburst nuclei . the obscuring dust instead contributes to the continuum underlying the 6.2@xmath0 m pah feature , reducing its ew . we have constructed a diagnostic diagram of two mid - infrared spectral quantities , the equivalent width of the 6.2@xmath0 m pah feature and the apparent 9.7@xmath0 m silicate strength , for the purpose of classifying infrared galaxies according to their mid - infrared spectral shape ( fig.[fig1 ] ) . the large majority of galaxies ( @xmath990% ) are dispersed around two branches : a horizontal one extending from continuum - dominated spectra to pah - dominated spectra and a diagonal one spanned between absorption - dominated spectra and pah - dominated spectra . seyfert galaxies and quasars ( square plotting symbols in fig.[fig1 ] ) are found exclusively on the horizontal branch characterized by the absence of pronounced silicate features . starburst galaxy spectra ( triangular plotting symbols ) are concentrated toward the extreme right tip of the two branches , the locus of pah - dominated spectra . of the ulirgs and hylirgs in our sample ( round plotting symbols ) about a quarter occupy the same space as the agns and starbursts in our sample . the large majority , however , are distributed along the diagonal branch , characterized by increasingly apparent silicate absorption and less pronounced pah emission features , with deeply obscured galactic nuclei such as ngc4418 @xcite and ngc1377 @xcite as the only non - ulirg , non - seyfert end members on this branch . note that low - metallicity galaxies ( not included in this study ) would be distributed along the horizontal branch . the existence of two distinct regimes of apparent silicate absorption , as suggested by the two distinct branches in fig.[fig1 ] , is intriguing and likely points to clear differences in the nuclear dust distribution between galaxies on the two branches . clumpy distributions produce only shallow absorption features @xcite , and are the likely explanation for the horizontal branch . the deep silicate absorption found on the slanted branch requires the nuclear source to be deeply embedded in a smooth distribution of dust that is both geometrically and optically thick @xcite . the transition from clumpy geometry to one dominated by a smooth dust distribution is a possible explanation for the sources located between the branches . the distribution of ulirg spectra along the full length of both the horizontal and the diagonal branch once again illustrates the diverse nature of the ulirg family . it , however , also raises the issue of ulirg evolution . for instance , what evolutionary path has taken iras08572 + 3915 to its extreme class 3a position in which the central power source is deeply buried ? did the interaction start with both nuclei classified as class 1c/2c starburst galaxies , which then gradually moved up the diagonal branch as the interaction stengthened and more dust accumulated on the remnant nuclei ? and how will iras08572 + 3915 evolve from there , once the obscuring screen breaks up and the hidden power sources are revealed ? in case of a hidden starburst the source likely will move back diagonally toward the starburst locus . however , if the dominant power source is an agn , will the source cross into the sparsely populated class 2a regime , evolving directly toward the agn - dominated class 1a regime , or will it first undergo a starburst before settling somewhere along the horizontal branch ? the sparse population of the class 2a domain either indicates that the crossing time is brief with few ulirgs caught in transformation or that deeply obscured ulirg nuclei mostly evolve to starbursts ( first ) . in a more extensive paper ( spoon et al . in preparation ) we will present correlations of additional parameters within the diagram ( e.g. mid - ir line ratios , l@xmath10 , l@xmath11/l@xmath10,absorption features , nuclear separation , optical classification ) . the authors wish to thank jernimo bernard - salas , nancy levenson , hlne roussel , matthew sirocky , alexander tielens and dan weedman for discussions , and vandana desai for help with the data reduction . support for this work was provided by nasa through contract number 1257184 issued by the jet propulsion laboratory , california institute of technology under nasa contract 1407 . hwws was supported under this contract through the spitzer space telescope fellowship program .	we present a new diagnostic diagram for mid - infrared spectra of infrared galaxies based on the equivalent width of the 6.2@xmath0 m pah emission feature and the strength of the 9.7@xmath0 m silicate feature . based on the position in this diagram we classify galaxies into 9 classes ranging from continuum - dominated agn hot dust spectra and pah - dominated starburst spectra to absorption - dominated spectra of deeply obscured galactic nuclei . we find that galaxies are systematically distributed along two distinct branches : one of agn and starburst - dominated spectra and one of deeply obscured nuclei and starburst - dominated spectra . the separation into two branches likely reflects a fundamental difference in the dust geometry in the two sets of sources : clumpy versus non - clumpy obscuration . spectra of ulirgs are found along the full length of both branches , reflecting the diverse nature of the ulirg family .	4,586	274
the observable properties of active galactic nuclei ( agn ) and black hole x - ray binaries ( bhxrbs ) are consequences of accretion on to a black hole at a variety of rates , in a variety of ` states ' , and within a variety of environments . the major difference between the aforementioned classes of object is the black hole mass . bhxrbs typically have a black hole mass @xmath010m@xmath5 while for agn it is @xmath6 . theoretically , the central accretion processes should be relatively straightforward to scale with mass , and this is supported by several observed correlations . these include a relation between the x - ray and radio luminosities and the black hole mass ( merloni , heinz & di matteo 2003 ; falcke , krding & markoff 2004 ) , and between x - ray variability timescales , mass accretion rate and mass ( mchardy et al . more quantitative similarities between accretion ` states ' and radio jet production have also been demonstrated ( krding , jester & fender 2006 ; for the current picture of accretion states in bhxrbs and their relation to radio jets see fender , belloni & gallo 2004 ) . studying the delays between different emission regions gives us a further handle on the scalability of black hole accretion , as signals propagate from , for example , the accretion flow to the jet . variability studies have so far shown that a correlation exists between the x - ray and optical emitting regions of both bhxrbs and agn , typically reporting small lags , which are consistent with at least some of the optical variations being due to x - ray heating of the disc @xcite . a recent study by @xcite has shown that a correlated time lag of @xmath0 100 ms exists between the x - ray and ir regions ( ir lagging x - rays ) for the bhxrb gx339 - 4 , indicating a close coupling between the hot accretion flow and inner regions of the jet . in the case of the bhxrb grs 1915 + 105 a variable x - ray to radio lag of @xmath7 mins ( radio lagging x - ray ) has been measured @xcite . discrete ejection events have been resolved in both the agn 3c120 @xcite and grs 1915 + 105 @xcite . the linear scaling with mass of the characteristic timescale around a black hole means that there are advantages to studying each class of object . in bhxrbs we can track complete outburst cycles , from the onset of disc instabilities through major ejection events , radio - quiet disc - dominated states , and a return to quiescence , on humanly - observable timescales ( typically years ) . for a typical agn the equivalent cycle may take many millions of years . however , for an agn we are able to resolve individual variations of the source on time - scales that are comparable to or shorter than the shortest physical time - scales in the system ( e.g. the dynamical time - scale ) , something which is currently impossible for bhxrbs . in ` black hole time ' we are able to observe the evolution of sources in fast - forward for bhxrbs and in detailed slow - motion for agn . in this paper we present the results of a long term ( @xmath8 years ) regular monitoring campaign in the x - ray and radio bands of the low luminosity active galactic nucleus ( llagn ) ngc 7213 . previous x - ray studies show that ngc 7213 is accreting at a low rate @xmath9 l@xmath2 @xcite . the hard state in bhxrbs is typically observed at bolometric luminosities below @xmath10 eddington , and seems to be ubiquitously associated with a quasi - steady jet . above @xmath10 , sources can switch to a softer x - ray state , the jets are suppressed @xcite ; furthermore transition to this softer state is usually associated with major transient ejection events . as ngc 7213 is considerably below l@xmath11 1% we therefore consider it a good candidate for comparison with other bhxrbs in the low / hard state . if we consider agn to be ` scaled up ' versions of bhxrbs by exploring the time lag between the x - ray and radio emitting regions we can compare , contrast and hopefully relate the accretion and jet production scenarios for agn and bhxrbs . a correlation has been established by @xcite and gallo et al . ( 2003 , 2006 ) relating the radio luminosity ( @xmath12 ) and x - ray luminosity ( @xmath13 ) for bhxrbs in the low / hard and quiescent states , where @xmath14 . @xcite - hereafter mhdm03 and @xcite extended the bhxrb relationship using two samples of agn to form the ` fundamental plane of black hole activity ' . by accounting for the black hole mass ( m ) the relationship @xmath14 has been extended to cover many orders of magnitude in black hole mass and luminosity . further refinements were made to the fundamental plane by @xcite - hereafter kfc06 , using an augmented and updated sample to examine the fitting parameters . throughout this paper we define the ` intrinsic ' behaviour of agn and bhxrbs as multiple measurements ( in the radio and x - ray ) of the _ same _ source . we define the ` global ' behaviour as single ( or average ) measurements of _ multiple _ sources , both with respect to the fundamental plane . for the bhxrbs in the low / hard state the relationship described above has not only been established globally but in some cases intrinsically , i.e gx 339 - 4 , v404 cyg and a small number of other systems have been shown to move up and down the correlation seen in the fundamental plane @xcite . however , in recent years an increasing number of outliers have been found below the correlation , i.e. less radio - loud then expected ( @xcite ; @xcite ; @xcite ) as well as some sources which move in the plane with a different slope ( e.g @xcite ) . to date the correlation found from the fundamental plane has only been measured globally for agn , not intrinsically . note , with respect to the global measurements of the agn population , the specific measurements of the radio and x - ray flux used in the correlation are sometimes taken at different times and thus could be a source of error in the correlation @xcite . as well as establishing the time lags , another goal of this work , is to establish , through quasi - simultaneous observations the intrinsic relationship between @xmath12 , @xmath13 and m observed in an llagn and its relevance to the fundamental plane of black hole activity . we use the mhdm03 and kfc06 samples for comparison both with an updated bhxrb sample taken from fender , gallo & russell ( 2010 ) - hereafter fgr10 . we also explore the possible scatter in agn data points away from the fundamental plane and place limits on this deviation . ngc 7213 is a face - on sa galaxy hosting a seyfert 1.5 nucleus located at a distance of 25 mpc assuming h@xmath15=71kms@xmath16mpc@xmath16 . low ionisation emission lines are observable in its nuclear spectrum , also making it a member of the liner class @xcite . the radio properties of ngc 7213 are intermediate between those of radio - loud and radio - quiet agn . previous radio studies at 8.4 ghz have not resolved any jet emission from the nucleus on scales 3 mas to 1 arcsec ( @xcite , @xcite,@xcite).the long baseline array ( lba ) is an australian six station vlbi instrument @xcite ; lba observations at 1.4 and 0.843 ghz have shown some evidence for larger scale emission that could be due to jet - fed radio lobes @xcite . however , as the nucleus remains unresolved it could be that the jet is oriented to some degree in the direction of the observer . as suggested by @xcite , the lower frequency emission could possibly be associated with a ` kink ' in the jet at a larger distance . this could be consistent with a general model proposed by @xcite where radio - intermediate objects are in fact radio -quiet objects whose jets are to some extent aligned in the observers direction and relativistically boosted . previous x - ray studies have so far failed to show significant evidence for a soft x - ray excess ; compton reflection ; or a broad fe k@xmath17 line in the ngc 7213 spectrum e.g. see @xcite , @xcite , @xcite . a narrow fe k@xmath17 line is observed and @xcite show evidence that it is produced in the broad line region ( blr ) . @xcite also report that the expected uv bump is either absent or extremely weak . the weakness or absence of these signatures suggests that the inner accretion disc is missing , perhaps replaced by an advection dominated accretion flow ( adaf ) ( as suggested by @xcite ) or similar hot flow , consistent with the low / hard state interpretation of ngc 7213 . @xcite estimate the black hole mass of ngc 7213 using the nuclear velocity dispersion calculation of @xcite and the velocity - dispersion versus black - hole - mass relationship of @xcite to obtain @xmath18 . bi - weekly monitoring was obtained with the australia telescope compact array ( atca ) . the interferometer setup was such that 128 channels of 1 mhz bandwidth were used to form two continuum channels centred at 4.8 and 8.4 ghz respectively . the radio observations have been reduced using the miriad radio reduction package @xcite . flux and bandpass calibration was achieved using ( in most cases ) pks j1939 - 6342 ( b1934 - 638 ) and for the phase calibration pks j2218 - 5038 ( b2215 - 508 ) . a variety of fitting techniques were then tested to extract the flux density of the source from the observations . these included testing point and gaussian fitting to the source in the image and uv plane . as atca is an east - west array and observations were short in duration ( @xmath02 hours ) an elongated synthesised beam was typically produced . image - plane fitting often failed to converge and/or produced non - physical results . therefore fitting was discarded in the image plane for the more reliable uv plane method . the final radio light curves and spectral indexes ( between the radio bands only ) are shown in figure [ lightcurve ] . to complement the snapshot observations a full 12 hour integration was performed on 2008 june 29 ; the purpose of which was to explore the polarisation properties of the jet , and to detect any faint extended structure . additional steps were taken to ensure adequate polarisation calibration was performed . this observation was also used to make a deep image of the source ; beyond the 0.5@xmath19 beam no weak extended radio emission was observed above a noise level @xmath20=78@xmath21j at 4.8 ghz . daily monitoring was obtained using 1 ksec snapshots obtained with the rossi x - ray timing explorer ( rxte ) proportional counter array ( pca ) , allowing a long - term light curve to be obtained in the 2 - 10 kev band . the data were reduced using ftools v6.8 , using standard extraction methods and acceptance criteria . the background was calculated from the most recent background models which corrects for the recent problems with the @xmath22 saa ( south atlantic anomaly ) history file . the final 2 - 10 kev fluxes were calculated by fitting a power law to the observed spectra . this allows us to take into account changes in the instrumental gain over the duration of the monitoring . the final un - binned x - ray light curve is shown in figure [ lightcurve ] and is used in all subsequent analysis . versus time . center bottom : radio flux @xmath23 versus time . the errors are too small to show on both radio flux figures but are typically @xmath00.4 mjy / beam . bottom : spectral index ( 8.4 - 4.8 ghz ) @xmath17 versus time where @xmath24 . ] figure [ lightcurve ] shows x - ray flux ( top panel ) , radio flux ( middle panels ) and radio spectral index @xmath17 where @xmath25 ( bottom panel ) versus time . the x - ray light curve shows a general decrease in flux until mjd 54600 . two distinct x - ray flares are observed at mjd 53920 and mjd 54390 , both appear to be correlated with events in the radio bands . we use the discrete correlation function ( dcf ) method of @xcite to calculate the cross - correlation coefficients between the entire x - ray and radio bands to find the lag . to calculate the centroid lag @xmath26 , we use a weighted mean of the positive lags above an 85% threshold of the peak dcf value . we use @xmath26 , rather than the peak value @xmath27 , because the centroid has been shown to better represent the physical lag @xcite . it has been shown that it is difficult to interpret @xmath27 as a physical quantity , and if it is used to calculate the lag , it usually offers an underestimate when compared with @xmath26 @xcite . the centroid width is calculated as [ cols="^,^,^,^,^",options="header " , ] the long term radio variability at 8.4 ghz is plotted in the lower panel of figure [ archiveplot ] . the flux values were taken from a variety of instruments and publications ( which are referenced on the plot ) . in order of increasing time the first point on the plot was taken from @xcite using the parkes - tidbindilla interferometer ( pti ) . then near simultaneous data points were taken with the vla @xcite and then atca @xcite . the flux from the long baseline array ( lba ) observation by @xcite was the last point plotted prior to the atca monitoring which is presented in this paper . when comparing the flux expected from the start of our atca monitoring with the lba data point , the lba point seems too low . the core is un - resolved with all of the radio telescopes described above . the pti observations constitutes the first data point in figure [ archiveplot ] and the lba the last ( before atca monitoring ) . note , the parkes - tidbindilla baseline is part of the lba network . @xcite comment that in the lba observations , no decrease in flux was observed with respect to baseline length i.e it was not resolved ; they conclude from this that the decrease in flux with respect to the other observations is accurate . however , both the pti and the lba are not sensitive to the same spatial resolutions as the vla and atca ( the largest atca baseline is 6 km and the shortest lba baseline used in the archival observation is 113 km ) . it is possible that some flux ( on atca equivalent baselines ) is not accounted for : which could plausibly give a decrease in flux i.e indicating brighter , larger scale structure . the logical consequence of this is that the pti data point is also missing some flux as it samples on a singular baseline of 275 km . we will discuss these measurements within the context of the fundamental plane and attempt to draw some conclusions in section 5.2 a full 12 hour synthesis atca observation was performed on 2008 june 29 at 8.4 and 4.8 ghz . the purpose of this observation was to perform a reliable polarisation calibration to ascertain an accurate measurement of the percentage linear polarised ( lp ) flux from the source . @xcite showed that from a sample of 11 llagn the mean lp was @xmath28 . @xcite from a survey of 22 blazars report a mean lp @xmath29 with a standard deviation @xmath30 . at the time of observation the polarisation of ngc 7213 was @xmath31 at 4.8 and 8.4 ghz respectively , which is more consistent with the reported value of typical llagn and not blazars . the percentage polarisation was calculated at other epochs , however without a full 12 hour synthesis parallactic angle calibration the values were often suspect and will not be presented here . we have shown that a weak but statistically significant delay exists between the x - ray and radio emitting regions , with the radio lagging behind the x - ray . a number of models have been proposed to explain and interpret the x - ray to radio lag in both bhxrbs and agn . the most notable of these are the ` internal shock model ' @xcite and a ` plasmon model ' @xcite . we will briefly explore these models and comment on the relevance - if any - to our data . we first consider a plasmon model . after the accretion mechanism(s ) have pushed matter into the jet , an adiabatically expanding initially self - absorbed synchrotron emitting plasmon travels at relativistic speeds down the path of the jet . at a given time and frequency the matter becomes optically thin and is ` detectable ' at that given frequency ( in this case either 8.4 and 4.8 ghz ) . if the jet is fed at a constant mass rate , density and velocity the delay time for material to become optically thin will be constant . if these parameters are not constant the delay between x - ray and radio will be variable ( e.g. see @xcite ) assuming that the disk and the jet are indeed coupled @xcite . note , the higher frequency radio emission ( 8.4 ghz ) will become optically thin first , thus a delay from 8.4 to 4.8 ghz is always expected . another model for explaining the emission seen in jets is the ` internal shock model ' @xcite . the synchrotron lifetime of an emitting region is too short to adequately explain the scales of jets observered in agn . these emitting regions - commonly referred to as ` knots ' - are often displaced from the central core emission . localised shocks within these regions are needed to explain the time - scales of variation observed @xcite . jet shock scenarios have also been used to model the common flat spectrum jet observered in bhxrbs and agn ( e.g. see @xcite ) . figure [ lightcurve ] shows the evolution of spectral index ; indicating that during a flare event @xmath17 increases ( @xmath32 ) , flattens and steepens ( @xmath33 ) shortly afterwards . the initial re - energisation given by a shock would push / compress the plasma into the optically thick regime ; subsequently this process moves the material into the optically thin regime . the variable time lag seen in the dcf could be interpreted as the time taken for newly injected matter to ` catch up ' with older matter expanding adiabatically in the jet to shock and produce a flare . the lag would be dependent on the lorentz factor of the newly injected material which in turn is related to the accretion rate @xcite . although both of these models can adequately explain the variations observed in our light curves , we can not completely disentangle them . it is possible that both of these models are in some part responsible for the behaviour . as the jet remains unresolved we can not identify an area of localised shocks or indeed discrete resolved events that we could use to support a particular model . to explore - in a very simplistic sense - how the time lags seen in bhxrbs scale up to agn we offer two comparisons ; that of a simple mass scaling and also mass - eddington ratio scaling . for example , discrete resolved ejection events have been seen in grs 1915 + 105 with a variable time lag between x - ray and radio ( ghz ) , with the clearest examples of events having lags of @xmath34 20 - 30 mins @xcite . taking the characteristic timescale of x - ray - radio lags measured in grs 1915 + 105 and scaling up to ngc 7213 with mass only ( assuming the mass of grs 1915 + 105 as @xmath0 10m@xmath5 and the mass of ngc 7213 as 9.6 @xmath35m@xmath5 and using @xmath36 ) we infer @xmath37 2 @xmath38 days , much longer than we observe . grs 1915 + 105 is however accreting at a much higher rate , therefore including a scaling by the ratio of the eddington luminosities ( using @xmath39 and taking @xmath40 ) we find @xmath41 140 days , reasonably comparable to our measured lag . the difficulty in comparing the actual x - ray - radio time lag measured for ngc 7213 , and that of the grs 1915 + 105 scaled time lag is that the radio emitting regions being probed are significantly different : in a broader view the spectral energy distributions are different . in fact until the structure of jets and how it scales with accretion rate and mass at a given frequency are better understood , such attempts at quantitative comparison are of marginal value , although the qualitative comparison is valuable . the fundamental plane of black hole activity @xcite shows that the correlation @xmath42 holds over many orders of magnitude of x - ray and radio luminosities , and black hole masses . figure [ full_plane ] shows ; at high luminosities a sample of agn taken from mhdm03 , and at the low luminosities an updated bhxrb sample taken from fgr10 . the updated bhxrb x - ray data were observed in the 0.5 - 10 kev energy range , therefore careful steps were taken ( using webpimms ) to correctly convert into the 2 - 10 kev range to compare with the mhdm03 and ngc 7213 data points . we plot on figure [ full_plane ] the best fit parameters @xmath43 log @xmath44 log @xmath45 as defined by mhdm03 . note , we do not re - fit the best fit line for the updated bhxrb sample . we paired the x - ray and radio data points to calculate the correlation by finding the closest x - ray point ( in time ) , to the radio , because the x - ray sampling was more frequent . the average for ngc 7213 sits well on the predicted correlation . the right hand panel of figure [ full_plane ] shows a close up of the ngc 7213 data . a least squared best fit for the ngc 7213 data points only is shown and parametrised by @xmath46 log @xmath47 ( the constant includes the mass term in the context of mhdm03 ) . we include in figure [ full_plane ] the correlation found from the archival x - ray and radio data ; however , as they were not simultaneous we do not include them in the best fit . we paired the x - ray and radio data points to calculate the correlation according to the closest in time . if either two x - ray , or two radio points were very close in time , both are included in the correlation . also note that these data were calculated between 8.4 ghz and 2 - 10 kev and not 4.8 ghz . assuming that the same correlation ( i.e @xmath0 flat spectrum ) holds between the 8.4 and 4.8 ghz observations the plot seems to show that two of the archival points do not follow the trend predicted by the fundamental plane . to assess whether the deviation / scatter in the archival points ( and indeed the other mhdm03 agn ) away from the fundamental plane can be explained due to a delay in sampling the x - ray and radio fluxes ; we take the fraction rms scatter in the rxte / atca x - ray and radio data and plot this errorbar with the archival points . we find the fractional x - ray rms to be @xmath48=26% and the fractional radio rms @xmath49=11% . the full light curve presented in this paper spans @xmath0 1000 days and the longest change in time between the archival observations is @xmath0 500 days ( see table [ achival_table ] ) . although this method only offers an estimate of the error between sampled data points , it does allow us to assess outliers . within this framework it appears that the first ( pti correlation ) and last ( lba correlation ) data points that were used in the correlation are outliers : the atca and vla correlation points sit close to the best fit . even when taking into account the full rms from the 1000 day light curve the error bars do not bring the data points close to the mhdm03 best fit . we have also speculated earlier that there could be a certain amount of missing flux associated with these long baseline observations . moving both the pti and lba points up by some set amount will still leave the pti point away from the correlation . although applying the intrinsic variation of the source to the archival points can not bring all points onto or close to the correlation ; the scatter in these points is within that permitted by the other mhdm03 points shown on the plot . by using the fractional rms of x - ray and radio variability from our study we have placed a constraint in the deviation of the archival data points away from the fundamental plane . these constraints suggest that other forms of scatter ( apart from bad sampling and missing flux ) could be affecting the data . as summarised in kfc06 other forms of scatter could be attributed to beaming , source peculiarities and spectral energy distribution - but note , these should remain relatively constant for the @xmath50 source . in figure [ kfc ] we plot a llagn @xmath51 sample and best fit parameters taken from kfc06 with the updated bhxrb sample . note , the x - ray data in the kfc06 sample are taken in the 1 - 10 kev band . as the ngc 7213 data points are in the 2 - 10 kev band we assume a photon index of @xmath52=1.8 and add a correction factor to the ngc 7213 data points to make them comparable . we apply a similar correction to the bhxrb sample . from this plot it is clear that the ngc 7213 data points are positioned slightly above the best fit line . considering the postion of ngc 7213 on the fundamental plane with respect to other llagn ( see figure 6 ) , we calculate its radio loudness parameter to assess the differentiation . we calculate the radio loudness parameter @xmath53 ( where @xmath54 and @xmath55 are the radio and optical luminosities ) ; we use a @xmath56 band magnitude of 16.3 , and find the optical flux @xmath57 using @xmath58log@xmath59 @xcite : giving @xmath60 = 134.8 . in this scheme radio - loud sources are typically defined as having an @xmath60 parameter @xmath61 10 , while radio - quite range between 0.1 - 1 @xcite . using the alternative radio loudness parameter of @xcite which utilises the x - ray instead of optical luminosity , @xmath62 , we find log@xmath63 -3.28 . @xcite show that for a sample of low - luminosity seyfert galaxies log@xmath64 while for a sample of low - luminosity radio galaxies ( llrgs ) log@xmath65 . therefore , with respect to the x - ray radio loudness , ngc 7213 is only slightly higher than that of a sample of low - luminosity seyfert galaxies ; while under the standard definition of radio loudness it is indeed radio loud . these results are consistent with the position of ngc 7213 on figure [ kfc ] . as was discussed earlier in this paper there is an apparent time lag between events in the x - ray and radio . therefore comparing the fitting parameters found from the ngc 7213 data with the mhdm03 relationship without correcting for the lag might give rise to errors as we are not matching the correct data points . the width in the cross - correlation peaks shows that the time lag is variable . thus , for example , the two radio flares could have different lag times associated with them . therefore shifting the entire radio light curve back by a set amount to match the x - ray could still give a scatter . to simplify this problem we separated out the data for the first flare only because we have a more accurate measurement of the lag in this specific case . we then shifted the radio data -35 days which was the time lag measured for this singular flare using the dcf at 4.8 ghz ( see table [ lag_table ] ) . the top panel of figure [ shifted ] shows the uncorrected data on the mhdm03 plot while the middle panel shows the corrected data ; for completeness the bottom panel shows all radio data points shifted back . for the first flare correcting the data appears to reduce the scatter and increase the gradient more in line with the mhdm03 best fit . it is now described by @xmath66 log @xmath67 . to check the statistical significance of this we measured the gradient for a variety of shifts . from 0 - 25 days the gradient gradually steepens until it gets close to 1 ( giving a coefficient of @xmath0 0.6 ) . from 30 - 50 days it plateaus @xmath0 1 and from 50 days the gradient decreases towards 0 . thus it does appear that moving the flare back by the amount given from the dcf function does seem to better represent the data with respect to the mhdm03 fit . however , it should be noted that as this is a log / log plot , measuring gradients from such a small range of values should be treated with care . it would , however , be interesting in future studies to assess the importance of this parameter . we have used the australian telescope compact array and the rossi x - ray timing explorer to conduct a long term study of agn variability in the llagn ngc 7213 . we have used the cross - correlation function to show that a complex and weakly significant correlated behaviour exists between the x - ray and radio emitting regions . although the statistics only show a weak correlation , this study is the first definitive campaign to probe this type of behaviour . we have shown that ngc 7213 sits well on the predicted fundamental plane of black hole activity plot when compared with the mhdm03 sample . however we have shown that when comparing ngc 7213 with a revised bhxrb and llagn sample that the data points are above the expected correlation ; which is however consistent with the calculated radio and x - ray loudness parameters . we have also shown some support that by correcting for the time lag between events in x - ray and radio the gradient of data points agree better with the best fit derived from the mhdm03 sample on the fundamental plane . m.e.bell would like to thank sera markoff , anthony rushton and sadie jones for their useful comments and discussion . the australia telescope is funded by the commonwealth of australia for operation as a national facility managed by csiro . this research has made use of the tartarus ( version 3.1 ) database , created by paul oneill and kirpal nandra at imperial college london , and jane turner at nasa / gsfc . tartarus is supported by funding from pparc , and nasa grants nag5 - 7385 and nag5 - 7067 .	we present the results of a @xmath0 3 year campaign to monitor the low luminosity active galactic nucleus ( llagn ) ngc 7213 in the radio ( 4.8 and 8.4 ghz ) and x - ray bands ( 2 - 10 kev ) . with a reported x - ray eddington ratio of @xmath1 l@xmath2 , ngc 7213 can be considered to be comparable to a hard state black hole x - ray binary . we show that a weak correlation exists between the x - ray and radio light curves . we use the cross - correlation function to calculate a global time lag between events in the x - ray and radio bands to be 24 @xmath3 12 days lag ( 8.4 ghz radio lagging x - ray ) , and 40 @xmath3 13 days lag ( 4.8 ghz radio lagging x - ray ) . the radio - radio light curves are extremely well correlated with a lag of 20.5 @xmath3 12.9 days ( 4.8 ghz lagging 8.4 ghz ) . we explore the previously established scaling relationship between core radio and x - ray luminosities and black hole mass @xmath4 , known as the ` fundamental plane of black hole activity ' , and show that ngc 7213 lies very close to the best - fit ` global ' correlation for the plane as one of the most luminous llagn . with a large number of quasi - simultaneous radio and x - ray observations , we explore for the first time the variations of a single agn with respect to the fundamental plane . although the average radio and x - ray luminosities for ngc 7213 are in good agreement with the plane , we show that there is intrinsic scatter with respect to the plane for the individual data points . [ firstpage ] agn jets radio x - ray lag	8,324	496
nonlocality is an important phenomenon in nature , particularly in quantum world . the direct recognition of quantum nonlocality comes from the fact that a quantum wave is spatially extended , in contrast to the point model for classical particles . in this paper we mainly discuss how the nonlocality affects the interactions between material particles of spin-@xmath0 . the problem is intriguing since the nonlocality has been gleamingly implied by the renormalization of conventional quantum field theory ( cqft ) , whence most relevant calculations have to be regulated by momentum cutoff to contain the non - point effect . the technique however , is usually available only at high energy scale , the case where the wavelengths of particles are ultra short . here we take into account the nonlocal effect emerging within the range of interactions possibly a few wavelengths ; but we do nt get involved in the hotly discussed long distance effects relating to entangled states and their applications such as quantum information , quantum communication and quantum computation etc .. up to date , we have recognized that one can not accurately measure the spatial coordinates of a proton by making an accelerated electron as probe , unless its wavelength is much shorter than the diameter of the proton . but the proton would be smashed and some other hadrons will be involved in the final state ( and thus the scattering becomes inelastic ) if making the electron s wavelength short enough . in the case of elastic scattering , the detected proton becomes a * singularity * for the electron s wave . the reason may be that , in the measurements , the quantity ( coordinates ) we inquire is not at the same spatial level as that the quantum entities settled in the coordinate is a four - dimension quantity but the electron s or proton s wave is eight - dimension , or put it in mathematical terminology , the quantity we inquire is real but a quantum object is complex . it is concluded from purely mathematical point of view that , only located in a space with dimension equal to or larger than that of the detected objects can an observer get complete information of direct measurement . as a tentative method and a starting point , in this paper we propose an * equal observer * , e.g. an electron , is also put into the hilbert space to observe another electron or other fermions such as protons . presumably , they are at the same spatial level . therefore the electron can use the metric ( gauge ) appropriate for the observed objects to measure physical observables . the method of * equal observer * is conducive to describing the observed quantum wave ( * nonlocal entity * ) as a whole with possibly less interaction - information lost , unlike in conventional quantum mechanics ( cqm ) where quantum wave is expressed on the basis of space - time points . the dynamics for the equal observer of a quantum wave is believed to be different from cqm . in this paper we employ the similarity between quantum * singularity * and gravitational * singularity * to describe how one fermion observes ( interacts with ) another fermion , and dynamically apply the formalism of general relativity ( gr ) by generalizing its space from real to complex [ fig . 1 ] . as for the elastic scattering of electron and proton , in calculating the radiative corrections to the lowest order of scattering process by employing quantum electrodynamics ( qed ) , we encounter the divergence rooted from leading - order potential form @xmath10 while making momentum @xmath11 . in calculating the collision of two heavy celestial bodies by using gr , the similar singularity rooted also from the form @xmath10 is encountered , but there the puzzle of divergence is automatically circumvented by carrying out a horizon , the outer of which is physical region , and the inner of which , now known as black hole region , is unphysical . quantum mechanically , the nonlocal region is usually unobservable in our space - time , and thus unphysical . enlightened by such physical scenario , we expect to define physical region for elemental fermions in complex space . in analogy to gr , the principle of nonlocality for two interacting fermions is : * there always exists a complex frame for observer * ( one fermion ) * in which the observed fermion * ( another fermion ) * looks like a plane wave , no matter the existence of interaction . * cqft itself can also lead us to perceive the implicit and profound relationship between nonlocality ( quantum wave ) and complex - curvature . generally , we interpret the scattering matrix between initial state @xmath12 and final state @xmath13 as @xmath14 , where @xmath15 @xmath16 can be any state of a complete set . in this formalism , the operator @xmath17 ( or alternatively , the hamiltonian ) is assumed known . then the matrix elements @xmath18whose square is proportional to transition rate between initial and final states can be evaluated . whereas from an equal observer angle , all the states @xmath19 are known and the state @xmath20 can be observed , so the operator @xmath17 can be carried out formally @xmath21 , consequently the interaction becomes known . this latter opposite thought reminds us of the physics in gr , where the force can be realized by the curving of space - time . so , if only the @xmath17matrix is defined locally in complex - space ( a quantum wave is viewed as a point in such space , and as a whole ) , the differential geometry for * nonlocal entity * would definitely occur . [ note : for convenience , in what follows we will not employ the language of @xmath17matrix , though relevant . ] the further understanding of the relationship between nonlocality and curvature is achieved in section 10 , where the local conservation laws do nt exist . in summary , one will gradually be aware of that all of the above intuitive knowledge on nonlocality could be _ transferred _ to and _ interpreted _ by the complex geometry we used , _ via _ the motion equation for fermions and the field equation for bosons in section 8@xmath229 . the main results of the paper are as follows : from our dynamical motion equation we can restore all the terms appearing in the quadratic form of dirac equation . for the @xmath23 field [ generalized qed ] , if electromagnetic fields @xmath4 and @xmath3 satisfy @xmath24 , the bosons will gain masses . based on the discussion of physical region , we can attain two qualitative understandings of quark confinement . in order not to make readers confused by mathematical details , we only list some main results of the real and complex geometry with detailed explanations and the necessary calculating techniques , then immediately apply them to physics problems discussion of dynamical equations and the definition of physical region . the remainder of this paper is arranged as follows . in sections 2 - 5 we introduce the necessary calculating techniques of geometry , the readers familiar with the contents can neglect this part and directly turn to section 6 . then in sections 6 - 9 we apply them to construct the motion equations for fermions and field equation for bosons . the conservation laws are sketched in section 10 . in section 11 , the physical region is defined for @xmath23 field [ generalized qed ] and @xmath6 field [ generalized qcd ] . consequently the qualitative understandings to confinement of quarks is presented under the approximation of interaction vertex @xmath25 . finally a concluding section is presented to summarize the paper and give some remarks on the applicability of the theory . in appendix a , the derivation of motion equation for fermions is elaborated . exterior product proves to be a powerful tool in calculating tensors . it obeys the rules of grassman algebra . one of its usages is to give rise to new tensors of higher order . for example , let @xmath27 be of coordinates for a space , and @xmath28 an arbitrary scalar function with respect to these coordinates , then the total differential of the function @xmath29 is d@xmath30 . we call @xmath31 a one - order tensor , or a vector , and @xmath32 the basis in the neighborhood of a given point @xmath33 . in terms of exterior product the d@xmath29 is called differential 1-form . now we can construct 2-form on the basis of the 1-form : dd@xmath34 . the sign @xmath35 denotes the exterior product . it satisfies the antisymmetric rule : @xmath36 , according to which one can easily conclude dd@xmath37 . conventionally , an @xmath38-form is written as @xmath39 the repeated index henceforth in this whole paper means summation . here the @xmath40 are components of the @xmath41order tensor , and the exterior products @xmath42 are the bases . the above forms can be readily extended to complex variables , e.g. , the two complex variables \{_z , z_}. and the only 2-form of these complex variables is @xmath43 @xmath44 @xmath35__@xmath45 where @xmath43 is a function of _ z _ and _ z. _ here _ z _ and _ z _ are independent variables . usually a space is called @xmath38-dimension manifold if it can be formed as smoothly as possible by affixing many infinitesimally flat patches of @xmath46(@xmath38-dimension real space ) , @xmath47(@xmath38-dimension complex space ) , or other sorts of @xmath38-dimension spaces . in fact in this paper we are only concerned about a particular sort of differential manifolds , but for convenience we will call it manifold without specification . above all , an important property for a given space is the existence of derivatives , which is essential to build up a description of differential geometry . as for hilbert space , the existence of derivatives may be determined by that of wave function @xmath48 in it . in general , the existence of derivatives of a real function can be understood intuitively . however , for a complex function , that demands the cauchy - riemann condition be satisfied . when the complex derivative can be defined everywhere , the function is said to be * analytic. now we review it in respect of complex geometry . conventionally , we identify an @xmath38-dimension complex manifold with a @xmath49-dimension real manifold . the complex coordinates and real coordinates have the relation @xmath50 then the following 1-form is written straightforwardly @xmath51 and subsequently @xmath52 @xmath53 the reverse of the above equations yields @xmath54 @xmath55 here the sign @xmath26 above the equals sign means a definition. perform exterior - product upon an arbitrary complex scalar function @xmath56 , it results in a 1-form @xmath57 make @xmath58 , we note that @xmath59 is just the cauchy - riemann relation @xmath60 another important property of a given space is its transformation . commonly , the vector fields spanned by basis @xmath61 [ real space ] or by basis \{@xmath62d@xmath63d@xmath64 @xmath65 } [ complex space ] on manifolds are the objects responsible for addressing the geometry of the manifolds . how do these fields transform from one point to another infinitesimal neighboring point of the manifolds ? the transformations are usually constrained by some groups as done in the well - known classical mechanics or quantum fields . we will return to this topic later . now let s turn to the affection of cauchy - riemann relation on a vector basis . ignoring some unnecessary details , we assume that the transformation of the @xmath38-dimension complex manifold has the general form @xmath66 as an application of eq.(3.6 ) , replacing the function @xmath29 by @xmath67 , and writing the @xmath67 of eq.(3.7 ) explicitly as @xmath68 , the form of the eq.(3.6 ) now reads @xmath69 make @xmath70 @xmath71 and introduce a @xmath72 matrix @xmath73@xmath74 which satisfies @xmath75 , @xmath76 denoting the identity matrix of rank @xmath49 , then the cauchy - riemann relation ( 3.8 ) can be written @xmath77 and correspondingly an identical relation holds @xmath78 write out the eq . ( 3.10 ) alternatively as @xmath79 and regardless of @xmath73 on @xmath80 , we obtain @xmath81 combining it with the equation ( 3.4 ) gives @xmath82 @xmath83 consistently the results ( 3.13)(3.14 ) can also be expressed using the basis @xmath84 and @xmath85d@xmath86,d@xmath87 , respectively as @xmath88 and @xmath89 @xmath90 from eq . ( 3.16 ) , we note that the d@xmath91 and d@xmath92 are eigen forms of the matrix @xmath73 , and the relations are independent of the choice of the coordinates . in this respect @xmath73 is called the complex structure * of the complex manifold . a manifold with complex structure is called * almost * complex manifold . the above mathematical formulae will become more meaningful if we replace @xmath93 and @xmath94 by the wave function @xmath95 and @xmath96 from the hilbert space ( manifold ) , which will be recognized in following sections . for convenience , henceforth we will put @xmath97 in a more general space , with hilbert space only as a special case . before giving the general definition of metric ( gauge ) , it is helpful to recall its definition in three - dimensional ( 3-d ) space . using the cartan method of moving frames [ 1 ] , let every point @xmath98 in 3-d space correspond to a frame @xmath99as following . if the point @xmath100 varies with a curvilinear net defined by @xmath101 i.e. , @xmath102 , the total differential of @xmath100 then takes the form @xmath103 if setting @xmath104 and @xmath105 , then from eq.(4.1 ) the point @xmath100 relates to frame @xmath106 as follows @xmath107 now define metric tensor as @xmath108 the parenthesis denotes the inner product of common sense . combining these definitions , the metric of @xmath109 reads @xmath110 from eq.(4.2 ) , the differential of basis @xmath111 can be formally written as @xmath112 where @xmath113 is also a 1-form similar to @xmath114 , then the differential for @xmath115 can be calculated as follows @xmath116 the above general forms is suitable for any frame @xmath117 . now let s turn to some special cases relevant to concrete frames . a well - known metric form for coordinates frames is the orthogonal one , i.e. @xmath118 . with this relation , eq.(4.4 ) becomes @xmath119 and eqs . ( 4.6 ) , ( 4.7 ) give rise to @xmath120 another general / important frame is the * natural frame * , in which @xmath121 is defined by @xmath122 then ( 4.1 ) is directly written as @xmath123 and ( 4.4 ) reads @xmath124 the natural frame is of elicitation in studying manifold , which can help us extend the above discussion to n - dimensional space . the space continuously generated by curvilinear net @xmath125 is a manifold . now let s turn to its natural bases @xmath126 . according to the terminology of quantum mechanics , @xmath111 is made by two parts : a matrix ( operator ) @xmath127 and a vector @xmath128 . since the operator parts @xmath127 are the same for different points @xmath100 , we usually write @xmath129 for general cases while @xmath130 for a given point @xmath100 . in this sense , the general form of inner product @xmath115 becomes @xmath131 for a given point @xmath33 on manifold , @xmath132 gives rise to a mapping that maps two arguments @xmath133 and @xmath134 to a real or complex number . then what is the relationship between the basis @xmath135 which has been mentioned in section ii and this @xmath136 ? that can be recognized by the integral @xmath137 where @xmath29 is an arbitrary function , the integrand @xmath138 is unaltered after integration . so it is reasonable to write @xmath139 the contravariant metric tensor @xmath140 is thus defined by @xmath141 with @xmath142 . the above arguments equally hold in complex space . for a given space , there may exist several ways to define its metric form , _ i.e. , _ the gauge to measure the space . in the physical respect , we are only interested in the forms like eq . ( 4.14 ) , even for complex manifold . the construction of metrics includes another important feature , i.e. the constructed metric @xmath143 should be of geometrically invariant quantity with respect to the transformation between frames of the different points of the manifold / space . for a given space , there may exist many ways to jump from one point to another point in its infinitesimal neighborhood . to address the manner of the jump is to determine the transformation group between the corresponding frames . the transformation group is also a key element to define manifold , different groups correspond to different manifolds . now let s turn to the general form of differential geometry . perform the infinitesimal group transformation upon the vector basis @xmath144 , the change of @xmath145 is then analogous to eq . ( 4.5 ) @xmath146 @xmath147 is 1-form ( the similar meaning as the above @xmath148 ) on the manifold . conventionally , we take the explicit form @xmath149 in which @xmath150 ( not a tensor ) is known as * connection * , referring to the manner how to affix the infinitesimal flat - space - patches to form a larger curved space . according to above definition eq . ( 4.15 ) , the variation of a vector @xmath151 under the transformation yields @xmath152 where @xmath153 is known as covariant differential . the above expressions indicate that the notations @xmath154 , @xmath155 and @xmath147 are identical:@xmath156 , which leads us to a conventional denotation @xmath157 where @xmath153 and @xmath158 are both viewed as operators . if for all components @xmath159 is met , the differential @xmath160 is called the parallel displacement for vector @xmath161 , and the path along which the displacement would occur is solvable from the equation @xmath162 . the connection @xmath147 can be carried out by demanding that it should preserve the inner product of two vector @xmath161 and @xmath163 @xmath164 if they are parallelly displaced , i.e. @xmath165 . in the above equation @xmath166 and @xmath167 and @xmath168 are 1-form . we see that @xmath169 leads to @xmath170 i.e. @xmath171 the parallel displacement @xmath159 and the conclusion that the connection preserves the metric @xmath172 , i.e. eq . ( 4.21 ) , are equivalent they can bring out each other . note that all the components of @xmath173 form a matrix @xmath174 , and likewise components of @xmath115 form a matrix @xmath175 , eq . ( 4.21 ) can be written in terms of matrices @xmath176 @xmath177 means the transpose of the matrix @xmath178 . the relation eq . ( 4.22 ) puts a way which we will elucidate later to carry out the connection @xmath179 with respect to the metric @xmath115 . let s turn to the most important quantity of manifold the curvature tensor which is an invariant 2-form under a certain transformation group that specifies the manifold . conventionally , the curvature matrix is defined by @xmath180 taking @xmath181 and bearing in mind that @xmath182 , we can express it explicitly @xmath183 the denotation @xmath184 in subscripts stands for the derivative with respect to the @xmath185-th variable . then by tr@xmath186 i.e. making @xmath187 in ( 4.24 ) [ summation convention is applied too . ] one gets the ricci tensor @xmath188 above formulae pertain to all manifolds with the metric form like eq . the analytic property of complex manifold is determined completely by the complex structure , so any connection possessed by complex manifold is demanded to preserve the complex structure . i.e. @xmath189 where @xmath190 means performing differential on a given basis @xmath191 , i.e. @xmath192 . apply eq . ( 5.1 ) to eq . ( 4.15 ) , remembering the eq.(3.14 ) [ please note : for a complex manifold , we use denotations @xmath193 and @xmath194 hereafter ] , then @xmath195 however , if we apply eq.(5.1 ) to @xmath196 , the result becomes @xmath197 which violates eq.(5.1 ) . therefore , eq.(5.1 ) holds only if @xmath198 replace @xmath199 by @xmath200 and repeat the same procedure , we have @xmath201 eqs.(5.4 ) suggest only the 1-form @xmath147 and @xmath202 are permitted in the complex manifold . if the connection of a manifold preserves the complex structure @xmath73 , then the manifold is called almost complex manifold , this statement is equivalent to aforementioned definition . furthermore if the repeat of the differential @xmath203 , i.e. @xmath204 always turns to null , @xmath205 ( other equivalent conditions are @xmath206 and @xmath207 ) , then the almost complex manifold becomes complex manifold . ( the proof of these statements is omitted in this paper . ) in this paper only complex manifolds are concerned . @xmath208 is called @xmath209 $ ] type connection , and @xmath210 is called @xmath211 $ ] type connection . the group transformation can not change the type of connection . in this paper , for simplicity we employ only @xmath209 $ ] type connections , and @xmath212 $ ] type connections are thus all trivial . it seems that the expression in section iv is only developed for real geometry , but in fact it also pertains to complex geometry . here we iterate the results in terms of complex geometry . first of all , let s assume that we have chosen a metric for the complex space , @xmath214 it is an additional requirement to real geometry that the indices @xmath215 and @xmath216 belong to two different types , one is normal component , the other is its complex conjugate , @xmath217 . the reason why we choose this particular metric form will be explained in the next section . following the process of the last section , let s make the connection preserve the metric @xmath218 . having chosen the @xmath219,@xmath220 as basis @xmath221 , the result is easily obtained from eq.(4.19 ) that @xmath222 in obtaining the above equation , we have used the following equations due to the parallel displacement @xmath223 @xmath224 where @xmath225and @xmath226 are components for the vector @xmath161 and @xmath227 @xmath228 @xmath229 . in analogy with the form ( 4.22 ) , eq.(5.6 ) can be expressed in the following matrix form @xmath230 where @xmath231 refers to the matrix with components @xmath232 . performing exterior product on the above equation turns the left hand trivial , and @xmath233 where @xmath234 , in which the minus sign comes from the transpose of matrices . ( 5.9 ) suggests the curvature matrix and metric matrix are of anticommute . in eq.(5.8 ) , if @xmath177 is the @xmath209 $ ] type , then @xmath231 is the @xmath211 $ ] type . bearing in mind that @xmath235 , eq.(5.8 ) gives rise to @xmath236 i.e. @xmath237 the inverse of matrix @xmath238 is so defined that @xmath239 then due to ( 5.10b ) the component form of @xmath178 is @xmath240 consequently the curvature form ( 4.23 ) and ( 4.24 ) certainly hold , with the variables now being complex @xmath241 from @xmath242 , and taking into account the definition of the components form of curvature , @xmath243 , it is straightforward to obtain @xmath244 here the meaning of @xmath245 is the same as that of @xmath246 , and likewise for @xmath247 . from this form it is obvious that the curvature is antisymmetric with respect to the last two indices @xmath185 , @xmath248 . except the requirement eq . ( 5.4 ) due to analytic condition , the manifold in virtue of @xmath249 adds no more constraint to the connection . on the basis of the above general complex manifold , if we demand the metric matrix @xmath238 satisfy @xmath251 with components form as @xmath252 i.e. if @xmath238-matrix is hermitian conjugate to itself , then the metric is called the hermitian metric . all the components of the hermitian metric form the hermitian matrix ( hermitian operator ) . this kind of operator is well known in quantum mechanics and has many good properties . to keep the hermitian metric @xmath253 and corresponding curvature invariant , the transformation group must be a unitary group @xmath250 . additionally , as for an hermitian manifold , if the connection @xmath254 is symmetric with respect to the indices @xmath215 and @xmath255 , the manifold then is called khler manifold . the transformation group for khler manifold is @xmath256 . to summarize the above results relevant to curvature , we note that it subjects to three constraints as the following . \(i ) as shown in eq . ( 5.4 ) , the form of connection like @xmath257 and @xmath258 is trivial , \(ii ) we only use [ 1 , 0 ] type connection , so the [ 0 , 1 ] type connection @xmath259 , \(iii ) the curvature form is [ 1 , 1 ] type , so the forms @xmath260 or @xmath261 vanish . under these constraints , one can calculate the following curvature components one by one according to the explicit form ( 4.24 ) : @xmath262 , @xmath263 , @xmath264 , @xmath265 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , @xmath271 , @xmath272 , @xmath273 . finally one concludes that only four of them are nonzero @xmath274 @xmath275 @xmath276 @xmath277 the above four equations hold independent of the torsion [ the torsion is defined as @xmath278 since the evaluating process has nothing to do with it . when the torsion is absent , the first two curvatures in eq . ( 5.14 ) display more symmetries @xmath279 , @xmath280 . the constraints certainly affect the ricci tensor . based on eq.(5.14 ) , the ricci tensors are obtained by tracing @xmath281 in eq . ( 4.23 ) , @xmath282 the equation suggests only two independent forms of ricci tensor , @xmath283 and @xmath284 , exist . furthermore , the curvature component @xmath285 is antisymmetric with respect to the indices @xmath286 and @xmath287 , i.e. @xmath288 , whence only one form of ricci tensor would occur . we will return to this antisymmetry in section ix . now the ricci tensor defined by ( 5.15 ) can be explicitly written as @xmath289 here the notation @xmath238 stands for determinant @xmath290 , and the last step is obtained by applying the differential relation of a matrix determinant @xmath291 to the derivatives of @xmath292@xmath293 the above discussions set no limit on transformation group , hence pertain to general situations , say , @xmath249 . for the hermitian or khler manifold , the independent components of curvature and ricci tensor should decrease . if @xmath294 is hermitian , then @xmath295 and @xmath296 , applying ( 5.14a ) , ( 5.14d ) or ( 5.14b ) , ( 5.14c ) to ( 5.18 ) , the following expression can be achieved @xmath297 the specification of khler manifold adds no more constraint to the form of ricci tensor except the symmetric indices of connection in equation ( 5.14a ) , ( 5.14b ) . now let s count the number of the independent components of the ricci tensor . apart from the fact that the two indices are antisymmetric , the group @xmath298 adds no constraints . so in this case the set @xmath299 , with @xmath300 being over @xmath301 , has totally @xmath302 elements . while @xmath303 need not be taken into account for antisymmetry . the freedom of @xmath294 , however , is totally @xmath304 components , so it requires @xmath305 gauge conditions to solve the field eq . furthermore , if the manifold is hermitian , i.e. the eq.(5.19 ) holds , @xmath306 , then the elements of set @xmath307 decrease to @xmath308 , and the freedom of @xmath294 also shrinks to @xmath309 , hence @xmath310 additional equations are needed to resolve the @xmath294 , almost the same as that of @xmath298 . a quantum fermion field is customarily expressed by dirac spinor @xmath311 , and the inner product of @xmath311 is prescribed as @xmath312 where @xmath313 to be consistent with conventional quantum field theory ( cfqt ) , here the @xmath314 is defined as @xmath315 , which is different from previous denotation @xmath316 . henceforth we distinguish the difference by whether or not the subscripts or superscripts are used : if used , then the latter definition is available , otherwise the former definition holds . the former definition @xmath317 is meaningful in making the inner product @xmath318 invariant under the transformation group @xmath319 , which is the spinor - representation of lorentz group . the invariance is a direct corollary of lorentz invariance of dirac equation [ 2 ] . to apply the differential geometry , let s generalize the inner product . to do so the definition of metric should be valid only within a very small region of the complex space , e.g. the inner product @xmath320 should be generalized to @xmath321 , and only in this sense does the inner product remain invariant under the transformation of group @xmath322 . any infinitesimal transformation is now performed on @xmath323 instead of on @xmath324 . @xmath325 can be viewed as plane wave locally and @xmath324 has conventional meaning on a larger scale . the above inner product holds when there is no interaction , if any interaction arises , the product has to be interpreted by a general metric form @xmath326 . the above form @xmath327 is only a special case while @xmath328 , @xmath329 for @xmath330 and @xmath331 for @xmath332 . in conclusion , the metric of the quantum field is demanded to be @xmath333 which does not violate experiences from qed . in a two - dimension ( 2-d ) plane , the curves of hyperbolic or elliptic type can be interpreted by the equations like @xmath334 or @xmath335 . general linear transformations in 2-d [ known to be @xmath336 group . they have the general form @xmath337 , with @xmath338 being real numbers ] ca nt change the types of curves , since the performance of linear transformations has two identical manners : one is to change the objective curves ; the other is to change the coordinate axes . applying the latter manner , the types of the curves are obviously reserved . in other words , under general linear transformations within 2-d , if a curve in 2-d is ever a type , it will forever be the type . in this paper we take into account only the quadratically homogeneous forms for variables , so considering these two types is enough . the above argument is expected to hold also in four dimension space ( 4-d ) . * * it becomes complicate in this case , since a 4-d curve being hyperbolic in one projected plane may be an ellipse in another . so the definition of hyperbolic or elliptic type has to be extended : for a 4-d curve , if there exists at least one 2-d plane so that the curve projected into it is hyperbolic , then curve is called hyperbolic , otherwise it is elliptic . in our concerns the 4-d interval [ in this paper metric means the same as interval ] @xmath339 in special relativity is hyperbolic type , because if the equation @xmath340 is projected into any 2-d subspace including the time axis , the resultant curve is a hyperbolic type . lorentz group preserves the interval @xmath341 , so the hyperbolic characteristic is also the main feature of lorentz group , which is called noncompact in terminology of group theory . in contrast , the group @xmath342 is compact for it preserve the interval @xmath343 , which is a elliptic type . instead of requiring the transformation groups to preserve the interval , now we only require the types of interval to be preserved . then we can find that hyperbolic characteristic of the interval @xmath339 will not change under the general linear group @xmath344 , similar to the above 2-d case . generally , to keep the type of an @xmath38-d interval of quadratic homogeneous form , that the transformation is linear is sufficient . here and hereafter the explanations of concepts relevant to group are rough and only for later application . the above definitions of hyperbolic and elliptic type in 2-d are for the global space . in a local region of the whole space , the two types should be interpreted by @xmath345 and @xmath346 respectively . correspondingly , if special relativity is treated locally , then the metric is @xmath347 in a small local region , the infinitesimal transformations of @xmath344 can not change the types of the curves . therefore , even though a transformation changes ( 7.1 ) to a general form @xmath348 , by which we can not tell its type directly , the type of the metric @xmath349 remains the same as eq . ( 7.1 ) . now we extend the above discussions and knowledge to complex space . in following three paragraphs , we construct the * parallelisms * between the metrics of real space and complex space , as well as the * parallelisms * between the groups preserving them , then generalize our understanding of hyperbolic and elliptic type in real space to the understanding of complex space . let s consider the complex - space metrics defined in the last section , @xmath350 , where @xmath351 is a complex spinor @xmath352 . this metric is invariant under the transformations of group @xmath322 , @xmath353 means that the elemental variables in the group are complex ones . . it can be verified [ 3 ] that there exists a two - to - one mapping between all the elements of the group @xmath319 and all the elements of the proper , orthochronous lorentz group . hereby the group @xmath322 is viewed as * complex parallellism * of the lorentz group ( which is born from physics in real space - time ) . now let s elucidate the relationship between group @xmath322 and aforementioned metric @xmath354 . the quantities which are transformed according to lowest - dimensional nontrivial representation of @xmath322 are called two - component _ spinors _ , which are doublets as states for spin . on the basis of higher dimensional representation of @xmath322 other spinors with more components can be constructed . in our case the four - component spinor @xmath355 is transformed according to named @xmath356 representation of @xmath322 [ involving @xmath357-matrices ] , their relationship can be derived from the covariance of dirac equation and we omit all the relevant details here . also it can be confirmed that the metric form @xmath358 remains unaltered under the transformation of @xmath359 representation of @xmath322 . in a word , the metric @xmath360 is the * complex parallellism * to the real interval @xmath339 . consequently the * complex parallelism * to eq . ( 7.1 ) should be @xmath361 now we know @xmath362and hence @xmath363is hyperbolic . however , we do nt know what is the largest group which includes @xmath322 as a subgroup that makes the type of metric @xmath364 unchanged . searching for what is exactly the largest group would be a tedious work and might simultaneously deviate us from the main line of developing this theory . in this paper we simply assume the latest group is @xmath365 ( in fact must be a subgroup of @xmath298 ) and bearing in mind that it is for hyperbolic type . the * metric parallelism * between elliptic types of real space and complex space can be constructed by directly generalizing each of real axes to a complex one . we are familiar with the metric defined in the 3-d space @xmath366 * , * which is invariant under the rotation group @xmath367 . suppose a complex space spanned by wave functions with only three components , @xmath368 , corresponding to metric @xmath369 * * and consistent with the probability in quantum mechanics , we conclude that its metric should be of the form @xmath370 , where the integration over the configuration space is implied . to preserve the probability in the whole complex space , the quantum mechanics requires any transformations performed on the wave function @xmath371 should be elements of special unitary group @xmath8 or its subgroups . @xmath1 groups are compact , just as @xmath372 . hereby the metric @xmath373 and group @xmath1 are viewed as the * complex parallellism * to that of elliptic type of real space . the caution should be practiced that it is not perfect to generalize the rotation transformation in real space directly to conservation of angular momentum as done in quantum mechanics to achieve the same result that metric @xmath374 and @xmath1 are elliptic type . since the lorentz group includes a spatial rotation part besides the boost part , from the noether theorem of quantum field theory one can also derive the conservation of angular momentum . then there will be confusions between the two types . the degrees of freedom of elliptic type presented in this paper are those degrees of freedom such as electro - charges , isospins , or colours etc . , other than spins and spatial angular momentum . the largest group to preserve the elliptic type of the metric @xmath375 is certainly @xmath376 groups . we have endowed the lorentz symmetry and special unitary symmetry with hyperbolic type and elliptic type , the corresponding types of metric defined according to these two symmetries remain stable even after the groups are appropriately extended . we know @xmath376 group will not change the type of metric @xmath377 , and a subgroup of @xmath298 will not change the type of @xmath378 . it is the extended groups that provide more degrees of freedom over which the complex space spanned by wave functions is curved . corresponding to the two symmetries , there are two types of curvings . since for a fermion , such as quark , it has both the curving characteristics hyperbolic type and elliptic type spinor and colour , we should combine the two types of metric into one general form @xmath379 where we generalize @xmath380 to @xmath381 are colour indices , @xmath215 , @xmath382-are spinor indices . the metric matrix @xmath383 may be written separately as a hyperbolic part multiplied by an elliptic part . after reviewing the geometrical method , let s turn to its application to physical problems . as an equal level observer , assumedly an electron can test another electron without losing any information . to an observer what is the motion equation of the observed electron ? for an observer in space - time the right equation is certainly the schrdinger equation or the dirac equation . but as an equal level observer , it constructs the equation as follows : there always exists a complex local frame for the observer in which the observed electron looks like a plane wave ( a free electron ) , in spite of the existence of interaction similar to that happens in general relativity saying that there always exists a local frame in space - time in which the observed particle looks like a free one moving along a straight line , in spite of the gravitation . in terms of geometry , it means that the motion of electron is just a geodetic line in complex space , i.e. , the parallel transport ( displacement ) . conventionally a free electron is described by plane wave @xmath384 , @xmath385 is dirac spinor . the complex space here should be of 4-dimension since @xmath385 possesses four components , furthermore , it should be hyperbolic as argued in the preceding section . in this respect the transformation group for the space should be @xmath298 . since the complex space is continuously spanned by components of local spinor @xmath386 , its natural bases are @xmath387 . consequently , @xmath388 , then the parallel displacement ( 5.6a ) @xmath389 turns out to be @xmath390 similarly , another equation for @xmath97 holds , @xmath391 above two equations are the general forms of motion equation for fermions . in eq . ( 8.2 ) the component @xmath392 is an infinitesimal change of @xmath393 relative to its neighboring points , which is determined by @xmath393 and the transformation performed on it . suppose that after a transformation the wave function @xmath393 changes to @xmath394 then resembling the form of eq.(3.7 ) , the wave function @xmath395 can be expressed by variables @xmath396 , @xmath397 now the differential form @xmath392 can be written explicitly as @xmath398 the derivative @xmath399 in the above equation however , is not operable in practical calculation . practically , we want to know how far the eqs . ( 8.1 ) , ( 8.2 ) deviate from the conventional dirac equation . or to put it alternatively , how can the terms in dirac equation be finally restored in local region by reducing aforementioned parallel displacement ? to obtain a form of dirac equation , let s first recall the replacement used in gr : @xmath400 . that reminds us to replace the differential operator @xmath26 in eqs . ( 8.1 ) , ( 8.2 ) by some forms of derivatives with respect to space - time . in view of the quadratic form of eqs . ( 8.1 ) , ( 8.2 ) , it is helpful to know the following quadratic form [ 2 ] of dirac equation before we do some replacement of operator @xmath26 , @xmath401 { \psi } = m^2 { \psi } \text { , } \tag{8.5}\ ] ] where @xmath402 is the field tensor of common sense . in what follows , we put @xmath403 . then the explicit form of eq . ( 8.5 ) is @xmath404 by using a weaker lorentz condition @xmath405 of gupta and bleuler [ 5 ] instead of the original one @xmath406 , the above equation changes to @xmath407 as a reasonable approximation to leading - order qed , the quadratic terms of @xmath408 , such as @xmath409 in the above equation and similar terms in our following calculation will be temporarily omitted . now let s turn to treating the equation ( 8.2 ) . first let s substitute the explicit form of @xmath410 , @xmath411 , into the equation ( 8.2 ) , concerning only the second term , it yields @xmath412 then the form of motion equation ( 8.2 ) becomes @xmath413 extending the discussion of eqs . ( 6.1)(6.3 ) straightforwardly by considering the vertex @xmath414 used in qed , we can interpret the metric tensor @xmath415 with interaction as @xmath416 ( henceforth we continue to use @xmath417 to express the field potential . take care not to confuse it with the metric tensor @xmath418 ) . it is easy to verify that @xmath419 . to obtain a form of dirac equation , comparing the second term of eq . ( 8.8 ) with the term @xmath420 in eq . ( 8.6 ) , we find it is effective to replace @xmath26 by @xmath421 . obviously , the replacement @xmath422 @xmath421 performed to the first term of eq . ( 8.8 ) directly induces @xmath423 . furthermore , by analyzing the dimensions of eq . ( 8.8 ) , we find the dimensions of the first term and that of the second term are not equal . in the natural units , the first term of the above equation has the dimension of energy square if assuming @xmath424 dimensionless . as for the second term , the form @xmath425 can be simplified as @xmath426 , and @xmath427 [ @xmath428 is energy square too , but the derivative performed on @xmath424 also contribute the dimension of an energy , so an extra energy dimension exists . to remedy the*unequal dimensions of two sides and according to our experience of treating the schrdinger equation with a nonlocal interaction - potential [ 6 ] , we add a line integral with respect to space - time @xmath429 to the second term . the integration is accompanied by the purely imaginary number @xmath430according to requirement restoring the dirac equation . now the form of equation ( 8.8 ) becomes @xmath431 consistently , the infinitesimal integral measurement @xmath432 should also be linked with @xmath255-matrix , assumed as @xmath433 . now let s evaluate the results of the second term @xmath434 in eq . it is convenient to write out the integral in terms of matrix @xmath435 here @xmath436 is a four - component vector , @xmath437 and @xmath438 all in their matrix forms , @xmath439 and @xmath440 . we will mainly deal with the first integral in ( 8.10 ) , since the second term includes two @xmath417 factors we temporarily omit it as second - order perturbation . the calculation involves the formulae of @xmath441matrices in simplifying the product @xmath442 . the detailed evaluation on the first integral is put to appendix a , here we only show the result in ( a. 7 ) @xmath443 where @xmath444 . submitting the integral result back into ( 8.9 ) yields @xmath445 it is found that the terms @xmath446and @xmath447 are just those required by ( 8.6b ) . however , quite a few other terms like @xmath448 , @xmath449 , etc . accompany the required ones . some of the redundant terms , such as @xmath450,@xmath451 , @xmath452 , with integrand relevant to local angular momentum @xmath453 , which is assumed second order small in perturbation theory according to our numerical experience , can be omitted temporarily . the contributions of remainder terms like @xmath454 , @xmath455 , @xmath456 , @xmath457 can not be judged reasonably , so these terms can not be thrown off and may show their significance in some situations when the wavelength of fermions is sufficiently long . for instance , @xmath458 is an operator , and it renders the results of @xmath459 dependent on the variation of the wave function @xmath424 . accordingly this sort of terms is possibly relevant to the nonlocal effect of aharonov - bohm type [ 7 ] . the terms @xmath460 , @xmath461 would behave in similar ways . but the term @xmath462 , without the factor @xmath463 before it , is the exceptional case as discussed in the following . comparing the eq . ( 8.12 ) with eq . ( 8.6 ) , it is noted that there lacks a mass term @xmath464 in the right hand of eq . ( 8.12 ) . there are two ways to remedy this flaw . first , one can directly add a mass term to the right hand of equation ( 8.9 ) , then the equation ( 8.2 ) would have a nonzero term in its right hand too . that obviously violates our original hypothesis that motion equation is just the geodetic line . another way out is to accept the term @xmath465 as the mass term . the reason is that according to the klein - gordon equation of the next section the double derivatives on potential @xmath466 would induce a mass factor and the integration @xmath467 would induce a phase factor depending on closed paths ( loops ) of fermions [ 7 ] , thus the total effect of the term @xmath468 seems equal to a mass term for motion equation of fermions . if the paths are not closed , then the mass of fermions would become dependent on potential @xmath466 . the claims of this paragraph are only qualitative , further confirmation is necessary . the relevant works are in process . apart from the above qualitative display , all of the four nontrivial terms , @xmath469 , @xmath470 , @xmath471 and @xmath472 , are surely nonlocal since they obviously ruin the local conservation law @xmath473 [ where @xmath474 , which can be directly derived from the conventional dirac equation ( 8.5 ) . if the approximation @xmath475 is not used here , the replacement @xmath422 @xmath476 would not be reasonable . in general case the matrix @xmath477 should be expanded to all possibilities of @xmath441matrices ( the total number of matrices is 16 ) without respecting only the vector form of interaction . but we think there leaves little arbitrariness to extend the theory to include more sorts of interaction . if surely the approximation @xmath478 holds for fundamental interactions , then accordingly we would extend the replacement @xmath479 @xmath421 to @xmath422 @xmath480 , @xmath481gellmann matrices , when we are concerned about the colour interaction between quarks . that will cause more integrations and other intricate terms to enter into eq . we will not detail that issue here . in conventional quantum field theory ( cqft ) , we only regard the part @xmath482 being responsible for the physical process . however , in view of the nonlocal ( curving ) effects , it has been stated in ref . [ 8 ] that the @xmath482 is not complete in describing all physics . even under our rough approximation , it can be seen in eq . ( 8.12 ) that more terms are present than required . it includes not only the normal terms @xmath483 but also some other terms on account of nonlocal effect . moreover , the definition of the connection and thus the physical meaning of @xmath408 now are not the same as those in cqft , since in cqft , @xmath417 ( potential ) is assumed as connection and @xmath483 as the curvature tensor . but here the @xmath417 is viewed only as components of metric tensor and @xmath483 as connection ( force ) . correspondingly the dimension of motion equation here is also different from that of schrdinger equation , the former is a force , and the latter a potential . a discrepancy of energy dimension appears under natural unit . the differential problem in fact has arisen in calculating the terms @xmath485 and @xmath486 . as a hypothesis , we have required the wave function to change with respect to space - time variables @xmath487 , with which the terms in dirac equation for electrodynamics are restored . however , in fact , since we are discussing the property in complex space , the differential @xmath488 should change with respect to complex variables as showed in eq . ( 8.3 ) , i.e. @xmath489 . making every coefficient of @xmath490 vanish , then eq.(8.2 ) yields @xmath491 this equation should be thoroughly respected by the electron as an equal level observer to another observed electron . although it is impossible for the electron to release all information it carries by projecting its complex space to our space - time , @xmath492 , we have to do so to get some physical observerbles in space - time . just for the physical reason and for consistence , in this section we directly make the replacements of the differential forms @xmath493 and @xmath494 , or the equivalent forms @xmath495 and @xmath496 , as follows : @xmath497 @xmath498 this subsection is devoted to explaining a subtle but important aspect in calculating the curvature tensor . it can be seen from the definition of eq.(4.24 ) that the curvature tensor @xmath500 is explicitly antisymmetric with respect to the indices @xmath185 and @xmath248 . however , the antisymmetry seems lost from the expression ( 5.14 ) . now let us impose that antisymmetry on the interpretation of eq.(5.14 ) and see what will be the result . substitute the explicit form of connection @xmath501 into eq.(5.14 ) , the components of curvature become explicit functions with respect to metric tensor : @xmath502 we only calculate the first one as an example , the others can be obtained by properly changing the indices . impose the antisymmetric property on the first equation of eq.(9.3 ) @xmath503 then the left hand and the right hand can be changed respectively to @xmath504=-\frac{\partial a^{\alpha \bar \gamma } } { \partial { z}^k}\frac{\partial a_{\bar \gamma \beta } } { \partial { \bar z}^j}-a_{\bar \gamma \beta } \frac \partial { % \partial { \bar z}^j}\frac \partial { \partial { z}^k}a^{\alpha \bar \gamma } \text { , } \tag{9.5a}\ ] ] @xmath505 the equality of the two sides directly gives @xmath506 i.e. @xmath507 the relation seems trivial in this form , but making the replacement of ( 9.2 ) results in @xmath508 which is a reasonable result telling that the operator @xmath499 is hermitian . so now we can insist on the antisymmetry in ricci tensor without worrying about any unexpected contradiction . as a byproduct of applying the above argument to ricci tensor ( 5.18 ) , a rule is gained that for any product of two derivatives like @xmath509 , @xmath238 arbitrary , the result should be antisymmetric if permuting @xmath510 and @xmath511 , @xmath512 . now let s introduce the field equation for bosons , e.g. photons in electrodynamics . for a boson field without any source , we are concerned about the case when the interaction is absent , i.e. , the connection ( force ) of complex space is trivial , and thus all the components of curvature tensor @xmath513 vanish . in that case the boson field satisfies the equation @xmath514 . the equation will not change under certain transformations in question ( e. g. @xmath515 group for photons ) since @xmath513 is a tensor . that is to say , the field equation will not change with the appearance of interaction . but as a general equation for a boson field the condition @xmath514 seems too strict , so we introduce a weaker constraint by merely demanding that ricci tensors vanish , @xmath516 that not only holds for free fields , but also is expected to hold while a source appears at a point infinitely far away . as in gr , we can add the source term to the right hand of equation , whose physical meaning will be clarified later , @xmath517 the coefficient @xmath518 can be determined by comparing it with the klein - gordon equation . next we will study what can be derived from this field equation , and what are its differences with the klein - gordon equation as well . we had better resolve the field equation ( 9.8 ) precisely before discussing the field property . but here we expect to understand the field properties by substituting some good approximations into the field equation . as argued in section vi , in the absence of interaction the metric matrix is of the form @xmath519 and after a period of interaction , according to the perturbative theory ( in this whole paper we respect the perturbative results of cqft ) , the matrix evolves into @xmath520 if only the electronic part is present , eq . ( 9.10 ) gives @xmath521 moreover , if we choose the large components approximation , which is validated by qed , then the form of @xmath522 yields @xmath523 now we briefly review the reasonability of the above approximation . the large components and the small components appear in solution of dirac equation when both the electric part ( scalar potential ) and magnetic part ( vector part ) of the boson field are small and the kinetic energy of fermion is very low : in general , the dirac equation can be written [ 2 ] , @xmath524 where @xmath525 is kinetic energy , @xmath526 the scalar potential and @xmath527 the vector potential . divide the four components @xmath528 into two parts composed respectively of the first two components and the last two components , @xmath529 with this denotation the dirac equation is readily reduced into two equations , @xmath530 after carrying out @xmath531 as @xmath532 , we see that when @xmath533 , @xmath526 and @xmath527 are all small , the two components @xmath534 and @xmath535 obey the relation @xmath536 so in the large component approximation , * the potential @xmath537 in the last two diagonal elements in ( 9.11 ) can be ignored reasonably since it contributes to @xmath538 the terms two orders less than that of the first two diagonal elements . consequently the approximation ( 9.12 ) does hold . under this approximation , the ricci tensor reads @xmath539 replacing the differential @xmath540 following eq . ( 9.2 ) , and assuming that the scalar field does not vary with time , then up to @xmath541 order the field equation @xmath542 turns out to be @xmath543 which is just the poisson equation satisfied by the electric field in vacuum . in the next subsection it will be shown that the approximation without the electric field is also heuristic . now let s turn to the approximation without electric field or source term , i.e. , the case that only radiation field exists . computing the similar form of ricci tensor in ( 9.17 ) by using the metric form ( 9.10 ) and making @xmath544 , we have @xmath545 we will treat the equation explicitly to construct its relation to the electric and magnetic energy forms @xmath546 and @xmath547 , where @xmath548 and @xmath549 . the calculation of ( 9.19 ) includes two parts @xmath550 and @xmath551 where the underlines denote the inner products between the vectors . and the sign @xmath552 is the tensor product for dyads , for example , @xmath553 @xmath554 . if an inner product is emphasized by underlines then the regular product order between dyads is not followed , e.g. @xmath555 . usually we only use the underlines while the gradient operator @xmath556 appears . now let s express the magnetic energy @xmath557 in terms of vector potential @xmath558 to further simplify the eq . ( 9.22 ) , the following relations are useful , @xmath559 @xmath560 \nonumber \\ \ & = & ( \vec \nabla \underline{\vec a}):(\vec \nabla \underline{\vec a})+\vec a% \cdot \vec \nabla ^2\vec a-\vec \nabla \underline{\vec a}:\underline{\vec \nabla } \vec a-(\vec a\cdot \vec \nabla ) ( \vec \nabla \cdot \vec a)\text { , } { ( 9.23b ) } \nonumber\end{aligned}\ ] ] @xmath561=(\vec a\cdot \vec \nabla ) ( \vec \nabla \cdot \vec a)-\vec a\cdot \vec \nabla ^2\vec a\text { . } \tag{9.23c}\ ] ] applying relations in ( 9.23 ) to ( 9.22 ) , we obtain @xmath562 the last step holds for transverse fields in the coulomb gauge , @xmath563 . now the eq . ( 9.20 ) and eq . ( 9.21 ) can be simplified to @xmath564 @xmath565 combine them into ( 9.19 ) @xmath566 consequently , the following equation holds , @xmath567 heuristically , the eq . ( 9.27 ) suggests if the inequality @xmath568 holds , then the equation for field @xmath569can automatically gain a mass term @xmath570 where the klein - gordon equation takes shape . for electrodynamics , since @xmath571 coincidentally and accurately vanishes , the net mass of photon is zero . in qed , the lagrangian density for photon field is @xmath572 the field tensor @xmath573 is defined in the preceding section . in terms of @xmath3 and @xmath4 , the lagrangian can be transformed to @xmath574 coincident with lagrange undetermined multiplier method , now it is natural to extend the lagrangian of qed by adding a mass term @xmath575 from which the klein - gordon equation for massive bosons can be obtained by using variational method . in the frame of this paper we do nt respect gauge invariance of the lagrangian . but the variational method would always be valid from mathematical viewpoint . now let s clarify the meaning of source term in eq . ( 9.8b ) . in quantum field theory the klein - gordon equation with source term is @xmath576 , hence according to eq . ( 9.29 ) @xmath577 the right hand is just the interaction hamiltonian . remembering that we have in eq . ( 9.2 ) replaced the derivatives with respect to @xmath578 by the derivatives with respect to space - time , and noticing the form @xmath579 , the above equation and eq . ( 9.17 ) suggest that the source term corresponding to eq . ( 9.8 ) should be of a double integrals with @xmath580 as integrand and @xmath581 as its infinitesimal integral measurement . we define @xmath582 , then @xmath583 here the @xmath584 refers to a source field induced by fermions , though it looks formally like from @xmath585 . the term @xmath586 should not be associated with @xmath584 since it is derived from the equation for free boson field . in above subsections the discussions are applicable in fact only to @xmath23 case , in which @xmath585 has no intrinsic degrees of freedom except spatial polarization . as stated in the previous section , if other intrinsic degrees are involved [ e. g. see ( 7.3 ) , @xmath587 , the replacement @xmath422 @xmath421 will not hold any longer . consequently , the replacement of eq . ( 9.2 ) becomes abated . instead we should employ the replacement @xmath422 @xmath588 , @xmath589 being generators of a unitary group . when the group is @xmath590 , whose representation can use pauli matrices , the replacement performing on double differential @xmath493 will induce the similar form to @xmath23 case , @xmath591 . in this case the field equation in space - time would be @xmath592 however , in @xmath8 case , the gellmann matrices @xmath593 have not the property of @xmath594 , so that the cross - terms do nt vanish . therefore , in @xmath8 case , the field equation in space - time can written only as @xmath595 make exterior differential to both sides of eq . ( 4.23 ) , one gets the following bianchi identity @xmath596 and its matrix form is @xmath597 write it out in components form @xmath598 on the other hand , the covariant differential of @xmath599 can be obtained by definition @xmath600 combining the above two , one gets @xmath601 in the case that torsion is absent , the right side is equal to zero , hence @xmath602}^a=0\;\text{. } \tag{10.6}\ ] ] the bracket @xmath603 $ ] means the left side also includes the terms with the indices of cyclic permutation . substituting the nonzero components in eq.(5.14 ) one by one to the above equation , for instance substituting the ( 5.14a ) , it becomes @xmath604}^\alpha & = & r_{\beta \bar jk;m}^\alpha + r_{\beta m\bar j% ; k}^\alpha + r_{\beta km;\bar j}^\alpha \nonumber \\ \ & = & r_{\beta \bar jk;m}^\alpha + r_{\beta m\bar j;k}^\alpha \stackrel{\text{% contracting } \alpha \text { and } \beta } { = } r_{\bar jk;m}+r_{m\bar j;k}=0\;% \text{. } { ( 10.7 ) } \nonumber\end{aligned}\ ] ] the other three forms of eq.(5.14 ) will give rise to the same form of the ricci identity @xmath605 . it has been proved in sec.v that the metric is invariant under covariant differential , @xmath606 . so with the definition of scalar curvature @xmath607 , the ricci identity possesses the following form @xmath608 i.e. , @xmath609 applying the relation @xmath610 to above equation , with @xmath611 , the eq . ( 10.9 ) can be changed to @xmath612 now the field equation can be rewritten by regarding the bianchi identity , @xmath613 its equivalence to eq . ( 9.8 ) can be demonstrated as follows : contracting the indices of eq . ( 10.11 ) results in @xmath614 , then substituting @xmath615 back to eq . ( 10.11 ) , finally the eq . ( 9.8 ) is restored . furthermore , the above equation can be extended to include source term @xmath616 as mentioned in the last section , the approximation ( 9.10 ) is good for perturbative theories . in such case , the metric reads [ 9 ] @xmath617 where the @xmath618 is required to be small comparing with @xmath619 , whose nonvanishing elements are only those diagonal ones , being @xmath620 . here the linear metric @xmath619 is used to raise or lower the indices of metric and curvature of linear part , e.g. @xmath621 . only in this sense , the following defined @xmath622 is different from the normal definition of the ricci tensor @xmath623 , though they have almost the same form . the linear part of the ricci tensor is @xmath624 the rest part then is @xmath625 similar to the treatment in general * * relativity [ * 7 * ] , we separate the left hand of the field eq . ( 10.12 ) into two parts @xmath626 , then eq . ( 10.12 ) can be written @xmath627 we can prove that the differential performed on the left hand of eq . ( 10.16 ) gives zero @xmath628 the first term is zero since the antisymmetric tensor @xmath629indices @xmath630 are antisymmetric@xmath631 is multiplied by symmetric tensor @xmath632 , i.e. , @xmath633 ; and the second term is trivial since the symmetric tensors @xmath634,@xmath635 and @xmath636 are accompanied with the antisymmetric tensor @xmath637 , so @xmath638 is directly zero before differential is performed . from ( 10.10 ) , ( 10.12 ) we conclude that the source term @xmath639 is conservative in covariant sense , @xmath640 . whereas from ( 10.16 ) , ( 10.17 ) we conclude that the sum @xmath641 is conservative in sense of common differential , @xmath642 . the two sorts of conservation directly bring out three conclusions : \(i ) as for the existence of the conserved equation @xmath643 it is natural to view the sum @xmath644 as the total energy - momentum tensor including both the fermion part @xmath645 and the boson part @xmath646 , which suggests in a complex infinitesimal local region the energy and momentum may transfer between @xmath639 and @xmath646 . \(ii ) since the @xmath647 will not be a tensor for the loss of nonlinear terms , it can be inferred from ( 10.16 ) that the boson part @xmath648 does not satisfy @xmath649 ( this equation and ( 10.18 ) ca nt be satisfied at the same time ) , and thus ca nt be a locally conserved quantity . partly for this reason the theory in this paper is * nonlocal. \(iii ) it seems that the aforementioned mass terms @xmath650 should be included in the boson self - energy term @xmath646 , but the last word on that requires more investigation . if that is the case , what about the remaining parts in @xmath646 ? the conclusions of this section are obtained in the absence of torsion . if torsion is present , then the forms of formulae will be more complicated . from now on we will work in the approximation of metric tensor shown in ( 9.10 ) : @xmath651 which is justified by the success of cqft for perturbative interaction . the metric , as stated in section vi , is @xmath652 correspondingly the ricci curvature for it is @xmath653 the present goal is to determine the physical region on the basis of the above formulae . substituting eq . ( 9.10 ) to the eq . ( 11.2 ) , we find the main quantity to be evaluated is the determinant of matrix @xmath654 . the determinant of ( 9.10 ) is @xmath655 suggested by the horizon of black hole in gr , the horizon - like boundary of the physical region may exist also for those fields with the same theoretical frame as gr . as a tentative step , let s first examine the physical region ( or alternatively , the singularity ) for electrons ( whose dynamics is governed by qed ) to get some rules and then apply them to quarks ( the dynamics is mostly governed by qcd ) . from the form of eq.(11.2 ) , it is noted that if @xmath656 then the logarithm function in ricci tensor becomes divergent . the solutions of the equation are defined as the singularities of ricci tensor . in the large component approximation and @xmath657 , we obtain @xmath658 for the electron in a hydrogen atom , @xmath537 is the coulomb form @xmath10 , then eq . ( 11.5 ) leads to @xmath659 . in the atomic ( natural ) unit , that means the length of @xmath660 is the average radius of the ground state of the hydrogen atom , which suggests that the zero singularity of logarithm function possibly symbolizes one side of physical region . on the other hand , if @xmath661 , then @xmath662 , the electron tends to be free ( asymptotically free ) . in summary we achieve the following conclusion held for electrons , @xmath663 now let s take this conclusion as a rule : even in other situations , @xmath664 and @xmath665 represent the two sides of boundary of physical region * , beyond which there are singularities [ fig . 2 ] . as for a electron in hydrogen atom the singularity region ( * nonlocal region * ) is the region within the range of proton s wavelength . as for the bound - state side of physical region , even in electrodynamics [ 10 ] , one may encounter alternatives in which a particular approximation of @xmath666 seems unnecessary . for instance , that @xmath667 resulting in @xmath668 looks also reasonable . however , in a perturbative theory there is no chance for any components of potential to have a value larger than @xmath665 . so in perturbative case , the only way to meet @xmath669 is to make @xmath670 and @xmath671 . but the requirement to the asymptotically free side is loose , because in any case the alternative solution @xmath672 ( additional to aforementioned @xmath673 , @xmath674 ) of @xmath675 is by no means forbidden . in section vi , we have noted that the metric form is relevant to the interaction vertex of cqft . in qcd , we adopt the colour - spin - independent form to express its interaction vertex . apart from a coupling constant , the vertex reads @xmath676 in which @xmath481 ( @xmath677 ) are gell - mann matrices . correspondingly , the wave function for quarks can be also written in a separable form @xmath678 where @xmath679 is the familiar spinor part [ being transformed under the group @xmath298 ] and @xmath681 the colour part [ being transformed under the group @xmath682 . in this respect , the complete form of metric for colour interaction can be written @xmath683 where the metric tensor @xmath684 may be further decomposed into @xmath685 we have mentioned in section iv how the connection , curvature and the ricci tensor are derived from @xmath686 . being multiplied by an additional matrix element @xmath687 , the forms of the aforementioned geometry quantities constructed from @xmath686 , particularly the ricci tensor , will not be changed since we can make the indices @xmath688 , @xmath689 fixed temporarily so that the factor @xmath687 becomes a constant . the same procedure is equally applicable to fixing the indices @xmath215 , @xmath690 and letting @xmath691 form the corresponding curvature and ricci tensor . combining the two procedures we obtain the ricci tensor for the whole metric @xmath692 where @xmath693 is for spinor and @xmath694 for colour. in view of section vii , we recognize that the metric tensor @xmath294 is for the hyperbolic space and tensor @xmath687 is for ellipse colour space , i.e. , @xmath693 and @xmath694 are subject to different geometries ( different curvings ) though their forms are similar . consequently , the field equation @xmath695 now becomes @xmath696 for simplicity in form , we assume that the constants or other variables before @xmath697 and @xmath698 have been absorbed in @xmath699 and @xmath694 . the above equation provides us with a way to give rise to mass term alternatively , or reversely to eliminate the mass term in eq . ( 9.29 ) . in section - ix we have obtained the field equation in space - time @xmath700 , which can be derived from @xmath693 in eq.(11.12 ) . on the other hand , @xmath701 can contribute a nontrivial term to the field equation in space - time , and roughly we denote such a term by @xmath702 . so far the field equation for colour field is given by @xmath703 from this equation one recognizes that even when @xmath704 can a mass term be provided by @xmath705 , if only the dimension of boson field is larger than @xmath665[@xmath23 ] . possibly the term @xmath705 can offer a mechanism to give rise to mass for gluons . but the reverse possibility exists simultaneously : if @xmath24 , the term @xmath702 may happen to eliminate the contribution from @xmath586when their signs are the same . so at the present formula - level it ca nt be asserted whether the gluon owns mass or not . from both experiments and the theory of qcd , it has been well known that the colour interaction is asymptotically free , i.e. while the transferred momenta of quarks are very large , the coupling constant of colour interaction tends to zero and the interaction becomes a perturbative one . in this case , following the condition @xmath707 in eq.(11.6 ) , we have @xmath674 , @xmath673 or @xmath708 @xmath709 . the other side of boundary of quarks physical region , where the quarks are tightly confined , is assumed to correspond to the case of bound electron in hydrogen atom . now we begin to discuss this side of boundary of physical region for colour interaction . when two quarks are in their ground state ( a low momentum state ) , according to eq.(11.6 ) and the parallelism between confined quarks and bound electrons [ fig . 2 ] , the condition @xmath710 should hold . the eq.(11.11 ) means that either @xmath711 or @xmath712 is satisfied . among others , @xmath712 deserves more attention . obviously it makes the rank of the colour matrix decrease at least by one , from @xmath8 to @xmath590 or @xmath23 , which suggests that quarks condense to hadrons , or possibly others , and thus no free quark appears . this conclusion seems able to account for the confinement of quarks , at least as a mechanism . next we examine the meaning of @xmath710 in more details . under our approximation ( 11.7 ) , we have expressed the metric matrix as @xmath713 , in which the metric tensors of spinor part and colour part become separable . this separable form gives an understanding to colour confinement in the last paragraph . the understanding however , will not be abated even if the two parts of metric tensor get entangled , since anyway the condition @xmath710 makes the rank of matrix @xmath714 decrease at least by * one * , * which * belongs to either the spinor or colour space . therefore in what follows we take into account the general matrix form with two parts entangled together . the following approximation of @xmath715 can be of an entangled form @xmath716 the use of eq . ( 11.14a ) can free us from writing out the explicit space - time indices or colour indices in some special problems . for instance , in what follows we can only write @xmath527 to imply the inclusion of colour part , as @xmath717 , @xmath718 for a perturbative interaction , it is impossible for scalar potential and vector potential to satisfy the relation @xmath719with @xmath720 . to avoid the coincidence with perturbative interaction , for colour interaction we prefer the condition @xmath721 rather than @xmath722 , @xmath723 to give the solution of @xmath724 . so we turn to the case @xmath720 and @xmath725 . we recognize that this case does nt rule out the possibility that while keeping the difference @xmath726 equal to @xmath665 , we can at the same time make both @xmath569and @xmath537 continuously decrease until @xmath537 approaches the _ perturbative _ situation @xmath723with @xmath727(infinitesimal value ) . therefore @xmath720 seems also coincident with _ perturbative _ situation , then we have to consider @xmath728 . without affecting the main conclusion , let s assume that the limiting situation @xmath729 holds . under this situation , we have @xmath730 ( @xmath731 is the corresponding real vector ) . now let s take into account the explicit form @xmath717 . since every component of @xmath527 is a matrix , for example @xmath732 @xmath733 , the value of any component should be one of the eigenvalues ( of the corresponding matrix ) or their combination ( the coefficients @xmath734 , @xmath677 , are real ) after tracing out the degree of freedom of colour interaction by , e.g. @xmath735 where the subscript @xmath736 and @xmath29 represent initial and final states respectively . it can be asserted all values of components are real because the generators @xmath481 ( @xmath677 ) of group @xmath8 are hermitian ( we choose gellmann matrices as the generators ) . then it appears to be inconsistent with the previous conclusion @xmath730 which is a pure imaginary vector . one way to treat the inconsistency is to extend the @xmath737generators of @xmath8 to @xmath738generators of @xmath6 @xmath739 in which an additional pure imaginary generator @xmath740 is introduced , we denote it as @xmath741 . we find @xmath741 , with @xmath742 , is not a hermitian matrix , so that its eigenvalues are not real . now we conclude that when the confinement happens only the first coefficient @xmath743 is nontrivial , it yields @xmath744 on condition that @xmath745 , the same extension of @xmath746generators to @xmath747generators would be necessary , and @xmath527 is a complex number accordingly . the above extension of generators is reasonable because if a space is curved , then some extra dimensions begin to get involved . for instance , after a plane is curved , one must choose three - dimension space and the corresponding transformation group in it to describe the curving . we choose @xmath9 space as the curved quark space in order to satisfy the confining condition . in summary , in this subsection the equation @xmath748 leads us to two explanations of the colour confinement : first , it gives the decrease of the rank of colour matrix , which is a general conclusion independent of separable approximation . second , it induces the curving of colour space , which is a conclusion dependent on the approximation ( 11.14 ) and ( 9.10 ) . _ relationship with conventional quantum field theory ( cqft ) _ the geometry model presented in this paper can also be viewed as a theory for elemental quantum fields , but it lacks many characteristics of cqft . as for symmetries , the transformations responsible for the groups @xmath249 or @xmath250 are used in this paper . whereas the concept of gauge transformation , under which the invariance of lagrangian or action is respected in cqft , is assumed to be irrelevant here . so the conservation laws due to invariant lagrangian are not appropriate to discuss in this frame . however , from purely geometrical angle , we respect the invariance of geometric quantities such as metric and curvature etc . under the transformation of structure group . generally speaking , in a nonlocal theory any conservation law holds only when some integrations are carried out over certain spaces , and thus the square root of jaccobian should appear in the forms of conservation law , as that in gr and implied in sections 810 . furthermore , generally in cqft the lagrangian depends on the fields and their first derivatives only . but in differential geometric , it allows for up to two derivatives of fields , which can be noticed in the quadratic form of motion equation for fermions and field equation for bosons . in view of the distinguished success of cqft in describing the purterbative interactions , the results from these sectors are respected . first , the dirac equation and klein - gordon equation in qed , the asymptotically freedom in qcd etc . are employed to compare with the results of our model . in that respect , the quadratic forms of dirac equation and klein - gordon equation are regained under some approximations . second , the interaction vertices of qed and qcd are used to design appropriate approximations of metric tensor used in the second half of the paper . however , the technique of renormalization is assumed unnecessary in this theory . because the inclusion of the nonlocal characteristics of quantum wave in this theory would automatically make the value of particles momentum within a limited range . _ relationship with general relativity ( gr ) _ as stated in the introduction , the formalism developed in this work is similar to that of gr . the difference is that we have generalized the space from real to complex . the important characteristic of our model is that the base manifold is complex . if the space - time is initially used as the base manifold , it is impossible to clarify physical meanings of dynamics or to get the results of this paper . additionally , the manifolds under concern are not necessarily riemann manifold , i.e. the metric tensor is not necessarily symmetric , @xmath749 . in gr , only one type of curving riemann curving ( hyperbolic curving)appears , but in our theory two curvings are present both hyperbolic and elliptic curvings . they are represented by two different structure groups @xmath249 and @xmath250 , which correspond to two differential geometries . the geometry with structure group @xmath250 is relevant to riemann manifold [ e.g. @xmath750 gives riemann metric ] , but @xmath249 not . in non - abelian case , the metric tensors from two geometries may be entangled as shown in ( 11.14 ) . _ the approximations used in this paper _ all the meaningful results are dependent on the four approximations in eqs . ( 9.10 ) , ( 10.13 ) , ( 11.10 ) and ( 11.14 ) . the credibility of the results relies on the reasonability of the approximations . the approximations follow from our understanding of the interaction vertex in cqft . the appreciated aspect of these approximations is the restoration of the terms of quadratic dirac equation and klein - gordon equation by the aids of the replacement @xmath751 . on the other hand , the appearance of mass term for bosons and the understanding to quark confinement also follow from above approximations and the replacement . from complex space to real space we use @xmath751 as a projection , we have to do so to get some physical observables in space - time . in doing so some information is inevitably lost , but it can not be avoided the observed space is complex , the observing space is real . strictly speaking , the replacement @xmath752 is not an approximation , but rather a technique following the approximation @xmath753 . it helps us gain the form of motion equation for fermions , though with quite a few additional integrations involved . to resolve that intricate motion equation possibly requires us to convert it back to the dirac - equation form by using some special techniques . _ remarks on applicability of the theory _ although the model is designed for a low - energy nonperturbative interaction , at present almost all the approximations imposed on the model have perturbative forms and originations . and thus all the results here rely on both the theory and the approximations . we have noticed some results , such as the additional terms in motion equation and the meaning of @xmath754 etc . , do not completely fit to the perturbative cases , which is expected to be the nonperturbative signals of our model . so after the model is reduced to get the main essences of cqft , we hope it can be perfectly useful for nonperturbative interaction [ e. g. in phenomenological study of the light quark excited states . ] or , long wavelength low energy region of perturbative interaction [ e. g. strong correlations of fermions in ultra low temperature . ] , as anticipated initially . _ future development of the theory _ first , we should search for a smaller group than @xmath515 to accurately describe the hyperbolic curving ( lorentz violation of spinors transformation ) , as stated in sec . the search progress may involve some new physics , not a merely mathematical problem . second , we should present a more reliable way to give the mass term in fermion motion equation in sec . c , some numerical calculation may be involved . third , we hope the theory applicable to weak interaction too . relevantly , the final understanding of mass problem relies undoubtedly on the issues of weak interaction . a lot of thanks to prof . w. t. geng for his constructive suggestions in preparing this manuscript . in this appendix we give the detailed steps and methods of evaluating the first integration in eq . ( 8.10 ) . @xmath755 here @xmath436 is a four - component vector , @xmath437 and @xmath438 are all in their matrix forms , @xmath439 and @xmath756 . we will mainly deal with the first integration in ( 8.10 ) , since the second term includes two @xmath417 factors we temporarily omit it as second - order perturbation . @xmath757 henceforth in this appendix we use @xmath758 to express the product @xmath759 when @xmath760 . we evaluate the above two terms in ( a.1 ) separately . the first term @xmath761 to evaluate the second term in ( a.1 ) , the following relation is useful : @xmath762 substituting this relation to second term in ( a.1 ) yields , @xmath763 we will leave the first term of ( a. 4 ) as it is and turn to cope with the second term , @xmath764 where @xmath765 . then performing the integration by parts , it yields @xmath766 where @xmath444 . in the first term of the right hand we have converted the expansion @xmath767 back to its original form for there is no way integrating out the coordinates to simple form . now substituting eq . ( a.6 ) and eq . ( a.2 ) into eq . ( a.1 ) leads to @xmath768 [ 1 ] shiing - shen chern , w. h. chen , k. s. lan , _ lectures on differential geometry _ , world scientific publishing co. pte . ( 1999 ) . + [ 2 ] r.p.feynman , _ quantum electrodynamics_-a lecture note and reprint volume . w. a. benjamin , inc . + [ 3 ] moshe carmeli and shimon malin , _ theory of spinors : an introduction _ , world scientific publishing co. pte . ltd . ( 2000 ) , chapter 3 and chapter 4 . + [ 4 ] n. n. bogolubov , a. a. logunov , i. t. todorov , _ introduction to axiomatic quantum field theory _ , w. a. benjamin , inc ( 1975 ) : chapter 5 and chapter 7 , relavant part on group @xmath322 . + [ 5 ] s. gupta , proc . london a63 , 681 ( 1950 ) ; k. bleuler , helv . acta 23 , 567 ( 1950 ) . see also , s. weinberg , _ the quantum theory of fields _ , cambridge university press ( 1996 ) : volume ii , pp . + [ 6 ] hai - jun wang , hui yang , and jun - chen su , phys . c 68 , 055204 ( 2003 ) : eq . [ 7 ] y. aharonov and d. bohm , phys . 115 , 485 ( 1959 ) ; y. aharonov and l. vaidman , phys . a * 61 * , 052108 ( 2000 ) . + [ 8 ] t. t. wu and c. n. yang , phys . rev . d 12 , 3845(1975 ) . + [ 9 ] s. weinberg , _ gravitation and cosmology : principles and applications of the general theory of relativity _ , john wiley & sons , inc . , 1972 : pp.165 . + [ 10 ] we use electrodynamics to discriminate it from conventional quantum electrodynamics ( qed ) , though in fact they share the same objectives ; as for qcd , we use quarkdynamics to discriminate the similar situation . fig . 1 : to describe how one fermion observes ( interacts with ) another fermion , we employ the formalism of general relativity by generalizing its space from real to complex . in gragh , only three axes are displayed , in fact we work in four - dimension space . 2 : patterns of physical region in three - dimension space , respectively for electrons ( a ) and quarks ( b ) . likewise the singularity and physical region for complex space are defined in sec . 11 by adding constraints to interaction potential .	the natural recognition of quantum nonlocality follows from the fact that a quantum wave is spatially extended . the waves of fermions display nonlocality in low energy limit of quantum fields . in this _ ab initio _ paper we propose a complex - geometry model that reveals the affection of nonlocality on the interaction between material particles of spin-@xmath0 . to make nonlocal properties appropriately involved in a quantum theory , the special unitary group @xmath1 and spinor representation @xmath2 of lorentz group are generalized by making complex spaces which are spanned by wave functions of quantum particles curved . the curved spaces are described by the geometry used in general relativity by replacing the real space with complex space and additionally imposing the analytic condition on the space . the field equations for fermions and for bosons are respectively associated with geodesic motion equations and with local curvature of the considered space . the equation for fermions can restore all the terms of quadratic form of dirac equation . according to the field equation it is found that , for the u(1 ) field [ generalized quantum electrodynamics ( qed ) ] , when the electromagnetic fields @xmath3 and @xmath4 satisfy @xmath5 , the bosons will gain masses . in this model , a physical region is empirically defined , which can be characterized by a determinant occurring in boson field equation . applying the field equation to @xmath6 field [ generalized quantum chromodynamics ( qcd ) ] , the quark - confining property can be understood by carrying out the boundary of physical region . and it is also found out that under the conventional form of interaction vertex , @xmath7 , only when the colour group @xmath8 is generalized to @xmath9 is it possible to understand the strongly bound states of quarks . pacs : 02.40.tt , 12.20.-m , 12.38.aw , 11.15.tk	25,565	485
smart and responsive complex materials can be achieved by self - organization of simple building blocks . by now , a broad range of functionalized colloidal and polymeric building blocks have been proposed and designed . @xcite this comprises synthetic colloidal structures , e.g. , patchy or janus colloids @xcite or biological molecules such as dna duplexes . @xcite these building blocks are able to self - organized into gel - like structures , e.g. , hydrogels , which are able to undergo reversible changes in response to external stimuli.@xcite thereby , rodlike molecules , such as viruses @xcite or telechelic associative polymers , @xcite exhibit novel scaffold - like structures , and theoretical and experimental studies have been undertaken to unravel their structural and dynamical properties in suspensions . here , polymer flexibility and end - interactions are the essential parameters to control the properties of the self - assembled network structures . @xcite the appearing structures can be directed and controlled by external parameters , specifically by the application of external fields such as a shear flow.@xcite here , a fundamental understanding of the nonequilibrium response of a network structure is necessary for the rational design of new functional materials and that of already existing synthetic and biological scaffold - like patterns . @xcite computer simulations are an extremely valuable tool to elucidate the self - organized structures of functionalized polymers . monte carlo @xcite and molecular dynamics simulation @xcite studies of coarse - grained models of end - functionalized flexible , semiflexible , and rodlike polymers in solution have shown that in thermal equilibrium self - organized scaffold - like network structures form above a critical attraction strength and within a range of concentrations . this network formation is strongly affected by the polymer flexibility , because flexible polymers can span a larger range of distances between connections points , even form loops , and deform easily thereby generating softer networks . the molecular dynamics simulation studies of telechelic polymers of ref . predict flower - like micellar aggregates for flexible polymers . for stiffer polymers , significant morphological changes appear , with liquid - crystalline - like order of adjacent polymers and inter - connected structures.@xcite recent nonequilibrium simulations of end - functionalized rodlike polymers exhibit further structural changes under shear flow . @xcite at low shear rates , the scaffold structure compactifies , while at intermediate shear rates novel bundle - like structures appear with nematically ordered rods . in the limit of very strong flows , all structures are dissolved and the rodlike polymers align in a nematic fashion . in this article , we extend the previous studies and investigate the influence of shear flow on the scaffold - like network structure of end - functionalized _ semiflexible _ polymers . both , the structure properties under shear flow as well as the rheological properties are analyzed for various shear rates . we find that an initial scaffold structure breaks up and densified aggregates are formed at low shear rates , while the structural integrity is completely lost at high shear rates . thereby , flexibility gives rise to particular compact aggregates at intermediate shear rates . in addition , the relaxation behavior of shear - induced structures after cessation of flow is analyzed in part in order to elucidate the reversibility of the shear - induced structures . we apply a hybrid simulation approach , which combines the multiparticle collision dynamics ( mpc ) method for the fluid , @xcite which accounts for hydrodynamic interactions , @xcite with molecular dynamics simulations for the semiflexible polymers . @xcite the mpc method has successfully been applied to study the equilibrium and nonequilibrium dynamical properties of complex systems such as polymers , @xcite colloids , @xcite vesicles and blood cells , @xcite as well as various active systems . @xcite the combination of coarse - grained modeling of end - functionalized polymers and a particle - based mesoscale hydrodynamic simulation technique is ideally suited for such a study . on the one hand , we want to elucidate the general principles of structure formation under nonequilibrium conditions . the achieved insight will be useful to understand the behavior of a broad spectrum of experimental systems , ranging from highly flexible synthetic polymers , e.g. , telechelics , to stiff biological macromolecules , such as dna segments . on the other hand , mesoscale hydrodynamic simulation approaches are essential , because only they allow to reach the large length and time scales , which are required to capture the long structural relaxation times in shear flow with typical shear rates of @xmath0 hz . @xcite in addition and most importantly , particle - based hydrodynamic simulation approaches naturally include thermal fluctuations , which are indispensable for a proper description of polymer entropy and entropic elasticity . of course , coarse - grained modeling has its limitations in predicting the behavior of particular experimental systems quantitatively . here , additional simulations of atomistic models are required to predict binding energies and bending rigidities . this paper is organized as follows . the simulation approaches are introduced in section [ sec2 ] . the deformation of the polymer network under shear and rheological properties are discussed in section [ sec3 ] , and the dependence on the polymer flexibility is addressed . relaxation of shear - induced structures is discussed as well . section [ sec4 ] summarizes our findings . our hybrid simulation approach combines the multiparticle collision dynamics method for the fluid with molecular dynamics simulations for the semiflexible polymers . @xcite in the mpc method , the fluid is represented by @xmath1 point particles of mass @xmath2 , which interact with each other by a stochastic process . @xcite the dynamics proceeds in two steps streaming and collision . in the streaming step , the particles move ballistically and their positions are updated according to @xmath3 here , @xmath4 and @xmath5 are the position and velocity vector of the _ _ i__th particle , and @xmath6 is the time between collisions . in the collision step , the particles are sorted into cells of a cubic lattice with lattice constant @xmath7 , and their velocities are rotated relatively to the center - of - mass velocity @xmath8 of the cell @xmath9,\ ] ] where @xmath10 is the rotation matrix for the rotation around a randomly oriented axis by the fixed angle @xmath11 . the orientation of the axis is chosen independently for every collision cell and collision step . a semiflexible polymer is modeled as a linear sequence of @xmath12 mass points of mass @xmath13 . these monomers are connected by harmonic springs with bond potential @xmath14 where @xmath15 is the position of monomer @xmath16 , @xmath17 is the equilibrium bond length , and @xmath18 is the spring constant . semiflexibility is implemented by the bending potential @xmath19 here , @xmath20 is the bending rigidity , where @xmath21 is the boltzmann constant , @xmath22 is the temperature , and @xmath23 is the persistence length . excluded - volume interactions between monomers are taken into account by the shifted and truncated lennard - jones potential ( lj ) @xmath24 & r < { r_c } \\ 0 & r \ge { r_c } \end{array } \right . , \ ] ] where @xmath25 is the diameter of a monomer and @xmath26 is the interaction strength . aside from the polymer ends , all monomer interactions are purely repulsive , with the cutoff distance @xmath27 and the shift @xmath28 . for the attractive ends , the cutoff is set to @xmath29 and @xmath30 is varied according to the desired attraction strength . the polymer - solvent coupling is implemented by including the monomers in the collision step . hence , the particle center - of - mass velocity of a cell containing monomers is @xmath31 where @xmath32 and @xmath33 are the number of solvent and monomer particles in the cell , respectively . @xcite we consider a cubic simulation box of side length @xmath34 . the parameters for the mpc fluid are @xmath35 , @xmath36 , and the mean number of fluid particles in a collision cell @xmath37 . we choose the bond length @xmath38 as length unit and set for the collision - cell size @xmath39 . moreover , we set @xmath40 , @xmath41 , and @xmath42 . the latter ensures that the bond lengths remain close to the equilibrium value even under shear flow for all considered shear rates . the equations of motion for the monomers are solved by the velocity - verlet algorithm with time step @xmath43 . @xcite in total , 2000 polymers of length @xmath44 are considered . approximating a polymer by a cylinder of length @xmath45 , the polymer - volume fraction is @xmath46 . initially , the polymers are distributed randomly in the simulation box and are equilibrated without end - attraction . then , the end - attraction is turned on and the system is again equilibrated until expectation values reach a steady state . shear flow is imposed on equilibrium structures by lees - edwards boundary conditions , @xcite with the flow direction along the @xmath47 axis and the gradient along the @xmath48 axis of the cartesian reference system . shear is characterized by the weissenberg number @xmath49 , where @xmath50 is the shear rate and @xmath51 is the end - to - end vector relaxation time of a polymer in dilute solution . @xcite explicitly , the values of the relaxation time are @xmath52 and @xmath53 for the persistence lengths @xmath54 and @xmath55 , respectively.@xcite in the following , we will refer to polymers with @xmath56 and @xmath55 as _ semiflexible _ and _ rodlike _ , respectively . for an efficient simulation of the polymer and mpc fluid dynamics , we exploit a graphics - processing - unit ( gpu ) based version of the simulation code . @xcite and the end - attraction strengths @xmath57 . open symbols correspond to the numbers at equilibrium without flow . error bars display the magnitude of the fluctuation in the steady state . the ines are guides for the eye.,width=321 ] , the end - attraction strength @xmath58 and various shear rates . the dashed lines are fits to guide the eye , with a gaussian function for @xmath59 and an exponential function for @xmath60.,width=321 ] for an end - end attraction strength @xmath61 , scaffold structures appear under equilibrium conditions . @xcite this equilibrium scaffold - like network structure undergoes severe structural rearrangement under shear flow . this is illustrated in fig . [ fgr : shear ] , where polymer configurations are shown for the persistence length @xmath56 , the end - attraction strength @xmath62 , and various shear rates . as shear flow is applied , the network breaks up and for low shear rates ( @xmath59 ) densified aggregates are formed . the scaffold structure persists , but the network phase separates into polymer - rich and polymer - poor domains . at intermediate shear rates @xmath63 , smaller , partially connected domains are formed , which are reminiscent to micellar structures . @xcite finally , for high shear rates @xmath60 , the structural integrity is completely lost and polymers are aligned in a nematic - like manner along the flow direction . the initial separation ( for @xmath64 ) into polymer - rich and polymer - poor domains appears in a similar fashion for rodlike polymers.@xcite hence , it seems to be a generic feature of such network structures . however , the shear - induced micellar structures are only observed for more flexible polymers . here , the flow is sufficiently strong to bend the polymers and induce an attraction between the ends of the same polymer . the nematic alignment at high shear rates is again similar to rodlike polymers . it is caused by the shear forces and appears also for dilute solutions of flexible polymers.@xcite to characterize these structures , we determine the average coordination number @xmath65 , which is defined as the number of end - beads in proximity of each other , i.e. , within distances @xmath66 . figure [ fgr : bond]a shows @xmath65 as a function of the shear rate for the end - attraction strengths @xmath67 and @xmath68 . for the lowest value @xmath69 , no scaffold is formed at equilibrium . @xcite in addition , the @xmath70 is independent of shear rate , which indicates that there is no shear - induced network structure either . naturally , the polymers are aligned by the flow , in a similar fashion as non - attractive polymers . @xcite in systems with scaffold structures , the equilibrium coordination number at zero shear exceeds that of disordered systems considerably , as discussed in more detail in ref . . this equilibrium scaffold structure is gradually broken by the shear flow for @xmath71 , and the average coordination number decreases . the number of free ends @xmath72 , which are not adjacent to any other end - bead , increases simultaneously . in contrast , for @xmath62 , the average coordination number first increases with increasing shear rate and passes through a maximum at @xmath73 . this is associated with the compactification of the scaffold structure visible in fig . [ fgr : shear ] . the attraction is evidently so strong that the shear - induced structural changes lead to an enhanced binding of polymer ends . the values of @xmath70 decrease rapidly with increasing shear rate for @xmath74 , and the value of an equilibrium non - attractive assembly of polymers is assumed . simultaneously , the number of free end - beads @xmath72 increases as the shear rate increases , as shown in fig . [ fgr : bond]b . we present the distribution of the coordination number for various shear rates in fig . [ fgr : bnd ] . the dashed lines are fits to guide the eye , with a gaussian function for @xmath59 and an exponential function for @xmath75 . evidently , nodes with a larger number of end - beads are induced at low shear rates ( @xmath59 ) compare to the distribution without flow ( @xmath76 ) . for high shear rates ( @xmath74 ) , the coordination number is significantly small . corresponding mean values are shown in fig . [ fgr : bond]a . a qualitative similar behavior of @xmath77 is found for systems at equilibrium and various attraction strengths.@xcite for attraction strengths @xmath78 , @xmath77 decreases exponentially with increasing @xmath79 . for larger values of @xmath30 , a maximum of the distribution function appears , as also shown in fig . [ fgr : bnd ] . the exponential decay indicates the lack of a network structure either due to too weak attraction or too strong external forces . ) for the end - attraction strength @xmath62 , and different persistence lengths , ( a ) @xmath56 , @xmath80 and ( b ) @xmath81 , @xmath82 . only beads with the slice @xmath83 are shown . the color code corresponds to the number of adjacent ends . ( multimedia view),width=321 ] = 4.0 and the persistence lengths @xmath84.,width=321 ] between bundles for the end - attraction strength @xmath85 = 4.0 and the persistence lengths @xmath84.,width=321 ] polymer flexibility strongly affects the appearing shear - induced structures . this is reflected in fig . [ fgr : shear_f ] ( multimedia view ) , where structures are displayed for the persistence lengths @xmath56 and @xmath55 . for semiflexible polymers ( @xmath86 ) , the original scaffold network breaks up and micellar structures are formed . in contrast , rodlike polymers ( @xmath87 ) are strongly aligned along the flow direction and form thick bundles , an effect already observed for various end - attraction strengths in ref . . in both cases , the end - beads assemble in nodes . for the semiflexible polymers , this can be achieved by significant shear - induced conformational changes of an individual polymer , which gives rise to micellar - like aggregates . the two ends of a polymer can even meet at the same node . @xcite this is not possible for rodlike polymers . their two ends can only participate in two different nodes.@xcite in consequence , more dense structures are formed with well aligned rods . the respective coordination number distributions are shown in fig . [ fgr : bnd_f ] . rodlike polymers form nodes with a large number of end - beads , in agreement with the thick bundles ( cf . [ fgr : shear_f]b ) . to further characterize the shear - induced structure , fig . [ fgr : theta_f ] presents the distribution @xmath88 of angles @xmath89 between bundles for the two different persistence lengths . here , a bundle is defined as a connection of two neighboring nodes by two or more polymers . for semiflexible polymers ( @xmath86 ) , the distribution exhibits a broad peak at @xmath90 . note that a peak at @xmath90 is a characteristics of a scaffold network , @xcite which is more pronounced in equilibrium structure without flow , @xcite while a peak at @xmath91 indicates parallel alignment of bundles along the flow direction . for rodlike polymers ( @xmath87 ) , the peak at @xmath91 is much more pronounced , as expected for bundles . peaks at @xmath90 and @xmath92 are also present for rodlike polymers , which implies that the initial scaffold - like connectivity is not completely lost . , the end - attraction strength @xmath62 , and various shear rates . , width=340 ] ) as a function of shear rate for different persistence lengths and end - attraction strengths . , width=321 ] in the small - strain region for the persistence length @xmath56 , the end - attraction strength @xmath62 , and various shear rates . , width=321 ] in the small - strain region for the persistence length @xmath56 , the end - attraction strength @xmath62 , and various shear rates . the inset shows the number including large - strain region . , width=321 ] the structural rearrangement under shear flow affects the rheological properties of the system . @xcite figure [ fgr : vel ] shows average monomer velocity profiles along the flow - gradient direction for @xmath56 and @xmath62 . the flow profiles are non - monotonic for shear rates @xmath59 , which has also been observed in previous studies of rodlike polymers . @xcite the bands in the velocity profile can be understood as a consequence of the structural inhomogeneity under shear flow . the low - shear - rate regions correspond to polymer - rich domains , where a densified network resists the applied shear . in contrast , polymer - poor domains can flow easily , which yields high - shear - rate regions . for higher shear rates , the velocity profile becomes smoother and we observe a linear monotonic profile for @xmath93 . here , the structural integrity is lost and polymers are aligned along the flow direction ( cf . [ fgr : shear ] ) . for both , rodlike@xcite and semiflexible polymers , a monotonic velocity profile is observed for weak end - attraction strengths ( @xmath94 ) , where the network is either not formed or not strong enough to resist flow . we present the polymer contribution to the shear viscosity @xmath95 as a function of shear rate in fig . [ fgr : strall ] . the polymer contribution to the shear stress @xmath96 is determined by the virial expression @xmath97 where the forces @xmath98 follow from the potentials of eqs . ( [ eq : eq3 ] ) , ( [ eq : eq4 ] ) , and ( [ eq : eq5 ] ) . @xcite the viscosity is then calculated as @xmath99 . for semiflexible polymers ( @xmath56 ) , the viscosity increases with increasing attraction strength for all shear rates ( cf . [ fgr : strall ] ) . in particular , for @xmath62 , the viscosity of systems of rodlike networks ( @xmath81 ) is somewhat larger than those comprised of semiflexible polymers ( @xmath56 ) . evidently , the rodlike nature enhances polymer end contacts , and thus , leads to more stable structures . the systems exhibit shear - thinning behavior for the range of applied shear rates , and a newtonian plateau is observed for weak end - attraction strength ( @xmath100 ) at low shear rates . the shear stress @xmath96 in the small - strain region @xmath101 is plotted in fig . [ fgr : str ] for @xmath62 . the stress increases initially in a linear manner . the end of this elastic regime is reached at the strain @xmath102 . for larger strains , the network deforms plastically and reaches its maximum strength for @xmath103 . for even larger strains , the stress decreases again . the initial elastic response and yield suggests that there is no newtonian viscosity plateau for large attraction strengths . to shed light on the structural change in the vicinity of the maximum strength , we present the average coordination number as a function of strain in fig . [ fgr : bondt ] . initially ( @xmath104 ) , @xmath70 is constant for low shear rates ( @xmath59 ) . in this regime , the network structure is stable and the deformation energy is stored , i.e. , the structure behaves elastically . as strain increases , @xmath70 starts to decrease and reaches a minimum at @xmath103 , where the network structure breaks up . for @xmath105 , @xmath70 increases again slowly ( cf . inset of fig . [ fgr : bondt ] ) , which implies that shear - induced aggregates form . for high shear rates ( @xmath74 ) , @xmath70 decreases monotonically and an asymptotic low steady - state value is assumed . here , the network breaks up continuously as shear flow is applied . and the end - attraction strength @xmath62 . polymers are relaxed without flow after sheared with ( a ) @xmath80 and ( b ) @xmath106 . only beads with the slice @xmath83 are shown . the color code corresponds to the number of adjacent ends . , width=321 ] and the end - attraction strength @xmath62 . polymers are relaxed without flow after sheared with different shear rates . the dashed lines are fits to guide the eye , with a gaussian function , width=321 ] in order to elucidate the uniqueness of the observed structures , we allow the shear - induced structures to relax after cessation of flow . figure [ fgr : relx ] shows snapshots of structures after relaxation from initially sheared states for @xmath56 and @xmath62 . shear - induced aggregates , which are formed at low shear rates ( @xmath107 ) remain after relaxation , and the initial scaffold - like network structure is not fully recovered . when the structural connectivity is fully destroyed for larger shear rates ( @xmath108 ) , the system relaxes back to a scaffold - like network . the coordination number distributions for the two structures are shown in fig . [ fgr : bnd_relx ] . in addition , the distribution of @xmath109 of the initial , non - sheared structure is displayed . the recovered structure after high shear rates ( @xmath110 ) shows a similar distribution of the coordination number as the initial scaffold - like network . however , the coordination number is clearly larger for the relaxed structure after application of a low shear rate ( @xmath111 ) . here , a new ( equilibrium ) structure is formed , which is at least metastable . our studies of networks with weak end - attraction strengths ( @xmath94 ) reveal that the scaffold - like network structure is recovered after relaxation regardless of pre - applied shear rate . the dependence of the network structure on the initial configuration for strong end - attraction ( @xmath112 ) has also been observed at equilibrium . @xcite hence , care has to be taken on equilibrated state of the system . the nonequilibrium structural and dynamical properties of end - functionalized semiflexible polymer suspensions have been investigated by mesoscale hydrodynamic simulations . under flow , the scaffold - like network structure of polymers breaks up and densified aggregates are formed at low shear rates , while the structural integrity is completely lost at high shear rates . we find that network deformation is strongly affected by the polymer flexibility . shear - induced aggregates , which are formed at low shear rates and strong end - attraction , show different structures depending on the polymer flexibility . for semiflexible polymers , the scaffold network breaks up under shear and micellar structures are formed . in contrast , rodlike polymers are more strongly aligned along the flow direction and form thick bundles of smectic - like stacks . for high attraction strengths @xmath113 , we find that shear - induced dense aggregates remain after relaxation , while the system relaxes back to a scaffold - like network when the structural connectivity is fully destroyed under high shear . for lower attraction strengths , the equilibrium structure is fully recovered . our studies shed new light on the nonequilibrium properties of self - organized scaffold structures , specifically their formation and deformation under flow . we expect this knowledge to be useful and provide the basis for further theoretical and experimental studies of such systems . 73ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.109.238301 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1039/c5sm01678a [ ( ) , 10.1039/c5sm01678a ] @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , , ) @noop * * , ( )	the nonequilibrium dynamical behavior and structure formation of end - functionalized semiflexible polymer suspensions under flow are investigated by mesoscale hydrodynamic simulations . the hybrid simulation approach combines the multiparticle collision dynamics method for the fluid , which accounts for hydrodynamic interactions , with molecular dynamics simulations for the semiflexible polymers . in equilibrium , various kinds of scaffold - like network structures are observed , depending on polymer flexibility and end - attraction strength . we investigate the flow behavior of the polymer networks under shear and analyze their nonequilibrium structural and rheological properties . the scaffold structure breaks up and densified aggregates are formed at low shear rates , while the structural integrity is completely lost at high shear rates . we provide a detailed analysis of the shear - rate - dependent flow - induced structures . the studies provide a deeper understanding of the formation and deformation of network structures in complex materials .	7,661	220
low - dimensional quantum systems are a formidable arena for the study of many - body physics : in one or two spatial dimensions ( 1d or 2d ) , the effects of strong correlations and interactions are enhanced and lead to dramatic effects . celebrated examples from condensed matter physics include such diverse cases as the fractionalization of charge and emergence of topological order in the quantum hall effect , high-@xmath0 superconductivity , or the breakdown of landau s fermi liquid theory in 1d , replaced by the luttinger liquid paradigm @xcite . in the past decade , breakthroughs in the field of optically trapped ultra - cold atomic gases @xcite have lead to a new generation of quantum experiments that allow to directly observe fundamental phenomena such as quantum phase transitions @xcite and coherent quantum dynamics @xcite in low - dimensional systems , including 1d gases @xcite . these revolutionary experiments are an ideal playground for the interplay with theory , as they allow to directly realize , in the laboratory , ideal setups that were previously regarded only as oversimplified thought experiments . on the theory side , many exact results in 1d can be obtained by a blend of methods that comprises lattice integrability @xcite and non - perturbative field theory approaches , in particular 2d ( 1 + 1d ) conformal field theory ( cft ) @xcite and integrable field theory @xcite . cft has been incredibly successful at making exact universal predictions for 1d condensed matter systems at a quantum critical point ; these include the kondo effect and other quantum impurity problems @xcite , and the many insights on quantum quenches @xcite as well as universal characterization of entanglement at quantum criticality @xcite . entanglement entropies , in particular , are currently in the limelight , as they have become experimentally measurable both in- @xcite and out - of - equilibrium @xcite , opening the route to a direct comparison between experimental data and many exact analytical results obtained by cft . indeed , entanglement entropies are usually difficult to compute in microscopic models @xcite , but their scaling limit is obtained within the powerful cft approach by solving elementary exercises on riemann surfaces @xcite . there is a caveat in the cft approach to 1d physics though : since it describes low - energy excitations around some fixed energy scale ( _ e.g. _ the fermi energy ) , cft does not accommodate strong variations of that scale throughout the system . this rules out _ a priori _ the possibility of tackling _ inhomogeneous _ systems , in which the relevant energy scale varies . this caveat is important , as inhomogeneous systems are the rule rather than the exception in the realm of quantum experiments : quantum gases at equilibrium always lie in trapping potentials ( often harmonic ) and therefore usually come with a non - uniform density ; this is also the case of many out - of - equilibrium situations , such as trap releases . in this paper , we make one step forward . we focus on the example of the free fermi gas , in a few illustrative in- and out - of - equilibrium inhomogeneous situations . the free fermi gas is technically simpler than interacting models , and yet it allows to draw interesting lessons that will hold more generally . we find that the varying energy scale is taken into account rather naturally in the effective field theory , in the form of varying parameters in the action . interestingly , the _ metric _ is one such parameter , so one generically ends up with a cft in curved 2d space . these conclusions are very general and hold under the reasonable assumption of separation of scales : there must exist an intermediate scale @xmath1 which is simultaneously large compared to the microscopic scale ( the inter - particle distance @xmath2 , where @xmath3 is the density ) , but small compared to the scale on which physical quantities vary macroscopically ( of order @xmath4 ) . indeed , at the intermediate scale @xmath1 , the system is well described by continuous fields because @xmath5 , and is locally homogeneous because @xmath6 , so one knows that it corresponds to a standard ( i.e. flat - space , translation - invariant ) field theory . from there , it is clear that unravelling the global theory for the _ inhomogeneous _ system is a problem of geometric nature : it is about understanding the global geometric data ( _ e.g. _ metric tensor , coupling constants , gauge fields , _ etc . _ ) that enter the action . this is the program we illustrate with the few simple examples below . we demonstrate the power of this formalism by providing new exact asymptotic formulas for entanglement entropies . let us start by considering a free fermi gas in 1d @xmath7 c(x ) , \ ] ] in the absence of an external potential , i.e. @xmath8 . for the reader s convenience , we briefly review the well known relation between this ( homogeneous ) 1d system and the cft of massless dirac fermions in 2d euclidean space - time . the question is : _ what is the proper field theory framework that captures the behavior of long - range correlations of arbitrary local observables @xmath9 _ ? here and below , the @xmath10-coordinate is imaginary time . the starting point to answer this question is the ground - state propagator @xmath11 } , \ ] ] where @xmath12 is the dispersion relation and @xmath13 is the fermi momentum . its large distance behavior is obtained by linearizing the dispersion relation around the two fermi points @xmath14 , @xmath15 with fermi velocity @xmath16 . one finds straightforwardly , for @xmath17 , @xmath18 } \\ % \nonumber & \simeq & \int_{-\infty}^{k_f } \frac{dk}{2\pi } e^{- i \left [ k x + i ( k - k_{\rm f } ) v_{\rm f } y \right ] } + \int_{-k_f}^{\infty } \frac{dk}{2\pi } e^{- i \left [ k x - i ( k+k_{\rm f } ) v_{\rm f } y \right ] } \\ & = & \frac{i}{2\pi } \left [ \frac{e^ { - i k_{\rm f } x } } { x + i v_{\rm f } y } - \frac{e^ { i k_{\rm f } x } } { x - i v_{\rm f } y } \right ] .\end{aligned}\ ] ] these two terms coincide with the right(r)-/left(l)-components of a massless dirac fermion in 2d euclidean spacetime , @xmath19 , that derive from the action ( with @xmath20 ) : @xmath21 .\ ] ] the action ( [ eq : dirac_flat ] ) is invariant under conformal transformations @xmath22 , @xmath23 , where @xmath24 is a holomorphic function of @xmath25 . the phase factors in ( [ eq : prop_transl ] ) may be incorporated into ( [ eq : dirac_flat ] ) with a chiral gauge transformation @xmath26 . here we do not keep track of these phase factors , as they are unimportant for our purposes , and simply discard them ; these aspects are discussed in appendix [ app:1 ] . from wick s theorem , it then follows that the large - scale behavior of arbitrary multi - point correlations of the original fermions can be obtained from correlators in the massless dirac theory . consider the fermi gas ( [ eq : fermi_gas ] ) in a harmonic potential @xmath27 . we ask : _ what is the underlying dirac action _ ? the system is now _ inhomogeneous _ , so there should be parameters in the effective action that vary with position . but _ what are these parameters _ in ( [ eq : dirac_flat ] ) ? in order to understand this , one needs to find the density profile first . the latter is obtained from the exact solution of the microscopic problem in the thermodynamic limit , which in this case is extremely simple . the single - particle eigenstates are just those of the harmonic oscillator , and the many - body ground state is obtained by simply filling up all eigenstates with negative energies . in the thermodynamic limit ( @xmath28 ) , the density profile follows the wigner semicircle law @xmath29 which is non - vanishing in the interval @xmath30 $ ] with @xmath31 . here and in the following , we set @xmath32 . the total number of particles is @xmath33 . away from the edges @xmath34 , there is an intermediate scale @xmath1 such that @xmath35 . at this scale , the system can be viewed as homogeneous , with a local fermi momentum @xmath36 : in a window of width @xmath37 around the position @xmath38 , the system consists of all states filled in the interval @xmath39 $ ] . thus , around a point @xmath40 \times \mathbb{r}$ ] in spacetime , the behaviour of the propagator must be the same as in the translation - invariant gas with @xmath41 : @xmath42 \ , , \quad\ ] ] where @xmath43 , @xmath44 , and @xmath45 is its complex conjugate . like in ( [ eq : prop_transl ] ) , we would like to view the terms @xmath46 as the r-/l - components of a massless dirac field . of course , in the neighborhood of @xmath47 , one can always do that . but the real question is : _ is there a consistent dirac theory defined globally on the entire domain @xmath48 \times \mathbb{r}$ ] , such that its propagator has the required local behavior everywhere _ ? if the reader is familiar with quantum field theories in curved background , they will probably have guessed that eq . ( [ eq : prop_local_harm ] ) is , in fact , related to the propagator of the massless dirac fermion _ in curved space_. the action of the latter theory in 2d is @xmath49,\ ] ] written in isothermal coordinates @xmath50 and in a fixed frame ( see appendix [ app:1 ] for all details ) . the underlying riemannian metric is @xmath51 to connect this theory to eq . ( [ eq : prop_local_harm ] ) , notice that , in the coordinate @xmath25 , the propagator behaves locally as @xmath52 thus , to prove that eq . ( [ eq : prop_local_harm ] ) is the propagator of a dirac fermion in a curved metric , it is sufficient to exhibit a map @xmath53 such that @xmath54 for some real - valued function @xmath55 . this equation can be solved easily . first , notice that it is equivalent to @xmath56 , @xmath57 . writing that @xmath58 , we find a constraint on @xmath59 : @xmath60 . looking for a solution that is independent of @xmath10 , we can set @xmath61 , which implies @xmath62 and @xmath63 . the solution is straightforward : up to an additive constant , we find the complex coordinate system @xmath64 , which lives on the infinite strip @xmath65 \times \mathbb{r}$ ] : @xmath66 this fixes the underlying geometry of the problem . to illustrate the power of this approach , we exhibit new exact results for the entanglement entropies of the fermi gas in external trapping potentials . such calculations are , in general , difficult . but within the framework we just developed , they boil down to elementary manipulations of complex analytic functions . the renyi entanglement entropies of a subsystem @xmath67 are defined as @xmath68 where @xmath69 is the reduced density matrix of @xmath67 and @xmath70 is an arbitrary real number . for @xmath71 , @xmath72 reduces to the von neumann entropy of the subsystem which is usually referred to as entanglement entropy . the main property we will use in the following is that the renyi entanglement entropies are related to the expectation values of the twist fields @xmath73 @xcite which under conformal mapping share the same transformation properties of primaries with dimension @xmath74 with @xmath75 the central charge of the cft ( @xmath76 for the free fermi gas ) . let us now apply eqs . ( [ eq : metric])-([eq : coordinates_harm ] ) to the problem of calculating the entanglement entropy in a harmonic trap . we start with the case of a bipartition @xmath77 consisting of two semi - infinite systems , @xmath78 $ ] , @xmath79 $ ] . in a homogeneous system , the renyi entanglement entropy is @xcite @xmath80 @xmath81 is a uv cutoff , sometimes dismissed in homogeneous systems , because it simply appears in the form of a non - universal constant offset . in _ inhomogeneous _ situations , however , it is crucial to have a closer look at this cutoff : since the energy scale changes throughout the system , why should nt @xmath81 vary as well ? and , if @xmath82 varies , then it must obviously affect the dependence of @xmath72 on @xmath38 . and indeed , there is a good reason why @xmath81 should vary with position : the continuous fermi gas is locally galilean invariant , and the only relevant microscopic scale is the inverse density @xmath83 , or equivalently @xmath84 . so the uv cutoff must simply be proportional to that scale : @xmath85 , for some dimensionless constant @xmath86 . coming back to the harmonic potential , we now make use of the coordinate system @xmath87 , with @xmath88 . the conformal mapping @xmath89 maps the @xmath90strip to the upper half plane . to evaluate @xmath91 , we first perform a weyl transformation @xmath92 , which changes @xmath91 into @xmath93 . next , we notice that , under the conformal mapping @xmath94 from the strip onto the upper half - plane , @xmath95 becomes @xmath96 . the latter factor , which is the one - point function in the upper half - plane , is equal to @xmath97 . putting everything together , @xmath98 hence , using eq . ( [ seps ] ) with @xmath85 , we finally have for the entanglement entropy @xmath99 } % \ , + \ , o(1 ) . s_{n } \ , = \ , \frac{n+1}{12 \,n } \ln \bigg [ k_f(x)\ , e^{\sigma ( z,\bar{z } ) } \ , \left\| \frac{dg(z)}{dz } \right\|^{-1 } \ , { \rm i m } \,g(z ) \bigg],\ ] ] up to an additive constant and subleading corrections , which we systematically drop from now on . this gives @xmath100 .%\ , + \ , o(1).\ ] ] a more complicated bipartition @xmath77 that can be considered in our framework is @xmath101 $ ] and @xmath102 \cup [ x_2 , + \infty]$ ] , where @xmath103 . the calculation is straightforward , but rather cumbersome and so we report it in appendix [ app : finite ] . the final result , setting @xmath104 , can be written as @xmath105 , % \ , + \ , o(1 ) \ , , \ ] ] which is a highly non - trivial generalization of recent results ( for @xmath106 ) obtained by means of random matrix theory @xcite . we checked the validity of this formula against exact finite size computations for lattice and continuous fermi gases using the approaches of refs . @xcite . of particles so as to have a finite density inside each well . our analytic prediction for the entanglement entropy is checked against numerical data for @xmath107 : the agreement is excellent , and improves further when increasing @xmath108 . inset : in phase space , the quasi - classical orbitals are equipotentials enclosing an area that is an integer multiple of @xmath109 . , scaledwidth=80.0% ] the generalization to arbitrary @xmath110 is very simple , even though the single particle problem is not always exactly solvable for a general potential . indeed , we are always interested in the thermodynamic limit , where the single particle states that matter are those up in the spectrum , and for which the semi - classical approximation becomes _ exact_. thus , focusing on the thermodynamic limit ( @xmath28 ) , we proceed as follows : semi - classically , the single - particle eigenstates correspond to the equipotentials @xmath111 that satisfy the bohr - sommerfeld quantization rule . to get the many - body ground state , one simply fills up all eigenstates with negative energies . the equipotentials can be locally parametrized as @xmath112 , where @xmath113 the position - dependent fermi velocity is @xmath114 . the metric that underlies the problem is @xmath115 , one can obtain a complex coordinate system on the worldsheet just like before ; the result for a general potential @xmath110 reads @xmath116 notice that the real part of @xmath117 is the time that a massless excitation takes to arrive at point @xmath38 , starting from some reference point @xmath118 with @xmath119 . the coordinate @xmath25 always lives on an infinite strip @xmath120 \times \mathbb{r}$ ] , whose width @xmath121 depends on @xmath110 and @xmath122 , and which can be conformally mapped onto the upper half - plane by @xmath123 . correlation functions are then evaluated exactly as in the harmonic case above . formula ( [ eq : s_kf ] ) holds , and can be used to derive new exact results for the entanglement entropy . for instance , considering @xmath124 one recovers the results from the so - called trap size scaling @xcite . one example that , to the best of our knowledge , can not be solved with other means is that of a double well potential @xmath125 in fig . [ fig1 ] we compare some exact finite size calculations for this potential with our novel prediction . the figure shows that ( apart from well - known finite size oscillations @xcite ) exact numerical results match perfectly our cft prediction . we now show how the above framework can be adapted to deal with out - of - equilibrium situations . the most natural problem to attack is the fermi gas ( [ eq : fermi_gas ] ) released from a harmonic trap . however , this problem may be solved by other methods , and it is known , for instance , that various observables obey a dynamical symmetry @xcite , including the entanglement entropy @xcite . this symmetry relates time - dependent quantities to their time - independent , ground - state , counterpart , simply by rescaling the coordinate @xmath38 to @xmath126 . since the dynamical symmetry allows to deal with this problem in an efficient way , it is not the most illustrative example for our purposes . instead , we turn to a lattice gas , released from a semi - infinite box : the initial state is such that all sites @xmath127 are filled , and all sites @xmath128 are empty ( this is also known as a quench from a domain wall initial state @xcite ) . at @xmath129 , one lets the system evolve with the hamiltonian @xmath130 \ , .\ ] ] in fourier space the hamiltonian is ( up to an additive constant ) @xmath131 with dispersion relation @xmath132 . is defined as ( [ eq : densitydef ] ) . green colors correspond to a density close to one , blue to a density close to zero . in both cases the system is said to be `` frozen '' : observables do not fluctuate at all . intermediate colors correspond to finite densities , and have non trivial fluctuations . the fluctuating region is a disk of radius @xmath133 , with density given by ( [ eq : densitydef]).,width=453 ] the relevant regime for an effective field theory description is that of large distances and late times , in a way such that the ratio @xmath134 is kept finite . in this limit the density profile at time @xmath135 is @xcite @xmath136 again , the question we want to answer is : _ what is the effective theory that captures long - range correlators @xmath137 in this inhomogeneous system ? _ we expect that it should be a ( lorentzian ) dirac theory in curved 1 + 1d spacetime . there are technical issues , however , associated with the lorentzian formulation of the problem for instance , the metric would be degenerate , @xmath138 , and there would be no clear distinction between right- and left - movers , so we chose to look at the problem in imaginary time , as routinely done in quench problems in cft @xcite . in this imaginary time approach to quantum quenches , the initial state becomes a boundary condition on the two sides of an infinite strip of width @xmath139 in imaginary time direction @xmath10 @xcite . real - time correlators are recovered by first performing a wick rotation @xmath140 , and then sending @xmath141 . we focus on correlators @xmath142 , where @xmath143 $ ] ( see fig . [ fig : arctic ] for the application to the domain wall quench ) . for example , the imaginary - time density profile is @xcite @xmath144 which gives back the real - time profile ( [ rhoreal ] ) after performing the wick rotation @xmath145 and taking the limit @xmath146 . the density is different from zero or one only inside the disc @xmath147 ; thus , there is a phase separation phenomenon , known as _ arctic circle _ @xcite . this is shown in fig . [ fig : arctic ] . in ref . @xcite , the field theory that describes long - range correlations inside the disc was unraveled : it is a dirac theory in curved space , with euclidean metric @xmath148 once we know this , it is again a straightforward exercise in cft to compute correlation functions . as an illustration , we calculate again the entanglement entropy , for the bipartition @xmath149 with @xmath150 $ ] , @xmath79 $ ] at times @xmath151 ( without loss of generality we assume @xmath152 ) . this can be done by performing elementary manipulations similar to those of sec . [ sec : eeharmtrap ] the main difference with the previous calculation is that , due to the presence of a finite lattice spacing , eq . needs to be modified . namely , the position - dependent cut - off is no longer simply proportional to @xmath153 . in the _ homogeneous _ problem , it is known from the exact lattice solution @xcite that the cut - off enters the formula for the renyi entropies as @xmath154 , instead of @xmath153 in the continuous gas . as a consequence of separation of scales , in the inhomogeneous case , we thus need to replace the local cut - off @xmath155 in eq . by @xmath156 . this leads to the following formula for the renyi entropies in imaginary time , @xmath157.\ ] ] here the complex coordinate @xmath158 must be compatible with the conformal structure of the metric ( [ eq : metric_arctic ] ) , _ i.e. _ one must have @xmath159 , for some function @xmath160 . both @xmath25 and @xmath160 are given in ref . @xcite , @xmath161 the function @xmath162 maps the interior of the disc @xmath147 , namely the interior of the critical region in the @xmath47 plane , onto the infinite strip @xmath163\times\mathbb r$ ] . the strip itself can be sent to the upper half plane by the conformal map @xmath164 . finally , recalling that @xcite @xmath165=\arccos \frac{x}{\sqrt{r^2-y^2 } } \,,\ ] ] elementary algebraic manipulations and the use of lead to the following expression for the renyi entropies in imaginary time @xmath166.\ ] ] the analytic continuation to real time is obtained by first substituting @xmath167 , and then sending @xmath168 . this gives @xmath169 \ , , \ ] ] a formula which was guessed from numerics in ref . @xcite , and was calling for an analytic derivation . we just provided this derivation , which crucially relies on the metric ( [ eq : metric_arctic ] ) that underlies the whole problem . in figure [ fig2 ] , we report a comparison of this prediction with numerical data and the agreement is perfect , up to the usual finite - size effects . the entanglement entropy for other bipartitions can be calculated as well , but the resulting formulas are more complicated and are therefore deferred to the appendix [ app : dwfinite ] . we mention that it is also possible to study different dispersion relations , but the results are rather technical ; they are reported in appendix [ app : dispersion ] . after a quench from the domain wall initial state . the numerical simulations are performed using a finite system with @xmath170 sites . data for @xmath171 are reported ( black circles ) and compared to our analytical prediction ( red line ) : the agreement is nearly perfect . inset : corresponding density profile @xmath172 after the quench.,scaledwidth=60.0% ] inhomogeneous 1d quantum systems are difficult to tackle and this is motivating an enormous activity in order to provide exact results in some regimes , as for example the recently developed integrable hydrodynamics @xcite ( anticipated in @xcite ) which may have important ramifications into transport in 1d systems @xcite such as quantum wires . to shed some light on these timely problems , we have shown here , with a few free fermion examples , that cft methods may be extended to attack this class of problems . the key assumption of our work is separation of scales : the system is locally homogeneous ( but only locally ) , which is also the working limit of all approaches constructed so far @xcite . in this regime , one can write a field theory action with varying parameters . in the examples tackled in this paper , we found that the metric tensor should vary , leading to cft in curved space . in particular , this new approach allows us to compute in a simple manner the entanglement entropies of these inhomogeneous systems ( both in- and out - of - equilibrium ) which are otherwise very difficult ( in most cases impossible ) to obtain with other methods . it is important to stress that the background metric and the inhomogeneous cut - off are non - universal functions and must be viewed as _ inputs _ of the formalism . they should be obtained a priori by a proper microscopic computation . we believe anyway that the results of this paper should open the door to several new developments . for example , by following the cft approach of ref . @xcite , our method could be used not only to determine the entanglement entropy , but also the entanglement negativity which is a proper measure of entanglement in mixed states @xcite and it is not yet known how to compute it exactly for free fermionic systems . another interesting development would be to recover by random matrix techniques @xcite the results we obtained for the entanglement entropy in a harmonic potential , possibly unveiling new structures of the random matrices . having exact results also for arbitrary trapping potential could also help understanding whether these general cases could be tackled by random matrix techniques . from a physical point of view , the main open problem is how to describe inhomogeneous interacting 1d systems , most importantly luttinger liquids such as heisenberg spin chains and lieb - liniger gases . in these cases , one expects also the coupling constant ( or luttinger parameter @xmath173 ) to vary with position in spacetime . fixing such parameters is a challenging problem . we hope that this paper will stimulate activity in that direction . we thank nicolas allegra and masud haque for collaboration on closely related topics @xcite , and viktor eisler for useful discussions . [ [ funding - information ] ] funding information + + + + + + + + + + + + + + + + + + + this work was supported partially by the erc under starting grant 279391 edeqs ( pc ) , and by the cnrs interdisciplinary mission and rgion lorraine ( jd ) . jd , jms , and jv thank sissa for hospitality . here we explain the form of the dirac action in a curved space - time , that was used in the main text . the most generic form of the dirac action in a curved two - dimensional eulidean space - time is given by @xmath174,\ ] ] where we chose @xmath175 , @xmath176 , @xmath177 and @xmath178 . the @xmath179 are the usual pauli matrices . in this representation the two components of the spinor @xmath180 are the chiral components @xmath181 and @xmath182 ; the function @xmath183 and @xmath184 are the vector and axial gauge field associated with the @xmath185 gauge symmetry and the ( anomalous ) chiral symmetry . the matrix @xmath186 is called a tetrad : it maps the tangent space of the manifold into @xmath187 while preserving the inner product . it satisfies @xmath188 , where @xmath189 is the metric tensor and @xmath190 the flat metric . in two dimensions , it is convenient to use complex coordinates @xmath191 and @xmath192 . the metric is conformally flat and off - diagonal , the line element thus reads @xmath193 . the tetrad is diagonal if one chooses complex coordinates both in the tangent space and in @xmath187 . its only non - vanishing components are complex conjugated and have with modulus @xmath194 . using finally @xmath195 we can then rewrite the dirac action in complex coordinates as @xmath196\psi_r+ \frac{1}{2\pi}\int d^2z~e^{\sigma - i\theta}[\psi_l^{\dagger}(\stackrel{\leftrightarrow}{\partial_{z}}+i\bar{a}^{(v)}-i\bar{a}^{(a)})]\psi_l,\ ] ] where @xmath197 and @xmath198 are the complex @xmath199 components of the vector and axial potential . the rotation of angle @xmath200 in between the tangent space of the manifold and the flat euclidean space does not alter any of the conformal maps discussed in the main text . this holds because the twist fields needed to compute the entropy are spinless@xcite . the extra phase factors in the fermionic propagators may be restored by performing a @xmath185 gauge transformation and a chiral transformation . the former acts on the chiral fermions as @xmath201 , and on the gauge field as @xmath202 , @xmath203 ; the chiral transformation instead acts as @xmath204 , and the gauge fields get modified as @xmath205 and @xmath206 . the entropies of a finite interval @xmath207 $ ] are slightly more complicated to calculate than the case of the semi - infinite line ( cf . eq . ( [ eq : s_harm ] ) ) . following ref . @xcite , the renyi entanglement entropy of order @xmath70 is related to the two - point function of the twist field @xmath73 on the desired geometry . here we work out the generalization to inhomogeneous systems and in particular for the case of fermions in a harmonic trap . in this case , the two - point function can be related to the one of the upper half plane by the combined action of a weil transformation and the conformal mapping @xmath208 from the strip @xmath65\times \mathbb{r}$ ] to the upper half - plane . this allows us to write such two - point function in the @xmath25 coordinate as @xmath209 the field @xmath210 is the conjugated twist field @xcite . the calculation of two - point functions of twist fields in the upper half plane is , in general , a very challenging problem ( indeed by images trick , it can be turned into a four point function in the complex plane which have been considered e.g. in refs . however , in the case of the massless free fermion field - theory , this two - point function in the half plane simplifies considerably and our desired object boils down to @xcite ( up to unimportant multiplicative constants ) @xmath211^{\delta_n}.\ ] ] in our setup @xmath212 and @xmath213 . using again @xmath208 and ( [ eq : coordinates_harm ] ) , we obtain @xmath214 , \label{harm - int}\end{aligned}\ ] ] where @xmath215 is given in eq . ( [ eq : s_harm ] ) . this is the result reported in the main text . for the case of a finite interval , the renyi entanglement entropy are easily obtained by using eq . and the conformal mapping from the strip @xmath163\times \mathbb{r}$ ] to the upper half - plane @xmath164 . setting @xmath216 ( i.e. equal imaginary times ) and using ( [ eq : zimag ] ) , we get @xmath217.\end{gathered}\ ] ] analytically continuing the result to real time and using the rescaled variables @xmath218 we obtain @xmath219.\ ] ] this formula is valid in the regime @xmath220 . notice that this is very similar to eq . ( [ harm - int ] ) , and indeed it has the same dependence of the rescaled variables @xmath221 . however , the leading dimensional term in one case scales like @xmath222 while in the other as @xmath223 ( times @xmath224 in both cases ) . this is somehow reminiscent of a similar anomalous scaling found in local quenches @xcite and it is unclear whether there is a connection between the two . here we come back to the entropy of a single interval @xmath225 $ ] following a domain wall quench , but with a more general dispersion relation . we consider hamiltonian with a dispersion relation of the form @xmath226 which contains only odd fourier modes . the reason for choosing this special form is mainly technical . as we shall see it makes explicit computations much simpler due to the symmetries @xmath227 and @xmath228 . in real time the stationary phase equation governing the long time dynamics is@xcite @xmath229 and may be used to compute systematically all correlation functions . now , due to the form ( [ eq : dispersion ] ) , if @xmath230 is a solution , then so is @xmath231 . in the following we will assume that there are at most two solutions @xmath230 and @xmath231 to the previous equation . under these assumptions , the density profile is @xmath232 as is emphasized in the main text and above , it is convenient to think of this problem in imaginary time . the stationary phase equation for the imaginary time problem reads@xcite @xmath233 with @xmath234 the hilbert transform of @xmath235 . in case there are only a finite number of fourier modes in the dispersion , solving ( [ eq : imstatphase ] ) amounts to solving algebraic equations in @xmath236 , which can be done in principle systematically . now we make the extra assumption that there are only two solutions @xmath237 and @xmath238 . the new variable @xmath25 lives in an infinite strip @xmath239 . the metric then reads @xmath240 the conformal distance to the boundary may be obtained by mapping the infinite strip to the upper half plane . such a mapping is once again provided by @xmath164 , so we obtain @xmath241 separating the real and imaginary parts @xmath242 , the above reads @xmath243 where @xmath244 and @xmath245 satisfy the system of equations @xmath246 & = & x,\\\label{eq : genstatphase2 } \sum_n ( 2n+1 ) a_{2n+1 } \sin ( 2n+1)\kappa \left[y\cosh ( 2n+1)q+r\sinh(2n+1)q\right ] & = & 0.\end{aligned}\ ] ] the reasoning to get the entanglement entropy should be clear at this point : first we get @xmath244 and @xmath245 from the above set of equations , and then we use them to compute the conformal distance to be plugged in the appropriate equation for the entropy . for example in the case of the standard cosine dispersion relation , using this procedure , we find the already known result @xmath247 , which can then be continued to real time @xmath248 . it is important to stress that the concept of conformal distance only makes sense in imaginary time : in principle only after computing @xmath249 in imaginary time we are allowed to make the analytic continuation to real time . in practice , one can avoid finding the general solution of the above system of equations . indeed , because of their analytic structure , we can perform the analytic continuation at the level of eqs . ( [ eq : genstatphase1 ] ) and ( [ eq : genstatphase2 ] ) , relaxing the requirement that @xmath244 and @xmath245 are real . plugging @xmath250 and taking the limit @xmath168 , we find that @xmath251 is a trivial solution to ( [ eq : genstatphase2 ] ) , due to the absence of even fourier modes . the variable @xmath244 is then the solution of @xmath252 = v(\kappa+\pi/2),\ ] ] and so we get @xmath253 . the analytic continuation of the conformal distance becomes @xmath254 putting back the contribution from the uv cutoff , @xmath255 , we finally obtain @xmath256 let us briefly comment on the case with more than two solutions to the stationary phase equation ( [ eq : realstatphase ] ) . with the form ( [ eq : dispersion ] ) of the dispersion they always come in pairs . we label them as @xmath257 and @xmath258 for @xmath259 . @xmath260 may be interpreted as a number of fermi seas . as before we compute the entanglement entropy using field theory , and add to this the contribution coming from the position - dependent uv cutoff . the generalization of our field - theoretical framework to account for several fermi seas is straightforward : we simply introduce @xmath260 different species of dirac fermions , one for each fermi sea ( pair of stationary modes ) . being non - interacting particles , all the dirac species contribute independently to the entanglement entropy . obtaining the position - dependent cutoff is more tricky , but can nevertheless be done using exact equilibrium results obtained for several fermi seas . we refer to ref . @xcite for the details . the lattice cutoff takes now the more complicated form @xmath261 the final result for the entropy reads @xmath262 after a quench from the domain wall initial state , for dispersion relation @xmath263 and @xmath264 . he numerical simulations are performed using a finite system with @xmath170 sites . data for @xmath171 are reported ( black circles ) and compared to our analytical prediction ( red lines ) : the agreement is nearly perfect . inset : corresponding density profile @xmath172 after the quench.,scaledwidth=60.0% ] let us now work out an example where the formula ( [ eq : main ] ) may be computed explicitely . we consider a dispersion of the form @xmath265 the restriction on @xmath266 ensures that there are at most two solutions to the stationary phase equation , @xmath230 and @xmath231 . @xmath267 may be obtained by solving a cubic equation : @xmath268,\qquad b(x , t)=\sqrt{9\alpha(x / t)^2+(1-\alpha)^3}-3\sqrt{\alpha}(x / t).\ ] ] the density profile in the inhomogeneous region @xmath269 is then given by @xmath270,\ ] ] and the entropy is obtained by plugging ( [ eq : statsolution ] ) in ( [ eq : main ] ) . in fig . 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we live in interesting times . in the world of high energy and nuclear physics the relativistic heavy ion collider ( rhic ) at brookhaven national laboratory is beginning its search into the new realm of high temperatures and low but nonzero chemical potentials . these experiments will surprise us . experiments have a habit of doing that . they humble us . they will show us new directions . they will make this talk obsolete . i want to emphasize that lattice gauge theory , which has become an evolutionary rather than a revolutionary field , should take a more active part in these developments . it should rise to the challenge of real data to find new methods and ideas and extend its comfortable euclidean ways to describe , predict and learn from scattering experiments . pioneering has the potential of reinvigorating the field . lattice gauge theory has provided a solid estimate for the critical temperature to make the quark gluon plasma as well as not - so - solid estimates of the required energy density , the magnitudes of screening lengths etc . but there is much more to predict ... hopefully before the experiments ... and doing so will be particularly exciting . and fun . i think that there are promising areas for such developments and i will discuss some throughout this talk . light - cone wavefunctions of nuclei , dense with gluons , quarks and anti - quarks , collisions with unexpectedly high multiplicities and signs of early development of a quark - gluon plasma will be reviewed . along the way suggestions for additional or new lattice based research will be made . other subfields of high energy physics are already interacting with the data . lattice gauge theorists should become more active participants . it hardly needs to be emphasized here that when lattice gauge theory develops a method to analyze a problem , it can do so from first principles , with ultimately no approximations . emphasis on the word `` ultimately '' , because our lattices are yet too small , our quark masses are yet too large , our statistics are yet too few , etc . but the field is making steady progress on all these issues , helped particularly by its use of improved but practical actions . lattice gauge theory is the only approach that can handle the physics of the three crucial length scales of hadronic dynamics at one time , in a consistent calculation . at sufficiently short distances lattice calculations and simulations confirm asymptotic freedom , which is so essential to exposing the underlying gluon and quark degrees of freedom of qcd as experiments enter the quark - gluon plasma phase . at more moderate distances where the running coupling is in the intermediate range and semi - classical instanton configurations are breaking the anomalous @xmath0 symmetry and , through properties of their ensemble , are breaking chiral symmetry and are producing the constituent quark masses , lattice gauge theory is at its best elucidating the complex crossover physics of this range of length scales . finally , at greater length scales , lattice methods confirm confinement , the fact that hadronic states are color singlets and the confining dynamics comes through thin , discernable but breakable flux tubes . qcd will not be fully understood until these three qualitatively different ranges of phenomena are incorporated into one tractable analytic approach . crucial hints needed to accomplish this will come from lattice studies . these are grand , and over - stated words . but progress is occurring . it is a pity that progress is not occurring on the challenge of producing a lattice simulation method for qcd at nonzero baryon chemical potential @xmath1 and vanishing temperature @xmath2 . the infamous sign problem of the fermion determinant continues to stand in our way . the same problem has bedeviled condensed matter physicists for almost 40 years . a theme of this talk is that studying extreme environments teaches us how qcd works under ordinary conditions . i believe that when we have licked the sign problem , or have found a new formulation of nonperturbative qcd free of it , we will be at a new level in our understanding of how qcd really works and makes its low lying baryonic excitations . in the second half of this review i will discuss several interesting contributions at this conference . these will include progress in mapping out the low @xmath1 ( chemical potential ) , high @xmath2 ( temperature ) part of the phase diagram of qcd , following the seminal work of fodor and katz . real time spectral functions for the production of lepton pairs will be discussed as will the dispersion relation of pions below but near the transition to the plasma . a first step toward simulating phenomenologically interesting cutoff four fermi models of the transition will also be mentioned briefly . as emphasized by e. shuryak @xcite , the mass scales of the quark gluon plasma are different and , importantly , smaller than those of the more familiar hadronic phase . the hadronic phase breaks chiral symmetry , the quark gluon plasma does not . the hadronic phase confines quarks , the quark gluon plasma does not . the binding mechanism in the hadronic phase is nonperturbative while the screening mechanism in the quark gluon plasma is perturbative . we know from deep inelastic scattering that the substructure scale in the hadronic phase is @xmath3 gev . at this @xmath4 the running coupling is becoming large enough that perturbation theory is failing to give accurate estimates and nonperturbative effects are competitive . by contrast perturbative screening masses in the quark gluon plasma are @xmath5 gev . and @xmath6 gev . for temperature above but near @xmath7 where the plasma first appears . the finer level spacings in the plasma act as a fine resolution grid to the dynamics in the hadronic phase . a collision which starts in the hadronic phase and ends in the plasma phase is especially sensitive to the levels and dynamics of the hadronic phase because of the relative wealth of open channels in the plasma phase . rhic collisions can be analyzed in four stages which are thought to be characterized by distinct time scales . first , viewed from the center of momentum frame , there are right moving ( positive @xmath8 ) and left moving ( negative @xmath8 ) nuclei . each of them can be described by infinite momentum frame wavefunctions which give the probability amplitudes for each nuclei to consist of ensembles of quarks and gluons . the parton model , suitably improved with asymptotic freedom to describe its short distance features , should provide a framework for these initial states . second , there is the collision between the constituents in each nuclei . experiments will shed light on the typical momentum transfers and multiplicities of these underlying collisions . third , there is the development of a thermalized plasma of quarks and gluons . and fourth , there is the development of the final states of hadronic debris from the hot soup of colliding and produced constituents . begin with the lightcone wavefunctions of the projectiles . the rightmover is , @xmath9 recall from infinite momentum frame @xcite or light cone perturbation theory @xcite that @xmath10 is the longitudinal fraction of the i@xmath11 constituent . it is a positive variable , @xmath12 and the constituents in each wavefunction @xmath13 account for the longitudinal fraction of the projectile , @xmath14 . the absence of vacuum structure in @xmath15 makes this wavefunction a useful formulation of the initial state of the collision . the bulk features of the wavefunctions of the projectiles are conveniently displayed on a rapidity plot , @xmath16 $ ] . the rapidity is particularly handy because it simply translates under a boost along the @xmath17 axis . one can plot the density of partons of both projectiles on a rapidity plot and deep inelastic experiments at a fixed @xmath4 indicate that the plot is essentially flat and boost invariant and of extent @xmath18 , where @xmath19 is the usual scattering variable , four times the square of the center of momentum energy . 3 in 3 in experiments indicate that parton rapidity distributions have some simple , statistical features . short range correlations in rapidity . the parton distributions lose memory , with a correlation length of @xmath20 units , in rapidity . screening in rapidity is an essential feature . therefore , features like the quantum numbers of each projectile are limited to the edges of the rapidity plot . unfortunately , this means that in very high energy heavy ion collisions , most of the partons are not influenced by the baryon number of the projectiles . the collisions can be characterized by a high energy density and temperature but not high chemical potential @xmath1 . the `` central region '' and is universal and `` vacuum like '' . the length of the central region grows like @xmath18 . although first principle calculations of light cone wavefunctions are beyond us , models having these features have been known for decades . multiperipheral graphs of qcd can populate the rapidity axis uniformly and have suggested that gluons are the most likely parton species in the central region @xcite . asymptotic freedom effects these considerations profoundly . recall that in the naive parton model , partons interact only softly with one another , never exchanging high transverse momenta . but if we consider the structure of the wavefunctions at high spatial resolution where the running coupling is small and perturbative asymptotic freedom applies , then high transverse momentum exchanges and short distance additional structure are inevitable @xcite . call the gluon distribution function in momentum space @xmath21 . this is the distribution function that would be resolved by a probe with resolving power @xmath22 . models and theoretical analysis of scattering data suggests that @xmath21 approaches a constant for small @xmath23 and fixed @xmath4 . however , as @xmath4 is increased one discovers that gluons can split perturbatively into three or four gluons of lesser longitudinal fractions @xmath10 . so , for sufficiently small @xmath23 , @xmath21 must be an increasing function of @xmath4 . this is a general feature of the asymptotically free parton model @xcite and perturbative evolution equations make this simple physical idea quantitative @xcite . for quark distribution functions which are accessible to deep inelastic electron and neutrino scattering , these ideas are well tested phenomenologically . if we view the parton density on the plane transverse to the collision axis @xmath8 , then at low @xmath4 we see just a few gluons of size @xmath22 , but as @xmath4 is taken larger we resolve many more gluons of smaller size . this progressive development has lead to the idea of `` gluon saturation '' in the light cone wavefunction of a nucleus @xcite . to understand this idea , consider the wave function and imagine that it is coming at you and you have a snapshot of the transverse plane . call the transverse size of the nucleus @xmath24 . each parton of transverse momentum @xmath25 in the wavefunction occupies an area @xmath26 , as suggested by the uncertainty principle . the parton s cross section is proportional to the square of the running coupling and its geometrical size , @xmath27 . but partons will overlap in the transverse plane when their number @xmath28 becomes comparable to @xmath29 which is @xmath30 . one supposes that when @xmath28 is this large , the overlap stops further growth in the density and `` saturation '' has been achieved @xcite . the transverse momentum of the partons at saturation is @xmath31 . the number of partons at the saturation level is @xmath32 and these are the number of partons of size scale @xmath33 which could materialize in the quark gluon plasma . in addition @xmath34 sets the scale for the transverse momenta of partons in the central region of the rapidity plot , @xmath35 . these formulas can be made more precise by considering longitudinal fractions @xcite . the discussion here is just a simplified , but , hopefully , intriguing introduction . when numbers for the various nuclei and the energies available at experimental facilities are substituted here , one finds that @xmath36 gev . this means that applying perturbative qcd is suspect and competitive nonperturbative effects should be expected . this is potentially the realm of lattice gauge theory . the relationship of euclidean lattice calculations and the minkowski scattering formulation appropriate to heavy ion collisions must be addressed . let s comment on the four different stages of a heavy ion collision separately . the physics issues in each are quite distinct . wave functions approach on another . as reviewed above , the phase space parton density is quite large @xmath37 . classical and semi - classical estimates should apply . in particular , the color per volume is large , so the lack of commutivity , @xmath38=if^{abc}q^c$ ] , appears to be a relatively small effect , and the gluon field can be treated classically for many purposes @xcite . \2 . interactions . the gluon saturation picture suggests that the characteristic time of interactions is quite small , @xmath39 fm./c and the energy density is very , very large , @xmath40 gev./fm.@xmath41 . such a large energy density would place the system deep in the quark gluon plasma phase . this is good news ! estimates of @xmath42 in the past were much more modest , on the order of several gev./fm.@xmath41 , at most . thermalization . after the collision the quark gluon matter expands and thermalizes . the experimental data and models suggest that the time scale for this to occur is @xmath43 fm / c . rhic data , especially that showing the spatial distribution of particles produced in non - head - on collisions , strongly suggests that the interactions at early times are strong and the quark gluon plasma appears very early in the evolution of the final state @xcite . the basic parton - parton interactions must be strong and involve high multiplicities . perturbative processes alone are not sufficient to explain the data . hydrodynamic expansion . the quark gluon plasma maintains thermalization and expands until decoupling sets in at @xmath44fm./c . in the expansion and production of the final state hadrons , the fastest particles are produced last , as in the `` inside - outside '' cascade of electron - positron annihilation processes @xcite . the expansion process is essentially one - dimensional , along the collision axis , until the latest stages of the development of the final state . a particle s rapidity turns out to be strongly correlated with the space - time rapidity , @xmath45 $ ] , of its point of creation . rhic experiments are hard and complicated and subject to many phenomenological models . typically several models , each in stark contradiction to the other , can describe a limited set of heavy ion data equally well . the `` smoking gun '' for the existence of the quark gluon plasma has yet to be found , but there are several pieces of data which are nicely explained , or at least , accommodated by it . consider a few pieces of the puzzle . jet quenching . the heavy ion collisions show little or no sign of jets for @xmath46 gev . this is interesting in light of the fact that if you scale single particle hadron spectra from proton - proton data to a - a data , you overshoot the jet data very significantly @xcite . it has been argued that this is evidence for the existence of an extended space - time region of the quark gluon plasma . the idea is that two colliding gluons in the plasma annihilate into an energetic quark anti - quark pair with sizeable tranverse momenta , but their interactions with the plasma medium saps the energy out of the energetic quark and its partner , eliminating ( `` quenching '' ) those jet - like characteristics . of course , very high energy jets are uneffected by the medium because of the short range nature of the most prevalent interactions on the rapidity axis , and jets with energies much larger than @xmath47 gev . are seen . suggestions of deconfinement and chiral symmetry restoration . lepton pairs have long been cited as effective probes into the internal dynamics of the quark gluon plasma . in particular , if the @xmath48 and @xmath49 existed in the plasma , then the leptonic decays , @xmath50 and @xmath51 would be easily seen , as in proton - a collisions . these peaks are missing , however @xcite . it is tempting to interpret this experimental result as evidence that the light hadrons have `` melted '' in the hot plasma . we will consider some lattice gauge theory data concerning these peaks in the second half of this talk . screening and j/@xmath52 suppression . one of the first theoretical suggestions for a signal of the existence of the quark gluon plasma concerned the heavy quark states j/@xmath52 . it was argued and backed up with potential model calculations , that quasi - free quark and gluon thermal screening would reduce the attractive forces between the heavy quarks sufficiently to eliminate the binding energies and the j/@xmath52 states themselves @xcite . in fact , the suppression of these states , again implied by the absence of the associated lepton pairs , is much more dramatic than that expected in hadron absorption models . rhic experiments suggest that there are early , strong , high multiplicity parton - parton collisions in heavy ion scattering processes . perturbation theory , such as the qcd multiperipheral diagrams @xcite which can fill the rapidity plot with a uniform density of gluons , fall far short of describing the data . this suggests that larger length scales where the running coupling is in the intermediate coupling range are involved . this is the domain where semi - classical collective excitations of qcd are important . e. shuryak has suggested that instantons and sphalerons @xcite may be causing some of the surprising scattering events seen at rhic . we have taken a short survey into topics of interest to heavy ion physicists . can lattice theorists have much impact here ? much of the physics here is stated in minkowski space , natural to scattering processes . employing euclidean lattice methods to tackle some of these questions will certainly take new initiatives and methods . as we will review in the second half of this talk , lattice methods do provide information on real time processes even within the present techniques we use . these include power spectra , maximum entropy methods and lepton pair rates . however , more comprehensive methods closer to first principles can probably be developed , too . let s look at a few such topics . the parton picture of qcd structure requires the use of the infinite momentum frame , or , equivalently light cone quantization . each parton has a particular tranverse and longitudinal momentum in a wavefunction expansion , as written above . infinite momentum frame hamiltonians have been written down in this language and two dimensional hamiltonians which control the transverse dynamics have been studied . lattice gauge theory versions of these systems can be constructed @xcite and equivalent three dimensional lagrangian systems derived . one can hope that formulations of this sort could be used to address issues of parton saturation in the context of lattice methods which can handle honestly the intermediate coupling character of these problems . the infinite momentum frame idea will eliminate vacuum fluctuations from the nonzero longitudinal fraction parts of the problem . in the context of continuum methods , there have been recent attempts along these lines by heavy ion physicists . an ambitious attempt to formulate deep inelastic scattering in euclidean terms has been pioneered by o. nachtmann @xcite . this looks like particularly fertile ground for a lattice incursion . as i will review in the second half of this talk , the region of the qcd phase diagram for small @xmath1 in the vicinity of the quark gluon transition @xmath7 is accessible to conventional simulation methods . this is a recent development that opens up experimentally relevant lattice calculations to the whole field . the thermodynamics of the formation of the quark gluon plasma for @xmath2 near @xmath7 and small baryon chemical potential @xmath1 should be studied very professionally now . some of the projects will be reviewed in the second half of this talk . we would certainly like to know how quickly the underlying physical mechanisms responsible for the transition from the hadronic phase to the quark gluon plasma phase change as @xmath1 is turned on for @xmath54 . we know that there are substantial interactions in the quark gluon plasma just above @xmath7 . how are these mechanisms and how are bulk thermodynamic quantities effected as @xmath1 is taken nonzero ? the role of instantons in the formation and character of the quark gluon plasma needs more elucidation even at the @xmath55 edge of the qcd phase diagram . according to the instanton liquid model @xcite , the instanton ensemble is disordered in the hadronic phase . this is essential for the model to produce chiral symmetry breaking and phenomenologically reasonable numbers . it is believed that as @xmath2 passes through @xmath7 , instantons and anti - instantons pair up and form `` neutral '' molecules . chiral symmetry is restored in such an ordered ensemble . it would be interesting to verify this `` binding - unbinding '' picture of the phase transition by lattice methods and to provide estimates of the scales of the phenomena involved . this will not be easy because instantons and anti - instanton molecules will tend to `` fall through '' the lattice and be hard to identify , let alone measure . improved actions will probably be needed . such studies would be worth doing because we know from other simulations that there are substantial interactions in the quark gluon plasma just above @xmath7 . we would also like to know how instanton dynamics change as @xmath1 is turned on for @xmath54 . these are interesting questions which are experimentally relevant . conventional lattice methods can address them now . perturbative @xcite and instanton @xcite methods have been used to predict color superconductivity for low temperatures and asymptotically large @xmath1 . a large gap has been found as well as interesting symmetries ( `` color - flavor locking '' @xcite ) . the large gap in the spectrum of the cooper quark - quark pairs suggests that this state of matter should occur in natural phenomena , such as the interiors of neutron stars , at high but not inaccessible densities . the lattice has been unable to contribute to this field because of the sign problem in the euclidean fermion determinant in @xmath56 lattice gauge theory . the weak coupling , perturbative and instanton , approaches to color superconductivity have not addressed the numerical impact of confinement on their considerations , so estimates of a critical density are suspect . much of the research has concentrated on @xmath57 color which is qualitatively different from qcd . cooper pairs in the @xmath57 model are color singlets and are degenerate with ordinary meson states because of the self - conjugate nature of quark and anti - quarks in this case . at chemical potentials greater than half the pion mass , the ground state of the model becomes a superfluid . instanton enthusiasts have noted that their model provides attractive forces which favor the appearance of scalar @xmath58 diquarks @xcite . the forces are only half as strong as the analogous ones in the @xmath57 model where diquark states are degenerate with mesons . this suggests one should find considerable scalar @xmath58 diquark correlations in ordinary hadronic physics . to my knowledge , there are none . perhaps lattice spectroscopists could scour their baryon wavefunctions and search for diquark correlations in their three body states . aside from the issue of color superconductivity , it would be good to settle the old controversy concerning the possible viability of diquark models of baryon structure . weak coupling descriptions of color superconductivity do nt need diquark correlation enhancements in ordinary hadronic states to be successful and accurate at nonzero chemical potentials . this is because the cooper pairs of the bcs theory have very large , `` floppy '' wavefunctions which are much larger than protons @xcite . the parallel sessions at lattice 2002 concerning qcd in extreme environments contained a number of interesting contributions . because of the lack of space i will discuss just four topics that generated the most discussion . i will have to point the reader to the writeups of the parallel talks for details . the four topics will be : 1 . the high @xmath2 , low @xmath1 edge of the qcd phase diagram , 2 . thermal dilepton rates , 3 . real - time pion propagation in hot qcd , and 4 . four dimensional four fermi models at nonzero @xmath1 . following the seminal work of fodor and katz last year @xcite , the field has realized that the high @xmath2 , low @xmath1 edge of the qcd phase diagram is accessible to a variety of mature lattice techniques . fodor and collaborators @xcite presented arguments that their multi - parameter generalization of the glasgow method has an efficiency which only falls off as a power , @xmath59 with @xmath60 , as the size of the lattice grows large . de forcrand and philipsen @xcite have done simulations at imaginary chemical potentials using conventional hybrid molecular dynamics algorithms and have found convincing , controlled results for @xmath61 . the bielefeld - swansea group @xcite have presented taylor expansions in @xmath1 of the fodor - katz method . sinclair @xcite simulated @xmath56 lattice gauge theory at nonzero isospin chemical potential @xmath62 and found that @xmath63 resembles the curve @xmath64 predicted by the other groups . before we consider the first two contributions in more detail , recall some elementary points . the transition from the hadronic phase to the quark gluon plasma phase at @xmath65 and @xmath66 is actually a rapid crossover for the range of quark masses presently under investigation . most lattice theorists believe that it is also a rapid crossover for realistic quark masses , @xmath67 , @xmath68 and @xmath69 , but larger scale simulations are needed here . as @xmath1 is turned on one expects that the crossover @xmath64 to decrease slowly . however , for sufficiently large @xmath1 one expects the crossover to become a first order transition and it is this point `` e '' , the critical endpoint , @xmath70 and @xmath71 , that one would like to know @xcite . present simulations indicate that @xmath71 is quite large @xcite . however , those simulations are on small lattices with relatively large quark masses . they should be repeated on large lattices with realistic quark masses to get a relevant answer for @xmath70 and @xmath71 . if @xmath71 turns out to be very small , there would be dramatic effects , potentially , in the projectile fragmentation regions at rhic . 3 in 3 in fodor and katz @xcite introduced multi - parameter reweighting to improve the overlap of lattice configurations on the edge of the phase diagram with particular points inside the phase diagram , @xmath72 this formula organizes the calculation of the partition function at a point inside the phase diagram , by making configurations on the edge of the phase diagram ( first line ) and reweighting ( second line ) in two variables , @xmath1 and @xmath73 . the idea is that configurations at @xmath7 experience both the hadron and the quark gluon plasma phase so there should be a considerable overlap with configurations nearby but at different @xmath1 and @xmath73 values . care must be taken here : as @xmath1 is increased , @xmath73 must be increased slightly so the system remains near the crossover coupling for that @xmath1 value , @xmath74 . as @xmath1 and @xmath73 are increased , one monitors the acceptance rate and keeps it near 50 percent @xcite . in this way lines essentially parallel to @xmath74 but displaced into the hadronic or the plasma phase have also been investigated and the equation of state has been calculated for a range of @xmath2 and @xmath1 values . they have shown that the crossover sharpens as @xmath1 grows and the endpoint `` e '' is approached @xcite . the position of the endpoint was found in last year s contribution but studying the finite size scaling properties of the lee - yang zeros of the partition function @xcite . 3 in 3 in following the paths in phase space where the authors monitor the efficiency of their overlap method , they are now computing the equation of state @xcite . the crossover is predicted to sharpen considerably as @xmath75 increases . the authors argue that at least for volumes up to @xmath76 , they can simulate chemical potentials up to @xmath77 with @xmath78 . and they argue that the efficiency of their multi - parameter overlap method scales as @xmath59 with @xmath60 , so the method should allow fairly large lattices to be simulated before its inefficiencies become overwhelming . the reader should consult the parallel session contribution by these authors for plots and details @xcite . the second contribution presented on this topic by de forcrand and philipsen @xcite simulated lattice at imaginary chemical potential using the conventional hybrid molecular dynamics algorithm . it followed the crossover , @xmath74 , by computing the plaquette susceptibility and monitoring its peak as a function of @xmath73 for fixed values of the imaginary chemical potetial . the authors argue that the curve @xmath74 is an even , analytic function of @xmath1 . writing @xmath79 , they find that only the first two terms are needed to describe the data accurately . the analytic continuation to real @xmath1 is , therefore , trivial . their resulting curve is in agreement with that of fodor and katz , although they havent investigated the order of the transition as a function of @xmath1 yet . the imaginary chemical potential method is very conservative and effective . it will work on large lattices with the efficiency of an ordinary hybrid molecular dynamics code . it can only treat small chemical potentials because imaginary chemical potentials induce tunnelling to other @xmath80 vacua when the imaginary chemical potential equals @xmath81 @xcite . for @xmath2 near @xmath82 mev . , the method is limited to @xmath83 mev . the reader is referred to the relevant contribution at the parallel session for quantitative details . in addition , similar work has appeared by delia and lombardo @xcite . we can look forward to considerable progress in understanding this strip of the qcd phase diagram now that several simulation methods and groups are involved . dilepton rates were discussed above as an informative probe into the dynamics of the quark gluon plasma . this topic also provides a nice example of traditional lattice gauge theory confronting new experimental scattering data . the dilepton rate , which is a function of frequency @xmath84 and momentum @xmath85 , can be written in the form @xcite , @xmath86 where the spectral function @xmath87 is related to the current - current correlation function , @xmath88 , on the periodic lattice as , @xmath89 on a @xmath90 lattice , one can measure the correlation function @xmath91 at each `` time '' slice , @xmath92 , @xmath93 and then attempt to invert the relation between the spectral function and the correlation function . there are two approaches to executing this inversion . first , one might use a model dependent functional form which depends on several parameters and determine @xmath87 . this approach has had only limited success . alternatively , one might use the maximum entropy method ( mem ) @xcite and find the `` most probable '' spectral function . since the lattice simulation gives the correlation function only at a discrete set of points @xmath94 , each with statistical errors , the problem of finding the continuous function @xmath95 of the continuous variable @xmath84 is ill - posed . the maximum entropy method can only produce a `` most probable '' answer with a measure of its reliability . the method , however , is well - known and quite successful in optics , where it is used for image reconstruction . again , the reader should consult the parallel sessions for a wealth of examples that illustrate the method s strengths and weaknesses . at past conferences the bielefeld group has presented data @xcite which have shown that the @xmath48 and @xmath49 peaks in the dilepton spectra are suppressed in the presence of the quark gluon plasma , in qualitative accord with the experimental trends . however , the dilepton signal is still somewhat larger than that found for free but hot quarks and gluons , at least for @xmath96 between 4 and 8 . these simulations were done above the transition , at @xmath97 and @xmath98 , so measureable attraction in these light quark states at these high temperatures is somewhat surprising . results such as these suggest that although bulk thermodynamic behavior of the quark gluon plasma might be close to the stephan - boltzmann limit , detailed hadronic structures are not completely washed out . at this conference s. datta @xcite presented bielefeld simulation results at @xmath99 and @xmath100 for the charmonium states . the s - wave states had a diminished , perhaps by a factor of 2 , signal above the transition where potential models predicted there would be no signal at all . the p - wave signal , however , was reduced by a factor of about 7 across the transition suggesting that screening has removed these charmonium states from the spectrum . a study of effective lagrangians by stephanov and son @xcite has uncovered curious features in the propagation of pions as the quark gluon plasma is approached by heating hadronic matter , @xmath101 . in fact , the pion dispersion relation is found to be determined by temperature dependent features of the system which are accessible to conventional euclidean lattice simulations . the three features one needs are 1 . the pion screening mass , 2 . the pion decay constant , and 3 . the axial isospin susceptibility . this work constitutes a nice , informative example where `` real - time '' quantities are accessible to static simulations . of course , the most popular example of this is the relation of the speed of sound @xmath102 in a system to its bulk thermodynamics , @xmath103 , where @xmath104 is the bulk pressure and @xmath105 is the energy density . other examples exist in condensed matter physics , as well . for example , the propagation features of spin waves in anti - ferromagnets are determined in large part by static correlation functions . these authors @xcite determine the real part of the soft pion dispersion relation , @xmath106 where we identify @xmath107 , the pion s speed , @xmath108 , the pion s screening mass , and @xmath109 , the pion s `` pole mass '' . the authors derive three low energy theorems at nonzero @xmath2 . first , @xmath110 , where @xmath111 is the pion decay constant and @xmath112 is the axial isospin susceptibility . these last two quantities are related by the other two theorems , which represent the generalizations of the gell - mann oakes renner relation to finite temperatures , @xmath113 . since these quantities are accessible to conventional lattice simulations , we have a nice illustration of determining `` real time '' physical quantities from static , euclidean simulations . the strategy of this development is interesting and quite general . one begins with the microscopic quark lagrangian underlying lattice qcd , @xmath114 as well as the low energy effective lagrangian that describes the properties of the low energy mesons , @xmath115 where @xmath116 , and @xmath117 . the effective lagrangian should be familiar from strong interaction phenomenology and its extension to nonzero chemical potential is determined by the symmetries of the underlying microscopic lagrangian without any numerical or phenomenological ambiguity @xcite . by matching the predictions of the microscopic theory with those of the low energy lagrangian , the authors determine their low energy theorems . note how curious pion propagation is as @xmath2 approaches @xmath7 . the pion propagation speed appraoches zero , as does the pion decay constant @xmath111 . a further investigation of the scaling laws governing the approach to the quark gluon transition reveals that @xmath107 vanishes faster as @xmath2 approaches @xmath7 than the screening mass @xmath108 diverges @xcite , so the pole mass @xmath118 approaches zero as @xmath2 approaches @xmath7 . since the pion s speed vanishes in this limit while its pole mass does also , perhaps we can call the pion an `` infinitely massive massless '' state ! hopefully , some of these features will show up in final state studies at rhic . as advertised in the first part of this talk , it would be good for all concerned if there were more interaction between the lattice community and the phenomenologists interested in the quark gluon plasma and in color superconductivity . one step in this direction has been taken by the swansea group @xcite which has found diquark condensation in strongly cutoff @xmath119 dimensional nambu jona - lasinio models , of the sort discussed by nuclear theorists , instanton phenomenologists and others . the value in the lattice simulations is , as usual , that their results are exact in principle , because they include all the fluctuations . the lagrangian which has been simulated is just one of the original nambu jona - lasinio models , @xmath120 + \frac{i}{2}[j(\bar{\psi}_i^{tr } c \gamma_5 \tau_2 \epsilon_{ij } \psi_j)+ \nonumber \\ \bar{j}(\bar{\psi}_i c \gamma_5 \tau_2 \epsilon_{ij } \bar{\psi}_j^{tr})]\end{aligned}\ ] ] where the last two terms are source terms for diquark condensation which are taken to zero ( @xmath121 ) at the end of a simulation meant to calculate the diquark condensate . these terms play the same role as a small bare fermion mass term in a lattice simulation of the chiral condensate the simulation must pick out a preferred direction in the space of the condensate , calculate the condensate and see if it survives in the limit where the source is taken away . in fact , simulations of this model indicate that as @xmath1 increases , a diquark condensate appears in the system and its @xmath1 dependence and magnitude are close to the predictions of the leading term in the large @xmath122 expansion . the reader should consult the parallel sessions of these conference proceedings for plots @xcite . one notable feature of this calculation is that it is the first lattice simulation in @xmath119 dimensions with a traditional fermi surface at nonzero chemical potential . future work in this direction will be the study of additional four fermi models suggested by the instanton liquid model . one complication will be the cutoff nature of these models these models are logarithmically trivial and they are useful only when strongly cutoff , with their parameters adjusted so that observables such as the pion decay constant and the chiral condensate take on physical values . luckily , the nuclear physics community has used such models for some time and know the `` best '' way to deal with these issues . the lattice model may reach an impasse , however , because when complicated four fermi interactions are entertained , it becomes impossible to avoid complex fermion determinants . in addition , these models do not confine quarks in their low temperature phase , so they are subject to criticisms discussed earlier . nonetheless , these studies should be informative and represent a test of some of the analytic , approximate approaches to the high @xmath1 , low @xmath2 superconductivity phase . lattice gauge theory has had much to say about the finite temperature behavior of qcd because it can handle the three length scales of qcd , perturbative behavior at weak coupling and short distances , semi - classical coherent behavior at intermediate couplings and intermediate distances , and confinement at strong couplings and large distances . it is now able to study the variable @xmath2-low @xmath1 strip of the phase diagram and make predictions relevant to heavy ion experiments . i hope that its lessons will lead to additional formulations and strategies to study real time physics relevant to collisions and nuclear structure in the not - too - distant future . 99 e. shuryak , hep - ph/0205031 . j.b . kogut and l. susskind , phys . 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lattice 2002 . p. de forcrand and o. philipsen , hep - lat/0205016 . allton , s. ejiri , s. hands , o. kaczmarek , f. karsch , e. laermann , ch . schmidt and l. scorzato , hep - lat/0204010 . sinclair , lattice 2002 . a. roberge and n. weiss , nucl . b275 , 734 ( 1986 ) . m. delia and m .- lombardo , hep - lat/0205022 . f. karsch , e. laermann , p. petreczky , s. stickan and i. wetzorke , hep - lat/0110208 . m asakawa , y. nakahara and t. hatsuda , prog . 46 , 459 ( 2001 ) . s. datta , lattice 2002 . d. son and m. stephanov , hep - ph/0111100 , 0204226 . kogut , m.a . stephanov and d. toublan , phys . lett . * b464 * , 183 ( 1999 ) . d. walters , lattice 2002 .	i review present challenges that qcd in extreme environments presents to lattice gauge theory . recent data and impressions from rhic are emphasized . physical pictures of heavy ion wavefunctions , collisions and the generation of the quark gluon plasma are discussed , with an eye toward engaging the lattice and its numerical methods in more interaction with the experimental and phenomenological developments . controversial , but stimulating scenarios which can be confirmed or dismissed by lattice methods are covered . in the second half of the talk , several promising developments presented at the conference lattice 2002 are reviewed .	11,667	144
knowledge - base systems must typically deal with imperfection in knowledge , in particular , in the form of incompleteness , inconsistency , and uncertainty . with this motivation , several frameworks for manipulating data and knowledge have been proposed in the form of extensions to classical logic programming and deductive databases to cope with imperfections in available knowledge . abiteboul , _ et al . _ @xcite , liu @xcite , and dong and lakshmanan @xcite dealt with deductive databases with incomplete information in the form of null values . kifer and lozinskii @xcite have developed a logic for reasoning with inconsistency . extensions to logic programming and deductive databases for handling uncertainty are numerous . they can broadly be categorized into non - probabilistic and probabilistic formalisms . we review previous work in these fields , with special emphasis on probabilistic logic programming , because of its relevance to this paper . * non - probabilistic formalisms * _ ( 1 ) fuzzy logic programming _ : this was essentially introduced by van emden in his seminal paper on quantitative deduction @xcite , and further developed by various researchers , including steger _ @xcite , schmidt _ _ @xcite . _ ( 2 ) annotated logic programming _ : this framework was introduced by subrahmanian @xcite , and later studied by blair and subrahmanian @xcite , and kifer and li @xcite . while blair and subrahmanian s focus was paraconsistency , kifer and li extended the framework of @xcite into providing a formal semantics for rule - based systems with uncertainty . finally , this framework was generalized by kifer and subrahmanian into the generalized annotated programming ( gap ) framework @xcite ) . all these frameworks are inherently based on a lattice - theoretic semantics . annotated logic programming has also been employed with the probabilistic approach , which we will discuss further below . _ ( 3 ) evidence theoretic logic programming _ : this has been mainly studied by baldwin and monk @xcite and baldwin @xcite ) . they use dempster s evidence theory as the basis for dealing with uncertainty in their logic programming framework . * probabilistic formalisms * indeed , there has been substantial amount of research into probabilistic logics ever since boole @xcite . carnap @xcite is a seminal work on probabilistic logic . fagin , halpern , and megiddo @xcite study the satisfiability of systems of probabilistic constraints from a model - theoretic perspective . gaifman @xcite extends probability theory by borrowing notions and techniques from logic . nilsson @xcite uses a possible worlds " approach to give model - theoretic semantics for probabilistic logic . @xcite notion of probabilistic entailment is similar to that of nilsson . some of the probabilistic logic programming works are based on probabilistic logic approaches , such as ng and subrahmanian s work on probabilistic logic programming @xcite and ng s recent work on empirical databases @xcite . we discuss these works further below . we will not elaborate on probabilistic logics any more and refer the reader to halpern @xcite for additional information . works on probabilistic logic programming and deductive databases can be categorized into two main approaches , annotation - based , and implication based . * annotation based approach * : ng and subrahmanian @xcite were the first to propose a probabilistic basis for logic programming . their syntax borrows from that of annotated logic programming @xcite , although the semantics are quite different . the idea is that uncertainty is always associated with individual atoms ( or their conjunctions and disjunctions ) , while the rules or clauses are always kept classical . in @xcite , uncertainty in an atom is modeled by associating a probabilistic truth value with it , and by asserting that it lies in an interval . the main interest is in characterizing how precisely we can bound " the probabilities associated with various atoms . in terms of the terminology of belief and doubt , we can say , following kifer and li @xcite , that the combination of belief and doubt about a piece of information might lead to an interval of probabilities , as opposed a precise probabilities . but , as pointed out in @xcite , even if one starts with precise point probabilities for atomic events , probabilities associated with compound events can only be calculated to within some exact upper and lower bounds , thus naturally necessitating intervals . but then , the same argument can be made for an agent s belief as well as doubt about a fact , they both could well be intervals . in this sense , we can say that the model of @xcite captures only the belief . a second important characteristic of this model is that it makes a conservative assumption that nothing is known about the interdependence of events ( captured by the atoms in an input database ) , and thus has the advantage of not having to make the often unrealistic independence assumption . however , by being conservative , it makes it impossible to take advantage of the ( partial ) knowledge a user may have about the interdependence among some of the events . from a technical perspective , only annotation constants are allowed in @xcite . intuitively , this means only constant probability ranges may be associated with atoms . this was generalized in a subsequent paper by ng and subrahmanian @xcite to allow annotation variables and functions . they have developed fixpoint and model - theoretic semantics , and provided a sound and weakly complete proof procedure . _ @xcite have proposed a sound ( propositional ) probabilistic calculus based on conditional probabilities , for reasoning in the presence of incomplete information . although they make use of a datalog - based interface to implement this calculus , their framework is actually propositional . in related works , ng and subrahmanian have extended their basic probabilistic logic programming framework to capture stable negation in @xcite , and developed a basis for dempster - shafer theory in @xcite . * implication based approach * : while many of the quantitative deduction frameworks ( van emden @xcite , fitting @xcite , debray and ramakrishnan @xcite , etc . ) are implication based , the first implication based framework for probabilistic deductive databases was proposed in @xcite . the idea behind implication based approach is to associate uncertainty with the facts as well as rules in a deductive database . sadri @xcite in a number of papers developed a hybrid method called information source tracking ( ist ) for modeling uncertainty in ( relational ) databases which combines symbolic and numeric approaches to modeling uncertainty . lakshmanan and sadri @xcite pursue the deductive extension of this model using the implication based approach . lakshmanan @xcite generalizes the idea behind ist to model uncertainty by characterizing the set of ( complex ) scenarios under which certain ( derived ) events might be believed or doubted given a knowledge of the applicable belief and doubt scenarios for basic events . he also establishes a connection between this framework and modal logic . while both @xcite are implication based approaches , strictly speaking , they do not require any commitment to a particular formalism ( such a probability theory ) for uncertainty manipulation . any formalism that allows for a consistent calculation of numeric certainties associated with boolean combination of basic events , based on given certainties for basic events , can be used for computing the numeric certainties associated with derived atoms . recently , lakshmanan and shiri @xcite unified and generalized all known implication based frameworks for deductive databases with uncertainty ( including those that use formalisms other than probability theory ) into a more abstract framework called the parametric framework . the notions of conjunctions , disjunctions , and certainty propagations ( via rules ) are parameterized and can be chosen based on the applications . even the domain of certainty measures can be chosen as a parameter . under such broadly generic conditions , they proposed a declarative semantics and an equivalent fixpoint semantics . they also proposed a sound and complete proof procedure . finally , they characterized conjunctive query containment in this framework and provided necessary and sufficient conditions for containment for several large classes of query programs . their results can be applied to individual implication based frameworks as the latter can be seen as instances of the parametric framework . conjunctive query containment is one of the central problems in query optimization in databases . while the framework of this paper can also be realized as an instance of the parametric framework , the concerns and results there are substantially different from ours . in particular , to our knowledge , this is the first paper to address data complexity in the presence of ( probabilistic ) uncertainty . * other related work * fitting @xcite has developed an elegant framework for quantitative logic programming based on bilattices , an algebraic structure proposed by ginsburg @xcite in the context of many - valued logic programming . this was the first to capture both belief and doubt in one uniform logic programming framework . in recent work , lakshmanan _ et al . _ @xcite have proposed a model and algebra for probabilistic relational databases . this framework allows the user to choose notions of conjunctions and disjunctions based on a family of strategies . in addition to developing complexity results , they also address the problem of efficient maintenance of materialized views based on their probabilistic relational algebra . one of the strengths of their model is not requiring any restrictive independence assumptions among the facts in a database , unlike previous work on probabilistic relational databases @xcite . in a more recent work , dekhtyar and subrahmanian @xcite developed an annotation based framework where the user can have a parameterized notion of conjunction and disjunction . in not requiring independence assumptions , and being able to allow the user to express her knowledge about event interdependence by means of a parametrized family of conjunctions and disjunctions , both @xcite have some similarities to this paper . however , chronologically , the preliminary version of this paper @xcite was the first to incorporate such an idea in a probabilistic framework . besides , the frameworks of @xcite are substantially different from ours . in a recent work ng @xcite studies _ empirical _ databases , where a deductive database is enhanced by empirical clauses representing statistical information . he develops a model - theoretic semantics , and studies the issues of consistency and query processing in such databases . his treatment is probabilistic , where probabilities are obtained from statistical data , rather than being subjective probabilities . ( see halpern @xcite for a comprehensive discussion on statistical and subjective probabilities in logics of probability . ) ng s query processing algorithm attempts to resolve a query using the ( regular ) deductive component of the database . if it is not successful , then it reverts to the empirical component , using the notion of _ most specific reference class _ usually used in statistical inferences . our framework is quite different in that every rule / fact is associated with a confidence level ( a pair of probabilistic intervals representing belief and doubt ) , which may be subjective , or may have been obtained from underlying statistical data . the emphasis of our work is on ( _ i _ ) the characterization of different modes for combining confidences , ( _ ii _ ) semantics , and , in particular , ( _ iii _ ) termination and complexity issues . the contributions of this paper are as follows . * we associate a _ confidence level _ with facts and rules ( of a deductive database ) . a confidence level comes with both a _ belief _ and a _ doubt _ ( in what is being asserted ) [ see section [ motiv ] for a motivation ] . belief and doubt are subintervals of @xmath0 $ ] representing probability ranges . * we show that confidence levels have an interesting algebraic structure called _ trilattices _ as their basis ( section [ lattice ] ) . analogously to fitting s bilattices , we show that trilattices associated with confidence levels are interlaced , making them interesting in their own right , from an algebraic point of view . in addition to providing an algebraic footing for our framework , trilattices also shed light on the relationship between our work and earlier works and offer useful insights . in particular , trilattices give rise to three ways of ordering confidence levels : the truth - order , where belief goes up and doubt comes down , the information order , where both belief and doubt go up , and the precision order , where the probability intervals associated with both belief and doubt become sharper , the interval length decreases . this is to be contrasted with the known truth and information ( called knowledge there ) orders in a bilattice . * a purely lattice - theoretic basis for logic programming can be constructed using trilattices ( similar to fitting @xcite ) . however , since our focus in this paper is probabilistic uncertainty , we develop a probabilistic calculus for combining confidence levels associated with basic events into those for compound events based on them ( section [ prob - calc ] ) . instead of committing to any specific rules for combining confidences , we propose a framework which allows a user to choose an appropriate mode " from a collection of available ones . * we develop a generalized framework for rule - based programming with probabilistic knowledge , based on this calculus . we provide the declarative and fixpoint semantics for such programs and establish their equivalence ( section [ prob - ddb ] ) . we also provide a sound and complete proof procedure ( section [ proof - theory ] ) . * we study the termination and complexity issues of such programs and show : ( 1 ) the closure ordinal of @xmath1 can be as high as @xmath2 in general ( but no more ) , and ( 2 ) when only _ positive correlation _ is used for disjunction , the data complexity of such programs is polynomial time . our proof technique for the last result yields a similar result for van emden s framework ( section [ termination ] ) . * we also compare our work with related work and bring out the advantages and generality of our approach ( section [ termination ] ) . in this section , we discuss the motivation for our work as well as comment on our design decisions for this framework . the motivation for using probability theory as opposed to other formalisms for representing uncertainty has been discussed at length in the literature @xcite . probability theory is perhaps the best understood and mathematically well - founded paradigm in which uncertainty can be modeled and reasoned about . two possibilities for associating probabilities with facts and rules in a ddb are van emden s style of associating confidences with rules as a whole @xcite , or the annotation style of kifer and subrahmanian @xcite . the second approach is more powerful : it is shown in @xcite that the second approach can simulate the first . the first approach , on the other hand , has the advantage of intuitive appeal , as pointed out by kifer and subrahmanian @xcite . in this paper , we choose the first approach . a comparison between our approach and annotation - based approach with respect to termination and complexity issues is given in section [ termination ] . a second issue is whether we should insist on precise probabilities or allow intervals ( or ranges ) . firstly , probabilities derived from any sources may have tolerances associated with them . even experts may feel more comfortable with specifying a range rather than a precise probability . secondly , fenstad @xcite has shown ( also see @xcite ) that when enough information is not available about the interaction between events , the probability of compound events can not be determined precisely : one can only give ( tight ) bounds . thus , we associate ranges of probabilities with facts and rules . a last issue is the following . suppose ( uncertain ) knowledge contributed by an expert corresponds to the formula @xmath3 . in general , we can not assume the expert s knowledge is perfect . this means he does not necessarily know _ all _ situations in which @xmath3 holds . nor does he know _ all _ situations where @xmath3 fails to hold ( @xmath4 holds ) . he models the proportion of the situations where he knows @xmath3 holds as his _ belief _ in @xmath3 and the proportion of situations where he knows @xmath4 holds as his _ doubt_. there could be situations , unknown to our expert , where @xmath3 holds ( or @xmath4 holds ) . these unknown situations correspond to the gap in his knowledge . thus , as far as he knows , @xmath3 is _ unknown _ or _ undefined _ in these remaining situations . these observations , originally made by fitting @xcite , give rise to the following definition . [ defn : cf ] ( _ confidence level _ ) denote by @xmath5 $ ] the set of all closed subintervals over @xmath6 $ ] . consider the set @xmath7 \times { \cal c}[0 , 1]$ ] . a _ confidence level _ is an element of @xmath8 . we denote a confidence level as @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] . in our approach confidence levels are associated with facts and rules . the intended meaning of a fact ( or rule ) @xmath3 having a confidence @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] is that @xmath10 and @xmath11 are the lower and upper bounds of the expert s _ belief _ in @xmath3 , and @xmath12 and @xmath13 are the lower and upper bounds of the expert s _ doubt _ in @xmath3 . these notions will be formalized in section [ prob - calc ] . the following example illustrates such a scenario . ( the figures in all our examples are fictitious . ) consider the results of gallup polls conducted before the recent canadian federal elections . \1 . of the people surveyed , between 50% and 53% of the people in the age group 19 to 30 favor the liberals . between 30% and 33% of the people in the above age group favor the reformists . + 3 . between 5% and 8% of the above age group favor the tories . the reason we have ranges for each category is that usually some tolerance is associated with the results coming from such polls . also , we do not make the proportion of undecided people explicit as our interest is in determining the support for the different parties . suppose we assimilate the information above in a probabilistic framework . for each party , we compute the probability that a _ randomly _ chosen person from the sample population of the given age group will ( not ) vote for that party . we transfer this probability as the _ subjective _ probability that _ any _ person from that age group ( in the actual population ) will ( not ) vote for the party . the conclusions are given below , where @xmath14 says @xmath15 will vote for party @xmath16 , @xmath17-@xmath18 says @xmath15 belongs to the age group specified above . @xmath19 , @xmath20 , and @xmath21 are constants , with the obvious meaning . @xmath22 @xmath23 , [ 0.35 , 0.41]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } $ ] @xmath17-@xmath18 . @xmath24 : @xmath25 , [ 0.55 , 0.61]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } $ ] @xmath17-@xmath18 . @xmath26 : @xmath27 , [ 0.8 , 0.86]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } $ ] @xmath17-@xmath18 . as usual , each rule is implicitly universally quantified outside the entire rule . each rule is expressed in the form @xmath28,~ [ \gamma,\delta]\rangle$}}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } body\ ] ] where @xmath29 $ ] . we usually require that @xmath30 and @xmath31 . with each rule , we have associated two intervals . @xmath32 $ ] ( @xmath33 $ ] ) is the _ belief _ ( _ doubt _ ) the expert has in the rule . notice that from his knowledge , the expert can only conclude that the proportion of people he _ knows _ favor @xmath20 or @xmath21 will not vote for @xmath19 . thus the probability that a person in the age group 19 - 30 will not vote for liberals , according to the expert s knowledge , is in the range @xmath34 $ ] , obtained by summing the endpoints of the belief ranges for reform and tories . notice that in this case @xmath35 ( or @xmath36 ) is not necessarily @xmath37 . this shows we can not regard the expert s doubt as the complement ( with respect to 1 ) of his belief . thus , if we have to model what _ necessarily _ follows according to the expert s knowledge , then we must carry both the belief and the doubt explicitly . note that this example suggests just one possible means by which confidence levels could be obtained from statistical data . as discussed before , gaps in an expert s knowledge could often directly result in both belief and doubt . in general , there could be many ways in which both belief and doubt could be obtained and associated with the basic facts . given this , we believe that an independent treatment of both belief and doubt is both necessary and interesting for the purpose of obtaining the confidence levels associated with derived facts . our approach to independently capture belief and doubt makes it possible to cope with incomplete knowledge regarding the situations in which an event is true , false , or unknown in a general setting . kifer and li @xcite and baldwin @xcite have argued that incorporating both belief and doubt ( called disbelief there ) is useful in dealing with incomplete knowledge , where different evidences may contradict each other . however , in their frameworks , doubt need not be maintained explicitly . for suppose we have a belief @xmath38 and a disbelief @xmath39 associated with a phenomenon . then they can both be absorbed into one range @xmath40 $ ] indicating that the effective certainty ranges over this set . the difference with our framework , however , is that we model what is _ known _ definitely , as opposed to what is _ possible_. this makes ( in our case ) an explicit treatment of belief and doubt mandatory . fitting @xcite has shown that _ bilattices _ ( introduced by ginsburg @xcite ) lead to an elegant framework for quantified logic programming involving both belief and doubt . in this section , we shall see that a notion of _ trilattices _ naturally arises with confidence levels . we shall establish the structure and properties of trilattices here , which will be used in later sections . [ conf - lattice ] denote by @xmath5 $ ] the set of all closed subintervals over @xmath6 $ ] . consider the set @xmath7 \times { \cal c}[0 , 1]$ ] . we denote the elements of @xmath8 as @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] . define the following orders on this set . let @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$}}$ ] , @xmath42,~ [ \gamma_{2},\delta_{2}]\rangle$}}$ ] be any two elements of @xmath8 . @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } \leq_t { \mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}$ ] iff @xmath43 , @xmath44 and @xmath45 , @xmath46 + @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } \leq_k { \mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}$ ] iff @xmath43 , @xmath44 and @xmath47 , @xmath48 + @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } \leq_p { \mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}$ ] iff @xmath43 , @xmath49 and @xmath47 , @xmath46 some explanation is in order . the order @xmath50 can be considered the _ truth _ ordering : truth " relative to the expert s knowledge increases as belief goes up and doubt comes down . the order @xmath51 is the _ knowledge _ ( or information ) ordering : knowledge " ( the extent to which the expert commits his opinion on an assertion ) increases as both belief and doubt increase . the order @xmath52 is the _ precision _ ordering : precision " of information supplied increases as the probability intervals become narrower . the first two orders are analogues of similar orders in bilattices . the third one , however , has no counterpart there . it is straightforward to see that each of the orders @xmath50 , @xmath51 , and @xmath52 is a partial order . @xmath8 has a least and a greatest element with respect to each of these orders . in the following , we give the definition of meet and join with respect to the @xmath50 order . operators with respect to the other orders have a similar definition . [ meet - join ] let @xmath53 be as defined in definition [ conf - lattice ] . then the meet and join corresponding to the truth , knowledge ( information ) , and precision orders are defined as follows . the symbols and denote meet and join , and the subscripts @xmath54 , @xmath55 , and @xmath56 represent truth , knowledge , and precision , respectively . @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \otimes_t$}}{\mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}~=~$ ] @xmath57,$ ] @xmath58\rangle$ ] . \2 . @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \oplus_t$}}{\mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}~=~$ ] @xmath59 $ ] , @xmath60\rangle$ ] . \3 . @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \otimes_k$}}{\mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}~=~$ ] @xmath61 $ ] , @xmath60\rangle$ ] . \4 . @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \oplus_k$}}{\mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}~=~$ ] @xmath62 $ ] , @xmath58\rangle$ ] . \5 . @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \otimes_p$}}{\mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}~=~$ ] @xmath63 $ ] , @xmath64\rangle$ ] . \6 . @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \oplus_p$}}{\mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$}}~=~$ ] @xmath65 $ ] , @xmath66\rangle$ ] . the top and bottom elements with respect to the various orders are as follows . the subscripts indicate the associated orders , as usual . @xmath67 , [ 0 , 0]\rangle$ ] , @xmath68 , [ 1 , 1]\rangle$ ] , + @xmath69 , [ 1 , 1]\rangle$ ] , @xmath70 , [ 0 , 0]\rangle$ ] , + @xmath71 , [ 1 , 0]\rangle$ ] , @xmath72 , [ 0 , 1]\rangle$ ] . @xmath73 corresponds to total belief and no doubt ; @xmath74 is the opposite . @xmath75 represents maximal information ( total belief and doubt ) , to the point of being probabilistically inconsistent : belief and doubt probabilities sum to more than 1 ; @xmath76 gives the least information : no basis for belief or doubt ; @xmath77 is maximally precise , to the point of making the intervals empty ( and hence inconsistent , in a non - probabilistic sense ) ; @xmath78 is the least precise , as it imposes only trivial bounds on belief and doubt probabilities . fitting @xcite defines a bilattice to be _ interlaced _ whenever the meet and join with respect to any order of the bilattice are monotone with respect to the other order . he shows that it is the interlaced property of bilattices that makes them most useful and attractive . we say that a trilattice is _ interlaced _ provided the meet and join with respect to any order are monotone with respect to any other order . we have [ lem : interlaced ] the trilattice @xmath53 defined above is interlaced . * proof . * follows directly from the fact that @xmath79 and @xmath80 are monotone functions . we show the proof for just one case . let @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } \leq_p { \mbox{$\langle[\alpha_{3},\beta_{3}],~ [ \gamma_{3},\delta_{3}]\rangle$}}$ ] and @xmath42,~ [ \gamma_{2},\delta_{2}]\rangle$ } } \leq_p { \mbox{$\langle[\alpha_{4},\beta_{4}],~ [ \gamma_{4},\delta_{4}]\rangle$}}$ ] . then @xmath41,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \otimes$}}_t { \mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$ } } = $ ] @xmath81 , [ max\{\gamma_1 , \gamma_2\ } , max\{\delta_1 , \delta_2\}]\rangle $ ] @xmath82,~ [ \gamma_{3},\delta_{3}]\rangle$ } } { \mbox { $ \otimes$}}_t { \mbox{$\langle[\alpha_{4},\beta_{4}],~ [ \gamma_{4},\delta_{4}]\rangle$ } } = $ ] @xmath83 , [ max\{\gamma_3 , \gamma_4\ } , max\{\delta_3 , \delta_4\}]\rangle$ ] since @xmath84 , we have @xmath85 , and @xmath86 . similarly , @xmath87 and @xmath88 . this implies @xmath89,~ [ \gamma_{1},\delta_{1}]\rangle$ } } { \mbox { $ \otimes_t$}}{\mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$ } } \leq_p { \mbox{$\langle[\alpha_{3},\beta_{3}],~ [ \gamma_{3},\delta_{3}]\rangle$ } } { \mbox { $ \otimes_t$}}{\mbox{$\langle[\alpha_{4},\beta_{4}],~ [ \gamma_{4},\delta_{4}]\rangle$}}\ ] ] other cases are similar . trilattices are of independent interest in their own right , from an algebraic point of view . we also stress that they can be used as a basis for developing quantified / annotated logic programming schemes ( which need not be probabilistic ) . this will be pursued in a future paper . in closing this section , we note that other orders are also possible for confidence levels . in fact , fitting has shown that a fourth order , denoted by @xmath90 in the following , together with the three orders defined above , forms an interlaced `` quadri - lattice '' @xcite . he also pointed out that this `` quadri - lattice '' can be generated as the cross product of two bilattices . intuitively , a confidence level increases according to this fourth ordering , when the precision of the belief component of a confidence level goes up , while that of the doubt component goes down . that is , @xmath89,~ [ \gamma_{1},\delta_{1}]\rangle$ } } \leq_f { \mbox{$\langle[\alpha_{2},\beta_{2}],~ [ \gamma_{2},\delta_{2}]\rangle$ } } \mbox { iff } \alpha_1 \leq \alpha_2 , \beta_2 \leq \beta_1 \mbox { and } \gamma_2 \leq \gamma_1 , \delta_1 \leq \delta_2\ ] ] in our opinion , the fourth order , while technically elegant , does not have the same intuitive appeal as the three orders truth , knowledge , and precision mentioned above . hence , we do not consider it further in this paper . the algebraic properties of confidence levels and their underlying lattices are interesting in their own right , and might be used for developing alternative bases for quantitative logic programming . this issue is orthogonal to the concerns of this paper . given the confidence levels for ( basic ) events , how are we to derive the confidence levels for compound events which are based on them ? since we are working with probabilities , our combination rules must respect probability theory . we need a model of our knowledge about the interaction between events . a simplistic model studied in the literature ( see barbara _ et al . @xcite ) assumes _ independence _ between all pairs of events . this is highly restrictive and is of limited applicability . a general model , studied by ng and subrahmanian @xcite is that of _ ignorance _ : assume no knowledge about event interaction . although this is the most general possible situation , it can be overly conservative when _ some _ knowledge is available , concerning some of the events . we argue that for real - life " applications , no single model of event interaction would suffice . indeed , we need the ability to parameterize " the model used for event interaction , depending on what _ is _ known about the events themselves . in this section , we develop a probabilistic calculus which allows the user to select an appropriate mode " of event interaction , out of several choices , to suit his needs . let * l * be an arbitrary , but fixed , first - order language with finitely many constants , predicate symbols , infinitely many variables , and no function symbols . we use ( ground ) atoms of * l * to represent basic events . we blur the distinction between an event and the formula representing it . our objective is to characterize confidence levels of boolean combinations of events involving the connectives @xmath91 , in terms of the confidence levels of the underlying basic events under various modes ( see below ) . we gave an informal discussion of the meaning of confidence levels in section [ motiv ] . we use the concept of _ possible worlds _ to formalize the semantics of confidence levels . [ defn : semantics ] ( _ semantics of confidence levels _ ) according to the expert s knowledge , an event @xmath3 can be true , false , or unknown . this gives rise to 3 possible worlds . let @xmath92 respectively denote _ true _ , _ false _ , and _ unknown_. let @xmath93 denote the world where the truth - value of @xmath3 is @xmath94 , @xmath95 , and let @xmath96 denote the probability of the world @xmath93 then the assertion that the confidence level of @xmath3 is @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] , written @xmath97,~ [ \gamma,\delta]\rangle$}}$ ] , corresponds to the following constraints : @xmath98 where @xmath10 and @xmath11 are the lower and upper bounds of the _ belief _ in @xmath3 , and @xmath12 and @xmath13 are the lower and upper bounds of the _ doubt _ in @xmath3 . equation ( [ eq : possibleworlds ] ) imposes certain restrictions on confidence levels . [ defn : consistentcf ] ( _ consistent confidence levels _ ) we say a confidence level @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] is _ consistent _ if equation ( [ eq : possibleworlds ] ) has an answer . it is easily seen that : [ prop : consistentcf ] confidence level @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] is _ consistent _ provided _ ( i ) _ @xmath30 and @xmath31 , and _ ( ii ) _ @xmath99 . the consistency condition guarantees at least one solution to equation ( [ eq : possibleworlds ] ) . however , given a confidence level @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] , there may be @xmath100 values in the @xmath32 $ ] interval for which no @xmath101 value exists in the @xmath33 $ ] interval to form an answer to equation ( [ eq : possibleworlds ] ) , and vice versa . we can `` trim '' the upperbounds of @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] as follows to guarantee that for each value in the @xmath32 $ ] interval there is at least one value in the @xmath33 $ ] interval which form an answer to equation ( [ eq : possibleworlds ] ) . [ defn : reducedcf ] ( _ reduced confidence level _ ) we say a confidence level @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] is _ reduced _ if for all @xmath102 $ ] there exist @xmath101 , @xmath103 such that @xmath100 , @xmath101 , @xmath103 is a solution to equation ( [ eq : possibleworlds ] ) , and for all @xmath104 $ ] there exist @xmath100 , @xmath103 such that @xmath100 , @xmath101 , @xmath103 is a solution to equation ( [ eq : possibleworlds ] ) . it is obvious that a reduced confidence level is consistent . [ prop : reducedcf ] confidence level @xmath9,~ [ \gamma,\delta]\rangle$}}$ ] is _ reduced _ provided _ ( i ) _ @xmath30 and @xmath31 , and _ ( ii ) _ @xmath105 , and @xmath106 . [ prop : reduction ] let @xmath107,~ [ \gamma,\delta]\rangle$}}$ ] be a consistent confidence level . let @xmath108 and @xmath109 . then , the confidence level @xmath110 , [ \gamma , min(\delta,\delta')]$ ] is a reduced confidence level . further , @xmath111 and @xmath112 are probabilistically equivalent , in the sense that they produce exactly the same answer sets to equation ( [ eq : possibleworlds ] ) . data in a probabilistic deductive database , that is , facts and rules that comprise the database , are associated with confidence levels . at the atomic level , we require the confidence levels to be consistent . this means each expert , or data source , should be consistent with respect to the confidence levels it provides . this does not place any restriction on data provided by different experts / sources , as long as each is individually consistent . data provided by different experts / sources should be combined , using an appropriate combination mode ( discussed in next section ) . we will show that the combination formulas for the various modes preserve consistent as well as reduced confidence levels . now , we introduce the various modes and characterize conjunction and disjunction under these modes . let @xmath3 and @xmath113 represent two arbitrary ground ( variable - free ) formulas . for a formula @xmath3 , @xmath114 will denote its confidence level . in the following , we describe several interesting and natural modes and establish some results on the confidence levels of conjunction and disjunction under these modes . some of the modes are well known , although care needs to be taken to allow for the 3-valued nature of our framework . _ ignorance : _ this is the most general situation possible : nothing is assumed / known about event interaction between @xmath3 and @xmath113 . the extent of the interaction between @xmath3 and @xmath113 could range from maximum overlap to minimum overlap . independence : _ this is a well - known mode . it simply says ( non-)occurrence of one event does not influence that of the other . positive correlation : _ this mode corresponds to the knowledge that the occurrences of two events overlap as much as possible . this means the conditional probability of one of the events ( the one with the larger probability ) given the other is 1 . negative correlation : _ this is the exact opposite of positive correlation : the occurrences of the events overlap minimally . mutual exclusion : _ this is a special case of negative correlation , where we know that the sum of probabilities of the events does not exceed 1 . we have the following results . let @xmath3 be any event , and let @xmath97,~ [ \gamma,\delta]\rangle$}}$ ] . then @xmath115,~[\alpha , \beta]\rangle$ ] . thus , negation simply swaps belief and doubt . * follows from the observation that @xmath97,~ [ \gamma,\delta]\rangle$}}$ ] implies that @xmath116 and @xmath117 , where @xmath100 ( @xmath101 ) denotes the probability of the possible world where event @xmath3 is _ true _ ( _ false _ ) . the following theorem establishes the confidence levels of compound formulas as a function of those of the constituent formulas , under various modes . [ thm : combinations ] let @xmath3 and @xmath113 be any events and let @xmath118,~ [ \gamma_{1},\delta_{1}]\rangle$}}$ ] and @xmath119,~ [ \gamma_{2},\delta_{2}]\rangle$}}$ ] . then the confidence levels of the compound events @xmath120 and @xmath121 are given as follows . ( in each case the subscript denotes the mode . ) @xmath122 @xmath123,$ ] @xmath124 , @xmath125 @xmath126\rangle$ ] . @xmath127 , [ max\{0 , \gamma_1 + \gamma_2 -1\}$ ] , @xmath128\rangle$ ] . @xmath129 , [ 1 - ( 1 - \gamma_1 ) \times ( 1 - \gamma_2 ) , 1 - ( 1 - \delta_1 ) \times ( 1 - \delta_2)]\rangle$ ] . @xmath130 , [ \gamma_1 \times \gamma_2 , \delta_1 \times \delta_2]\rangle$ ] . @xmath131 , [ max\{\gamma_1 , \gamma_2\ } , max\{\delta_1 , \delta_2\}]\rangle$ ] . @xmath132 , [ min\{\gamma_1 , \gamma_2\ } , min\{\delta_1 , \delta_2\}]\rangle$ ] . @xmath133 @xmath134 , [ min\{1 , \gamma_1 + \gamma_2\ } , min\{1 , \delta_1 + \delta_2\}]\rangle$ ] . @xmath135 @xmath136 , [ max\{0 , \gamma_1 + \gamma_2 - 1\ } , max\{0 , \delta_1 + \delta_2 - 1\}]\rangle$ ] . @xmath137 , [ min\{1 , \gamma_1+\gamma_2\ } , min\{1 , \delta_1+\delta_2\}]\rangle$ ] . @xmath138 , [ max\{0 , \gamma_1 + \gamma_2 - 1\ } , max\{0 , \delta_1 + \delta_2 - 1\}]\rangle$ ] . * proof . * each mode is characterized by a system of constraints , and the confidence level of the formulas @xmath139 are obtained by extremizing certain objective functions subject to these constraints . the scope of the possible interaction between @xmath3 and @xmath113 can be characterized as follows ( also see @xcite ) . according to the expert s knowledge , each of @xmath140 can be true , false , or unknown . this gives rise to 9 possible worlds . let @xmath92 respectively denote _ true _ , _ false _ , and _ unknown_. let @xmath141 denote the world where the truth - value of @xmath3 is @xmath94 and that of @xmath113 is @xmath142 , @xmath143 . , @xmath144 is the world where @xmath3 is true and @xmath113 is false , while @xmath145 is the world where @xmath3 is false and @xmath113 is unknown . suppose @xmath146 denotes the probability associated with world @xmath141 . then the possible scope of interaction between @xmath3 and @xmath113 can be characterized by the following constraints . @xmath147 the above system of constraints must be satisfied for all modes . specific constraints for various modes are obtained by adding more constraints to those in equation ( [ ign - eq ] ) . in all cases , the confidence levels for @xmath148 and @xmath149 are obtained as follows . @xmath150,\\ & & [ min(\sigma_{{\mbox { $ w_{ij}$}}\not \models f\circ g } { \mbox { $ w_{ij}$ } } ) , max(\sigma_{{\mbox { $ w_{ij}$}}\not \models f\circ g } { \mbox { $ w_{ij}$}})]\rangle\end{aligned}\ ] ] where @xmath151 is @xmath152 or @xmath153 . : _ ignorance_. the constraints for ignorance are exactly those in equation ( [ ign - eq ] ) . the solution to the above linear program can be shown to be @xmath154,$ ] @xmath124 , @xmath125 @xmath126\rangle$ ] , @xmath155 , [ max\{0 , \gamma_1 + \gamma_2 -1\}$ ] , @xmath128\rangle$ ] . the proof is very similar to the proof of a similar result in the context of belief intervals ( no doubt ) by ng and subrahmanian @xcite . : _ independence_. independence of events @xmath3 and @xmath113 can be characterized by the equation @xmath156 , where @xmath157 is the conditional probability of the event @xmath3 given event @xmath113 . more specifically , since in our model an event can be _ true _ , _ false _ , or _ unknown _ , ( in other words , we model belief and doubt independently ) we have : @xmath158 then the constraints characterizing independence is obtained by adding the following equations to the system of constraints ( [ ign - eq ] ) . @xmath159 the belief in @xmath160 , and doubt in @xmath161 can be easily verified from the system of constraints [ ign - eq ] and [ ind - eq ] @xmath162 to obtain the doubt in @xmath160 we need to compute the minimum and maximum of @xmath163 . it is easy to verify that : @xmath164 the belief in @xmath161 is obtained similarly ( in the dual manner . ) thus , we have verified that @xmath129 , [ 1 - ( 1 - \gamma_1 ) \times ( 1 - \gamma_2 ) , 1 - ( 1 - \delta_1 ) \times ( 1 - \delta_2)]\rangle$ ] . + @xmath130 , [ \gamma_1 \times \gamma_2 , \delta_1 \times \delta_2]\rangle$ ] . : _ positive correlation _ : two events @xmath3 and @xmath113 are positively correlated if they overlap as much as possible . this happens when either ( _ i _ ) occurrence of @xmath3 implies occurrence of @xmath113 , or ( _ ii _ ) occurrence of @xmath113 implies occurrence of @xmath3 . in our framework we model belief and doubt independently , and positive correlation is characterized by 4 possibilities : \(a ) occurrence of @xmath3 implies occurrence of @xmath113 , and non - occurrence of @xmath113 implies non - occurrence of @xmath3 . + ( b ) occurrence of @xmath3 implies occurrence of @xmath113 , and non - occurrence of @xmath3 implies non - occurrence of @xmath113 . + ( c ) occurrence of @xmath113 implies occurrence of @xmath3 , and non - occurrence of @xmath113 implies non - occurrence of @xmath3 . + ( d ) occurrence of @xmath113 implies occurrence of @xmath3 , and non - occurrence of @xmath3 implies non - occurrence of @xmath113 . each of these four condition sets generates its own equations . for example , ( a ) can be captured by adding the following equations to the system of constraints [ ign - eq ] . @xmath165 hence , for condition ( a ) , the system of constraints [ ign - eq ] becomes @xmath166 the analysis is further complicated by the fact that the confidence levels of @xmath3 and @xmath113 determine which of these cases apply , and it may be different for the lowerbound and upperbound probabilities . for example , if @xmath167 ( @xmath168 ) , then the lowerbound ( upperbound ) for belief in @xmath169 is obtained when occurrence of @xmath3 implies occurrence of @xmath113 . otherwise , these bounds are obtained when occurrence of @xmath113 implies occurrence of @xmath3 . the solution to these linear programs can be shown to be @xmath170 , [ max\{\gamma_1 , \gamma_2\ } , max\{\delta_1 , \delta_2\}]\rangle$ ] , and + @xmath132 , [ min\{\gamma_1 , \gamma_2\ } , min\{\delta_1 , \delta_2\}]\rangle$ ] . a more intuitive approach to the derivation of confidence levels for conjunction and disjunction of positively correlated events is to rely on the observation that these events overlap to the maximum extent possible . in our framework it means the worlds where @xmath3 is _ true _ and those where @xmath113 is _ true _ overlap maximally , and hence , one is included in the other . similarly , since we model belief and doubt independently , the worlds where @xmath3 is _ false _ and those where @xmath113 is _ false _ also overlap maximally . the combination formulas can be derived directly using these observations . : _ negative correlation _ : negative correlation is an appropriate mode to use whenever we know that events @xmath3 and @xmath113 overlap as little as possible . this is to be contrasted with positive correlation , where the extent of overlap is the greatest possible . mutual exclusion , is a special case of negative correlation where the sum of the probabilities of the two events does not exceed 1 . in this case the two events do not overlap at all . in the classical framework , mutual exclusion of two events @xmath3 and @xmath113 is characterized by the statement : ( _ i _ ) occurrence of @xmath3 implies non - occurrence of @xmath113 , and vice versa . on the other hand , if the two events @xmath3 and @xmath113 are negatively correlated but not mutually exclusive , we have : ( _ ii _ ) non - occurrence of @xmath3 implies occurrence of @xmath113 , and vice versa . in case ( _ i _ ) the sum of the probabilities of the two events is at most 1 , while in case ( _ ii _ ) this sum exceeds 1 and hence the two events can not be mutually exclusive . in our framework we model belief and doubt independently , and each of the above conditions translates to two conditions as follows . note that in our framework , `` not _ true _ '' means `` _ false _ or _ unknown _ '' , and `` not _ false _ '' means `` _ true _ or _ unknown _ '' . * event @xmath3 is _ true _ implies @xmath113 is not _ true _ , and vice versa . this condition generates the equation @xmath171 . * the dual of condition ( a ) , when the non - occurrence of the two events do nt overlap . event @xmath3 is _ false _ implies @xmath113 is not _ false _ , and vice versa . this condition generates the equation @xmath172 . * event @xmath3 is not _ true _ implies @xmath113 is _ true _ , and vice versa . this condition generates the equations @xmath173 , @xmath174 , @xmath175 , and @xmath176 . * the dual of ( _ c _ ) , @xmath3 is not _ _ implies @xmath113 is _ false _ , and vice versa , which generates the equations @xmath177 , @xmath178 , @xmath179 , and @xmath176 . similar to the case for positive correlation , the confidence levels of @xmath3 and @xmath113 determine which of these cases apply . for example , if @xmath180 , then case ( c ) should be used to determine the lowerbound for belief in @xmath181 . alternatively , and more intuitively , we can characterize negative correlation by observing that the worlds where @xmath3 is _ true _ and those where @xmath113 is _ true _ overlap minimally , and the worlds where @xmath3 is _ false _ and those where @xmath113 is _ false _ also overlap minimally . the confidences of @xmath148 and @xmath121 can be obtained using the equations , or directly from the alternative characterization : @xmath133 @xmath134 , [ min\{1 , \gamma_1 + \gamma_2\ } , min\{1 , \delta_1 + \delta_2\}]\rangle$ ] @xmath135 @xmath136 , [ max\{0 , \gamma_1 + \gamma_2 - 1\ } , max\{0 , \delta_1 + \delta_2 - 1\}]\rangle$ ] : _ mutual exclusion _ : mutual exclusion is a special case of negative correlation . the main difference is that it requires the sum of the two probabilities to be at most 1 , which is not necessarily the case for negative correlation ( see the previous case ) . in the classical framework , if two events are mutually exclusive , their negation are not necessarily mutually exclusive . rather , they are negatively correlated . in our framework , however , one or both conditions ( _ a _ ) and ( _ b _ ) , discussed in the previous case , can hold . the appropriate condition is determined by the confidence levels of the two mutually exclusive events , and the corresponding combination formula can be obtained from the combination formulas of negative correlation . the following formulas , for example , are for mutually exclusive events @xmath3 and @xmath113 where @xmath182 ( but no other restriction ) . @xmath137 , [ min\{1 , \gamma_1+\gamma_2\ } , min\{1 , \delta_1+\delta_2\}]\rangle$ ] . + @xmath138 , [ max\{0 , \gamma_1 + \gamma_2 - 1\ } , max\{0 , \delta_1 + \delta_2 - 1\}]\rangle$ ] . next , we show that the combination formulas for various modes preserve consistent as well as reduced confidence levels . the case for reduced confidence levels is more involved and will be presented first . the other case is similar , for which we only state the theorem . [ thm : reduced ] suppose @xmath3 and @xmath113 are any formulas and assume their confidence levels are reduced ( definition [ defn : reducedcf ] ) . then the confidence levels of the formulas @xmath148 and @xmath121 , obtained under the various modes above are all reduced . * let @xmath118,~ [ \gamma_{1},\delta_{1}]\rangle$}}$ ] and @xmath119,~ [ \gamma_{2},\delta_{2}]\rangle$}}$ ] . since the confidence levels of @xmath3 and @xmath113 are reduced , we have : @xmath183 + @xmath184 + @xmath185 + @xmath186 + the consistency of the confidence levels of the combination events @xmath148 and @xmath121 in different modes as derived in theorem [ thm : combinations ] follow from the above constraints . for example let us consider @xmath187 , [ max\{\gamma_1 , \gamma_2\ } , min\{1 , \delta_1 + \delta_2\}]\rangle\ ] ] we need to show \(1 ) @xmath188 + ( 2 ) @xmath189 + ( 3 ) @xmath190 + ( 4 ) @xmath191 to prove ( 1 ) : if @xmath192 then ( 1 ) holds . otherwise , assume , without loss of generality , that @xmath193 . we can write @xmath194 + @xmath195 and hence @xmath196 and ( 1 ) follows . inequality ( 2 ) follows easily from @xmath197 . to prove ( 3 ) : if @xmath192 then ( 3 ) holds . otherwise , we can write @xmath198 + @xmath199 and hence @xmath200 and ( 3 ) follows . note that if @xmath201 then @xmath202 follows from the above constraint . to prove ( 4 ) let @xmath203 and @xmath204 where @xmath205 . then @xmath206 . proving the consistency of the confidence levels of other combinations and other modes are similar and will not be elaborated here . [ thm : consistent ] suppose @xmath3 and @xmath113 are any formulas and assume their confidence levels are consistent ( definition [ defn : consistentcf ] ) . then the confidence levels of the formulas @xmath148 and @xmath121 , obtained under the various modes above are all consistent . * proof is similar to the previous theorem and is omitted . in this section , we develop a framework for probabilistic deductive databases using a language of probabilistic programs ( p - programs ) . we make use of the probabilistic calculus developed in section [ prob - calc ] and develop the syntax and declarative semantics for programming with confidence levels . we also provide the fixpoint semantics of programs in this framework and establish its equivalence to the declarative semantics . we will use the first - order language * l * of section [ prob - calc ] as the underlying logical language in this section . * syntax of p - programs : * a _ rule _ is an expression of the form @xmath207 , @xmath208 , where @xmath209 are atoms and @xmath210,~ [ \gamma,\delta]\rangle$}}$ ] is the confidence level associated with the rule ) . ] . when @xmath211 , we call this a _ fact_. all variables in the rule are assumed to be universally quantified outside the whole rule , as usual . we restrict attention to range restricted rules , as is customary . a _ probabilistic rule _ ( p - rule ) is a triple @xmath212 , where @xmath213 is a rule , @xmath214 is a mode indicating how to conjoin the confidence levels of the subgoals in the body of @xmath213 ( and with that of @xmath213 itself ) , and @xmath215 is a mode indicating how the confidence levels of different derivations of an atom involving the head predicate of @xmath213 are to be disjoined . we say @xmath214 ( @xmath216 is the mode associated with the body ( head ) of @xmath213 , and call it the _ conjunctive _ ( _ disjunctive _ ) mode . we refer to @xmath213 as the underlying rule of this p - rule . when @xmath213 is a fact , we omit @xmath214 for obvious reasons . a _ probabilistic program _ ( p - program ) is a finite collection of p - rules such that whenever there are p - rules whose underlying rules define the same predicate , the mode associated with their head is identical . this last condition ensures different rules defining the same predicate @xmath217 agree on the manner in which confidences of identical @xmath217-atoms generated by these rules are to be combined . the notions of herbrand universe @xmath218 and herbrand base @xmath219 associated with a p - program @xmath16 are defined as usual . a p - rule is ground exactly when every atom in it is ground . the herbrand instantiation @xmath220 of a p - program is defined in the obvious manner . the following example illustrates our framework . [ medical - ex ] people are assessed to be at high risk for various diseases , depending on factors such as age group , family history ( with respect to the disease ) , etc . accordingly , high risk patients are administered appropriate medications , which are prescribed by doctors among several alternative ones . medications cause side effects , sometimes harmful ones , leading to other symptoms and diseases . here , the extent of risk , administration of medications , side effects ( caused by medications ) , and prognosis are all uncertain phenomena , and we associate confidence levels with them . the following program is a sample of the uncertain knowledge related to these phenomena . @xmath221-@xmath222 @xmath223 , [ 0.1,0.1]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } $ ] @xmath224 , @xmath225-@xmath226 @xmath227 . @xmath228 @xmath229 , [ 0,0]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } $ ] @xmath230-@xmath222 , @xmath231 @xmath232 . @xmath233 @xmath234 , [ 0.12,0.12]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } $ ] @xmath230-@xmath235 . @xmath233 @xmath236 , [ 0.70,0.70]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } $ ] @xmath237 , @xmath238-@xmath239 @xmath240 . we can assume an appropriate set of facts ( the edb ) in conjunction with the above program . for rule 1 , it is easy to see that each ground atom involving the predicate @xmath230-@xmath241 has at most one derivation . thus , a disjunctive mode for this rule will be clearly redundant , and we have suppressed it for convenience . a similar remark holds for rule 2 . rule 1 says that if a person is midaged and the disease @xmath242 has struck his ancestors , then the confidence level in the person being at high risk for @xmath242 is given by propagating the confidence levels of the body subgoals and combining them with the rule confidence in the sense of @xmath243 . this could be based on an expert s belief that the factors @xmath244 and @xmath225-@xmath245 contributing to high risk for the disease are independent . each of the other rules has a similar explanation . for the last rule , we note that the potential of a medication to cause side effects is an intrinsic property independent of whether one takes the medication . thus the conjunctive mode used there is independence . finally , note that rules 3 and 4 , defining @xmath246 , use positive correlation as a conservative way of combining confidences obtained from different derivations . for simplicity , we show each interval in the above rules as a point probability . still , note that the confidences for atoms derived from the program will be genuine intervals . * a valuation based semantics : * we develop the declarative semantics of p - programs based on the notion of valuations . let @xmath16 be a p - program . probabilistic valuation _ is a function @xmath247 which associates a confidence level with each ground atom in @xmath219 . we define the satisfaction of p - programs under valuations , with respect to the truth order @xmath50 of the trilattice ( see section [ prob - calc ] ) . we say a valuation @xmath248 _ satisfies _ a ground p - rule @xmath249 , denoted @xmath250 , provided @xmath251 @xmath252 . the intended meaning is that in order to satisfy this p - rule , @xmath248 must assign a confidence level to @xmath253 that is no less true ( in the sense of @xmath50 ) than the result of the conjunction of the confidences assigned to @xmath254 s by @xmath248 and the rule confidence @xmath111 , in the sense of the mode @xmath214 . even when a valuation satisfies ( all ground instances of ) each rule in a p - program , it may not satisfy the p - program as a whole . the reason is that confidences coming from different derivations of atoms are combined strengthening the overall confidence . thus , we need to impose the following additional requirement . let @xmath255 be a ground p - rule , and @xmath248 a valuation . then we denote by @xmath256-@xmath257 the confidence level propagated to the head of this rule under the valuation @xmath248 and the rule mode @xmath214 , given by the expression @xmath258 . let @xmath259 be the partition of @xmath220 such that ( i ) each @xmath260 contains all ( ground ) p - rules which define the same atom , say @xmath261 , and ( ii ) @xmath261 and @xmath262 are distinct , whenever @xmath263 . suppose @xmath264 is the mode associated with the head of the p - rules in @xmath260 . we denote by @xmath265-@xmath266 the confidence level determined for the atom @xmath261 under the valuation @xmath248 using the program @xmath16 . this is given by the expression @xmath267-@xmath268 . we now define satisfaction of p - programs . [ def : satisfaction ] let @xmath16 be a p - program and @xmath248 a valuation . then @xmath248 satisfies @xmath16 , denoted @xmath269 exactly when @xmath248 satisfies each ( ground ) p - rule in @xmath220 , and for all atoms @xmath270 , @xmath265-@xmath271 . the additional requirement ensures the valuation assigns a strong enough confidence to each atom so it will support the combination of confidences coming from a number of rules ( pertaining to this atom ) . a p - program @xmath16 logically implies a p - fact @xmath272 , denoted @xmath273 , provided every valuation satisfying @xmath16 also satisfies @xmath272 . we next have let @xmath248 be a valuation and @xmath16 a p - program . suppose the mode associated with the head of each p - rule in @xmath16 is positive correlation . then @xmath269 iff @xmath248 satisfies each rule in @xmath220 . * proof. we shall show that if @xmath256-@xmath274 for all rules @xmath275 defining a ground atom @xmath253 , then @xmath265-@xmath271 , where the disjunctive mode for @xmath253 is positive correlation . this follows from the formula for @xmath276 , obtained in theorem [ thm : combinations ] . it is easy to see that @xmath277 . but then , @xmath256-@xmath274 implies that @xmath278-@xmath279 and hence @xmath265-@xmath271 . the above proposition shows that when positive correlation is the only disjunctive mode used , satisfaction is very similar to the classical case . for the declarative semantics of p - programs , we need something like the least " valuation satisfying the program . it is straightforward to show that the class of all valuations @xmath280 from @xmath219 to @xmath8 itself forms a trilattice , complete with all the 3 orders and the associated meets and joins . they are obtained by a pointwise extension of the corresponding order / operation on the trilattice @xmath8 . we give one example . for valuations @xmath281 , @xmath282 iff @xmath283 , @xmath284 ; @xmath283 , @xmath285 . one could investigate least " with respect to each of the 3 orders of the trilattice . in this paper , we confine attention to the order @xmath50 . the least ( greatest ) valuation is then the valuation false * ( * true * ) which assigns the confidence level @xmath74 ( @xmath73 ) to every ground atom . we now have [ lem : lvaluation ] let @xmath16 be any p - program and @xmath281 be any valuations satisfying @xmath16 . then @xmath286 is also a valuation satisfying @xmath16 . in particular , @xmath287 is the least valuation satisfying @xmath16 . * * we prove this in two steps . first , we show that for any ground p - rule + @xmath255 + whenever valuations @xmath288 and @xmath248 satisfy @xmath289 , so does @xmath286 . secondly , we shall show that for a p - program @xmath16 , whenever _ atom - conf_@xmath290 and _ atom - conf_@xmath291 , then we also have _ atom - conf_@xmath292 . the lemma will follow from this . \(1 ) suppose @xmath293 and @xmath294 . we prove the case where the conjunctive mode @xmath214 associated with this rule is ignorance . the other cases are similar . it is straightforward to verify the following . \(i ) @xmath295 . + ( ii ) @xmath296 . from ( i ) and ( ii ) , we have @xmath297 , showing @xmath298 . \(2 ) suppose @xmath281 are any two valuations satisfying a p - program @xmath16 . let @xmath299 be the set of all ground p - rules in @xmath220 whose heads are @xmath253 . let @xmath300 _ rule - conf_@xmath301 and @xmath302 _ rule - conf_@xmath303 . since @xmath304 and @xmath305 , we have that @xmath306 and @xmath307 , where @xmath215 is the disjunctive mode associated with @xmath253 . again , we give the proof for the case @xmath215 is ignorance as the other cases are similar . let @xmath308 _ rule - conf_@xmath309 . clearly , @xmath310 and @xmath311 . thus , @xmath312 and @xmath313 . it then follows that @xmath314 , which was to be shown . we take the least valuation satisfying a p - program as characterizing its declarative semantics . consider the following p - program @xmath16 . @xmath315,~[0.3,0.45]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } ~b;~~ind,~pc)$ ] . @xmath316,~[0.1,0.2]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } ~c;~~ign,~pc)$ ] . + 3 . @xmath317,~[0,0.1]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } ; ~~\_,~ind)$ ] . 4 . @xmath318,~[0.1,0.2]\rangle}}{{\mbox{$<\hspace{-9pt } \rule[2pt]{80pt}{0.5pt}$ } } } $ } } ; ~~\_,~ind)$ ] . in the following we show three valuations @xmath319 , of which @xmath320 and @xmath321 satisfy @xmath16 , while @xmath322 does not . in fact , @xmath321 is the least valuation satisfying @xmath16 . @xmath323,~[0,0]\rangle & \langle[0.8,0.9],~[0.05,0.1]\rangle & \langle[0.5,0.9],~[0,0]\rangle \\ v_2 & \langle[0.9,1],~[0,0]\rangle & \langle[0.9,1],~[0,0]\rangle & \langle[0.5,0.7],~[0.1,0.4]\rangle \\ v_3 & \langle[0.9,0.95],~[0,0.1]\rangle & \langle[0.7,0.8],~[0.1,0.2]\rangle & \langle[0.45,0.8],~[0.1,0.4]\rangle \end{array}$ ] for example , consider @xmath320 . it is easy to verify that @xmath320 satisfies @xmath16 . rules 1 through 4 are satisfied by @xmath320 since : @xmath324,~[0.3,0.45]\rangle \wedge_{ind } \langle[0.9,1],~[0,0]\rangle = \langle[0.45,0.7],~[0,0]\rangle \le_t \langle[0.5,0.9],~[0,0]\rangle$ ] @xmath325,~[0.1,0.2]\rangle \wedge_{ign } \langle[0.8,0.9],~[0.05,0.1]\rangle = $ ] @xmath326,~[0.1,0.3]\rangle \le_t \langle[0.5,0.9],~[0,0]\rangle$ ] @xmath327,~[0,0.1]\rangle \le_t \langle[0.9,1],~[0,0]\rangle$ ] @xmath328,~[0.1,0.2]\rangle \le_t \langle[0.8,0.9],~[0.05,0.1]\rangle$ ] further , the confidence level of @xmath253 computed by the combination of rules 1 and 2 is also satisfied by @xmath320 , namely , @xmath329,~[0.3,0.45]\rangle \wedge_{ind } \langle[0.9,1],~[0,0]\rangle ) \vee_{pc } ( \langle[0.6,0.8],~[0.1,0.2]\rangle$ ] @xmath330,~[0.05,0.1]\rangle ) = \langle[0.45,0.8],~[0,0 ] \le_t \langle[0.5,0.9],~[0,0]\rangle$ ] * fixpoint semantics : * we associate an immediate consequence " operator @xmath1 with a p - program @xmath16 , defined as follows . [ tp - defn ] let @xmath16 be a p - program and @xmath220 its herbrand instantiation . then @xmath1 is a function @xmath331 , defined as follows . for any probabilistic valuation @xmath248 , and any ground atom @xmath270 , @xmath332 there exists a p - rule @xmath333 such that @xmath334 . call a valuation @xmath248 _ consistent _ provided for every atom @xmath253 , @xmath335 is consistent , as defined in section [ lattice ] . [ thm : tpmonotone ] ( 1 ) @xmath1 always maps consistent valuations to consistent valuations . ( 2 ) @xmath1 is monotone and continuous . * proof . * ( 1 ) this fact follows theorem [ thm : consistent ] , where we have shown that the combination functions for all modes map consistent confidence levels to consistent confidence levels . ( 2 ) this follows from the fact that the combination functions for all modes are themselves monotone and continuous . we define bottom - up iterations based on @xmath1 in the usual manner . + @xmath336 ( which assigns the truth - value @xmath74 to every ground atom ) . + @xmath337 , for a successor ordinal @xmath10 . + @xmath338 , for a limit ordinal @xmath10 . we have the following results . [ prop : fixpoint ] let @xmath248 be any valuation and @xmath16 be a p - program . then @xmath248 satisfies @xmath16 iff @xmath339 . * proof . * _ ( only if ) . _ if @xmath248 satisfies @xmath16 , then by definition [ def : satisfaction ] , for all atoms @xmath270 , @xmath265-@xmath271 and hence @xmath339 . _ _ if @xmath339 , then by the definition of @xmath1 ( definition [ tp - defn ] ) for all atoms @xmath270 , @xmath265-@xmath271 and hence @xmath248 satisfies @xmath16 . the following theorem is the analogue of the van emden - kowalski theorem for classical logic programming . [ thm : lfplvaluation ] let @xmath16 be a p - program . then the following claims hold . + ( i ) @xmath340 the @xmath50-least valuation satisfying @xmath16 . + ( ii ) for a ground atom @xmath253 , @xmath341 iff @xmath273 . * * follows lemma [ lem : lvaluation ] , theorem [ thm : tpmonotone ] and proposition [ prop : fixpoint ] . proof is similar to the analogous theorem of logic programming and details are omitted . since confidences coming from different derivations of a fact are combined , we need a notion of disjunctive proof - trees . we note that the notions of substitution , unification , etc . are analogous to the classical ones . a variable appearing in a rule is _ local _ if it only appears in its body . [ dpt ] let @xmath113 be a(n atomic ) goal and @xmath16 a p - program . then a _ disjunctive proof - tree _ ( dpt ) for @xmath113 with respect to @xmath16 is a tree @xmath342 defined as follows . @xmath342 has two kinds of nodes : _ rule _ nodes and _ goal _ nodes . each rule node is labeled by a rule in @xmath16 and a substitution . each goal node is labeled by an atomic goal . the root is a goal node labeled @xmath113 . let @xmath288 be a goal node labeled by an atom @xmath253 . then every child ( if any ) of @xmath288 is a rule node labeled @xmath343 , where @xmath213 is a rule in @xmath16 whose head is unifiable with @xmath253 using the mgu @xmath344 . we assume that each time a rule @xmath213 appears in the tree , its variables are renamed to new variables that do not appear anywhere else in the tree . hence @xmath213 in the label @xmath343 actually represents a renamed instance of the rule . if @xmath288 is a rule node labeled @xmath343 , then whenever an atom @xmath345 occurs in the body of @xmath346 , @xmath288 has a goal child @xmath248 labeled @xmath345 . + 4 . for any two substitutions @xmath347 occurring in @xmath342 , @xmath348 , for every variable @xmath349 . in other words , all substitutions occurring in @xmath342 are compatible . a node without children is called a leaf . a _ proper _ dpt is a finite dpt @xmath342 such that whenever @xmath342 has a goal leaf labeled @xmath253 , there is no rule in @xmath16 whose head is unifiable with @xmath253 . we only consider proper dpts unless otherwise specified . a rule leaf is a _ node ( represents a database fact ) while a goal leaf is a _ failure _ node . * remarks : * \(1 ) the definition of disjunctive proof tree captures the idea that when working with uncertain information in the form of probabilistic rules and facts , we need to consider the disjunction of all proofs in order to determine the best possible confidence in the goal being proved . \(2 ) however , notice that the definition does _ not _ insist that a goal node @xmath253 should have rule children corresponding to all possible unifiable rules and mgu s . \(3 ) we assume without loss of generality that all rules in the p - program are standardized apart by variable renaming so they share no common variables . \(4 ) a goal node can have several rule children corresponding to the same rule . that is , a goal node can have children labeled @xmath350 , where @xmath213 is ( a renamed version of ) a rule in the program . but we require that @xmath351 , @xmath352 , be distinct . \(5 ) we require all substitutions in the tree to be compatible . the convention explained above ensures there will be no conflict among them on account of common variables across rules ( or different invocations of the same rule ) . \(6 ) note that a dpt can be finite or infinite . \(7 ) in a proper dpt , goal leaves are necessarily failure nodes ; this is not true in non - proper dpts . \(8 ) a proper dpt with no failure nodes has only rule leaves , hence , it has an odd height . confidences are associated with ( finite ) dpts as follows . [ dpt - conf ] let @xmath16 be a p - program , @xmath113 a goal , and @xmath342 any finite dpt for @xmath113 with respect to @xmath16 . we associate confidences with the nodes of @xmath342 as follows . each failure node gets the confidence @xmath353,~[1,1]\rangle$}}$ ] , the * false * confidence level with respect to truth ordering , @xmath74 ( see section [ lattice ] ) . each success node labeled @xmath343 , where @xmath213 is a rule in @xmath16 , and @xmath111 is the confidence of rule @xmath213 , gets the confidence @xmath111 . suppose @xmath288 is a rule node labeled @xmath343 , such that the confidence of @xmath213 is @xmath111 , its ( conjunctive ) mode is @xmath214 , and the confidences of the children of @xmath288 are @xmath354 . then @xmath288 gets the confidence @xmath355 . suppose @xmath288 is a goal node labeled @xmath253 , with a ( disjunctive ) mode @xmath215 such that the confidences of its children are @xmath356 . then @xmath288 gets the confidence @xmath357 . we recall the notions of identity and annihilator from algebra ( see ullman @xcite ) . let @xmath358 be any element of the confidence lattice and @xmath359 be any operation of the form @xmath360 or of the form @xmath361 , @xmath362 being any of the modes discussed in section [ prob - calc ] . then @xmath111 is an _ identity _ with respect to @xmath359 , if @xmath363 . it is an _ annihilator _ with respect to @xmath359 , if @xmath364 . the proof of the following proposition is straightforward . [ identity - annihilator ] the truth - value @xmath365,~[1,1]\rangle$}}$ ] is an identity with respect to disjunction and an annihilator with respect to conjunction . the truth - value @xmath366,~[0,0]\rangle$}}$ ] is an identity with respect to conjunction and an annihilator with respect to disjunction . these claims hold for all modes discussed in section [ prob - calc ] . in view of this proposition , we can consider only dpts without failure nodes without losing any generality . we now proceed to prove the soundness and completeness theorems . first , we need some definitions . a _ branch _ @xmath345 of a dpt @xmath342 is a set of nodes of @xmath342 , defined as follows . the root of @xmath342 belongs to @xmath345 . whenever a goal node is in @xmath345 , exactly one of its rule children ( if any ) belongs to @xmath345 . finally , whenever a rule node belongs to @xmath345 , all its goal children belong to @xmath345 . we extend this definition to the subtrees of a dpt in the obvious way . a _ subbranch _ of @xmath342 rooted at a goal node @xmath113 is the branch of the subtree of @xmath342 rooted at @xmath113 . we can associate a substitution with a ( sub)branch @xmath345 as follows . ( 1 ) the substitution associated with a success node labeled @xmath343 is just @xmath344 . ( 2 ) the substitution associated with an internal goal node is simply the substitution associated with its unique rule child in @xmath345 . ( 3 ) the substitution associated with an internal rule node @xmath288 in @xmath345 which is labeled @xmath343 is the composition of @xmath344 and the substitutions associated with the goal children of @xmath288 . the substitution associated with a branch is that associated with its root . we say a dpt @xmath342 is _ well - formed _ if it satisfies the following conditions : ( i ) @xmath342 is proper , ( ii ) for every goal node @xmath113 in @xmath342 , for any two ( sub)branches @xmath367 of @xmath342 rooted at @xmath113 , the substitutions associated with @xmath368 and @xmath369 are distinct . the second condition ensures no two branches correspond to the same classical proof " of the goal or a sub - goal . without this condition , since the probabilistic conjunctions and disjunctions are not idempotent , the confidence of the same proof could be wrongly combined giving an incorrect confidence for the ( root of the ) dpt . henceforth , we will only consider well - formed dpts , namely , dpts that are proper , have no failure nodes , and have distinct substitutions for all ( sub ) branches corresponding to a goal node , for all goal nodes . [ soundness ] [ soundness ] let @xmath16 be a p - program and @xmath113 a ( ground ) goal . if there is a finite well - formed dpt for @xmath113 with respect to @xmath16 with an associated confidence @xmath111 at its root , then @xmath370 . * proof . * first , we make the following observations regarding the combination functions of theorem [ thm : combinations ] : \(1 ) conjunctive combination functions ( all modes ) are monotone . + ( 2 ) disjunctive combination functions ( all modes ) are monotone . + ( 3 ) if @xmath3 and @xmath113 are confidence levels , then @xmath371 and @xmath372 for all conjunctive and disjunctive combination functions ( all modes ) . we prove the soundness theorem by induction on the height of the dpt . assume the well - formed dpt @xmath342 of height @xmath373 is for the goal @xmath113 . note that @xmath342 has an odd height , @xmath374 for some @xmath375 , since it is a proper dpt with no failure nodes ( see remark 7 at the beginning of this section ) . : @xmath376 . in this case the dpt consists of the goal root labeled @xmath113 and one child labeled @xmath377 , where @xmath213 is a rule in @xmath16 whose head is unifiable with @xmath113 . note that this child node is a success leaf . it represents a fact . obviously , in the first iteration of @xmath1 , @xmath378 , where @xmath379 is the confidence level of @xmath213 . it follows from the monotonicity of @xmath1 , that @xmath380 . : @xmath381 . assume the inductive hypothesis holds for every dpt of height @xmath382 , where @xmath383 . consider the dpt @xmath342 for @xmath113 . the root @xmath113 has rule children @xmath384 labeled @xmath385 . each @xmath384 is either a fact , or has goal children @xmath386 . consider the subtrees of @xmath342 rooted at these goal grand children of @xmath113 . by the inductive hypothesis , the confidence levels @xmath387 associated with the goal grand children @xmath388 by the dpt are less than or equal to their confidence levels calculated by @xmath1 , , @xmath389 . hence , by properties ( 1)-(3 ) above , the confidence level associated to @xmath113 by @xmath342 is less than or equal to the confidence level of @xmath113 obtained by another application of @xmath1 , @xmath390 . hence @xmath370 . note that @xmath342 must be well - formed otherwise this argument is not valid . [ completeness ] [ completeness ] let @xmath16 be a p - program and @xmath113 a goal such that for some number @xmath391 , @xmath392 . then there is a finite dpt @xmath342 for @xmath113 with respect to @xmath16 with an associated confidence @xmath111 at its root , such that @xmath393 . * let @xmath55 be the smallest number such that @xmath394 . we shall show by induction on @xmath55 that there is a dpt @xmath342 for @xmath113 with respect to @xmath16 such that the confidence computed by it is at least @xmath395 . : this implies @xmath397,~[1,1]\rangle$}}$ ] . this case is trivial . the dpt consists of a failure node labeled @xmath113 . : suppose the result holds for a certain number @xmath398 . we show that it also holds for @xmath399 . suppose @xmath253 is a ground atom such that @xmath400 . now , @xmath401 there exists a rule @xmath213 such that @xmath214 is the mode associated with its body , and @xmath215 is the mode associated with its head , and there exists a ground substitution @xmath344 such that @xmath402 . consider the dpt for @xmath253 obtained as follows . let the root be labeled @xmath253 . the root has a rule child corresponding to each rule instance used in the above computation of @xmath403 . let @xmath248 be a rule child created at this step , and suppose @xmath404 is the rule instance corresponding to it and let @xmath344 be the substitution used to unify the head of the original rule with the atom @xmath253 . then @xmath248 has @xmath405 goal children with labels @xmath406 respectively . finally , by induction hypothesis , we can assume that ( i ) a dpt for @xmath254 is rooted at the node labeled @xmath254 , and ( ii ) the confidence computed by this latter tree is at least @xmath407 , @xmath408 . in this case , it readily follows from the definition of the confidence computed by a proof - tree that the confidence computed by @xmath342 is at least @xmath409 is a rule defining @xmath253 , @xmath379 is the confidence associated with it , @xmath214 is the mode associated with its head , and @xmath215 is the mode associated with its body@xmath410 . but this confidence is exactly @xmath403 . the induction is complete and the theorem follows . theorems [ soundness ] and [ completeness ] together show that the confidence of an arbitrary ground atom computed according to the fixpoint semantics and using an appropriate disjunctive proof tree is the same . this in turn is the same as the confidence associated according to the ( valuation based ) declarative semantics . in particular , as we will discuss in section [ termination ] , when the disjunctive mode associated with all recursive predicates is positive correlation , the theorems guarantee that the exact confidence associated with the goal can be effectively computed by constructing an appropriate finite dpt ( according to theorem [ completeness ] ) for it . even when these modes _ are _ used indiscriminately , we can still obtain the confidence associated with the goal with an arbitrarily high degree of accuracy , by constructing dpts of appropriate height . in this section , we first compare our work with that of ng and subrahmanian @xcite ( see section [ intro ] for a general comparison with non - probabilistic frameworks ) . first , let us examine the ( only ) mode " for disjunction used by them . they combine the confidences of an atom @xmath253 coming from different derivations by taking their intersection . indeed , the bottom of their lattice is a valuation ( called formula function " there ) that assigns the interval @xmath0 $ ] to every atom . from the trilattice structure , it is clear that ( i ) their bottom corresponds to @xmath78 , and ( ii ) their disjunctive mode corresponds to @xmath411 . [ problem1 ] @xmath412 : @xmath413 { \mbox{$\leftarrow$}}e(x , z ) : [ v_1 , v_2 ] , ~~$]@xmath414 $ ] . + @xmath415 : @xmath416 { \mbox{$\leftarrow$}}e(x , y ) : [ v_1 , v_2]$ ] . + @xmath417 : @xmath418 $ ] . + @xmath419 : @xmath420 $ ] . + @xmath421 : @xmath422 $ ] . this is a pf - program in the framework of ng and subrahmanian @xcite . in a pf - rule each literal is annotated by an interval representing the lower - bound and upper - bound of belief . variables can appear in the annotations , and the annotation of the head predicate is usually a function of body literals annotations . the program in this example is basically the transitive closure program , with independence as the conjunctive mode in the first rule . the disjunctive function for the @xmath56 predicate , as explained above , is interval intersection . let us denote the operator @xmath1 defined by them as @xmath423 for distinguishing it from ours . it is not hard to see that this program is inconsistent in their framework , and @xmath424would assign an empty probability range for @xmath425 . this is due to the existence of two derivations for @xmath425 , with non - overlapping intervals . this is quite unintuitive . indeed , there is a definite path ( with probability 1 ) corresponding to the edge @xmath426 . one may wonder whether it makes sense to compare this approach with ours on an example program which is inconsistent according to their semantics . the point is that in this example , there is a certain path with probability [ 1,1 ] from 1 to 2 , and an approach that regards this program as inconsistent is not quite intuitive . now , consider the p - program corresponding to the annotated program @xmath427 @xmath428 , obtained by stripping off atom annotations in @xmath429 and shifting the annotations in @xmath430 to the associated rules . also , associate the confidence level @xmath431,~[0,0]\rangle$}}$ ] with @xmath429 . for uniformity and ease of comparison , assume the doubt ranges are all @xmath432 $ ] . as an example , let the conjunctive mode used in @xmath429 be independence and let the disjunctive mode used be positive correlation ( or , in this case , even ignorance ! ) . then @xmath433 would assign the confidence @xmath434,~[0,0]\rangle$ ] to @xmath435 , which agrees with our intuition . our point , however , is not that intersection is a wrong " mode . rather , we stress that different combination rules ( modes ) are appropriate for different situations . [ problem2 ] now consider the following pf - program ( @xmath412 and @xmath415 are the same as previous example ) : @xmath412 : @xmath413 { \mbox{$\leftarrow$}}e(x , z ) : [ v_1 , v_2 ] , ~~$]@xmath414 $ ] . + @xmath415 : @xmath416 { \mbox{$\leftarrow$}}e(x , y ) : [ v_1 , v_2]$ ] . + @xmath436 : @xmath437 $ ] . + @xmath438 : @xmath439 $ ] . in this case , the least fixpoint of @xmath423 is only attained at @xmath2 and it assigns the range @xmath440 $ ] to @xmath441 and @xmath425 . again , the result is unintuitive for this example . since @xmath423 is not continuous , one can easily write programs such that no reasonable approximation to @xmath424 can be obtained by iterating @xmath423 an arbitrary ( finite ) number of times . ( , consider the program obtained by adding the rule @xmath442 : @xmath443~{\mbox{$\leftarrow$}}~p(x , y ) : [ 0,0]$ ] to @xmath444 . ) notice that as long as one uses any arithmetic annotation function such that the probability of the head is less than the probability of the subgoals of @xmath412 ( which is a reasonable annotation function ) , this problem will arise . the problem ( for the unintuitive behavior ) lies with the mode for disjunction . again , we emphasize that different combination rules ( modes ) are appropriate for different situations . now , consider the p - program corresponding to the annotated program @xmath445 , obtained in the same way as was done in example [ problem1 ] . let the conjunctive mode used in @xmath429 be independence and let the disjunctive mode be positive correlation or ignorance . then @xmath433 would assign the confidence level @xmath446~[0,0]\rangle$ ] to @xmath435 . this again agrees with our intuition . as a last example , suppose we start with the confidence @xmath447,~[0,0]\rangle$ ] for @xmath448 instead . then under positive correlation ( for disjunction ) @xmath449,~[0,0]\rangle$ ] , while ignorance leads to @xmath450,~[0,0]\rangle$ ] . the former makes more intuitive sense , although the latter ( more conservative under @xmath52 ) is obviously not wrong . also , in the latter case , the @xmath451 is reached only at @xmath2 . now , we discuss termination and complexity issues of p - programs . let the _ closure ordinal _ of @xmath1 be the smallest ordinal @xmath10 such that @xmath452 . we have the following [ thm : ordinal ] let @xmath16 be any p - program . then the closure ordinal of @xmath1 can be as high as @xmath2 but no more . * proof . * the last p - program discussed in example [ problem2 ] has a closure ordinal of @xmath2 . since @xmath1 is continuous ( theorem [ thm : tpmonotone ] ) its closure ordinal is at most @xmath2 . [ defn : datacomplexity ] ( _ data complexity _ ) we define the _ data complexity _ @xcite of a p - program @xmath16 as the time complexity of computing the least fixpoint of @xmath1 as a function of the size of the database , the number of constants occurring in @xmath16 . it is well known that the data complexity for datalog programs is polynomial . an important question concerning any extension of ddbs to handle uncertainty is whether the data complexity is increased compared to datalog . we can show that under suitable restrictions ( see below ) the data complexity of p - programs is polynomial time . however , the proof can not be obtained by ( straightforward extensions of ) the classical argument for the data complexity for datalog . in the classical case , once a ground atom is derived during bottom - up evaluation , future derivations of it can be ignored . in programming with uncertainty , complications arise because we _ can not _ ignore alternate derivations of the same atom : the confidences obtained from them need to be combined , reinforcing the overall confidence of the atom . this calls for a new proof technique . our technique makes use of the following additional notions . define a _ disjunctive derivation tree _ ( ddt ) to be a well - formed dpt ( see section [ proof - theory ] for a definition ) such that every goal and every substitution labeling any node in the tree is ground . note that the height of a ddt with no failure nodes is an odd number ( see remark 7 at the beginning of section [ proof - theory ] ) . we have the following results . [ ddt - bu - eval ] let @xmath16 be a p - program and @xmath253 any ground atom in @xmath219 . suppose the confidence determined for @xmath253 in iteration @xmath453 of bottom - up evaluation is @xmath111 . then there exists a ddt @xmath342 of height @xmath454 for @xmath253 such that the confidence associated with @xmath253 by @xmath342 is exactly @xmath111 . * proof . * the proof is by induction on @xmath55 . : @xmath376 . in iteration 1 , bottom - up evaluation essentially collects together all edb facts ( involving ground atoms ) and determines their confidences from the program . without loss of generality , we may suppose there is at most one edb fact in @xmath16 corresponding to each ground atom ( involving an edb predicate ) . let @xmath253 be any ground atom whose confidence is determined to be @xmath111 in iteration 1 . then there is an edb fact @xmath455 in @xmath16 . the associated ddt for @xmath253 corresponding to this iteration is the tree with root labeled @xmath253 and a rule child labeled @xmath213 . clearly , the confidence associated with the root of this tree is @xmath111 , and the height of this tree is @xmath37 ( @xmath456 , for @xmath457 . : assume the result for all ground atoms whose confidences are determined ( possibly revised ) in iteration @xmath55 . suppose @xmath253 is a ground atom whose confidence is determined to be @xmath111 in iteration @xmath458 . this implies there exist ground instances of rules @xmath459 , @xmath460 ; @xmath461 such that ( i ) the confidence of @xmath254 @xmath462 computed at the end of iteration @xmath55 is @xmath463 ( @xmath464 ) , and ( ii ) @xmath465 @xmath466 @xmath467 @xmath468 , where is the disjunctive mode for the predicate @xmath253 . by induction hypothesis , there are ddts @xmath469 , @xmath470 , each of height @xmath471 or less , for the atoms @xmath406 , @xmath472 which exactly compute the confidences @xmath354 , @xmath473 respectively , corresponding to iteration @xmath55 . consider the tree @xmath474 for @xmath253 by ( i ) making @xmath475 rule children of the root and ( ii ) making the @xmath469 , ( @xmath476 ) subtrees of @xmath412 ( @xmath477 ) . it is trivial to see that @xmath474 is a ddt for @xmath253 and its height is @xmath478 . further the confidence @xmath474 computes for the root @xmath253 is exactly @xmath479 @xmath466 @xmath467 @xmath468 . this completes the induction and the proof . proposition [ ddt - bu - eval ] shows each iteration of bottom - up evaluation corresponds in an essential manner to the construction of a set of ddts one for each distinct ground atom whose confidence is determined ( or revised ) in that iteration . our next objective is to establish a termination bound on bottom - up evaluation . ddt branches are defined similar to those of dpt . let @xmath342 be a ddt . then a branch of @xmath342 is a subtree of @xmath342 , defined as follows . \(i ) the root belongs to every branch . + ( ii ) whenever a goal node belongs to a branch , exactly one of its rule children , belongs to the branch . + ( iii ) whenever a rule node belongs to a branch , all its goal children belong to the branch . let @xmath253 be a ground atom and @xmath342 any ddt ( not necessarily for @xmath253 ) . then @xmath342 is @xmath253-non - simple provided it has a branch containing two goal nodes @xmath288 and @xmath248 such that @xmath288 is an ancestor of @xmath248 and both are labeled by atom @xmath253 . a ddt is @xmath253-simple if it is not @xmath253-non - simple . finally , a ddt is simple if it is @xmath253-simple for every atom @xmath253 . let @xmath342 be a ddt and @xmath254 be a branch of @xmath342 in which an atom @xmath253 appears . then we define the _ number of violations of simplicity _ of @xmath254 with respect to @xmath253 to be one less than the total number of times the atom @xmath253 occurs in @xmath254 . the number of violations of the simplicity of the ddt @xmath342 with respect to @xmath253 is the sum of the number of violations of the branches of @xmath342 in which @xmath253 occurs . clearly , @xmath342 is @xmath253-simple exactly when the number of violations with respect to @xmath253 is 0 . our first major result of this section follows . [ term - bound ] let @xmath16 be a p - program such that only positive correlation is used as the disjunctive mode for recursive predicates . let @xmath480a@xmath481 , and @xmath482 is any simple ddt for @xmath483 = @xmath454 , @xmath453 . then at most @xmath458 iterations of naive bottom - up evaluation are needed to compute the least fixpoint of @xmath1 . essentially , for p - programs @xmath16 satisfying the conditions mentioned above , the theorem ( i ) shows that naive bottom - up evaluation of @xmath16 is guaranteed to terminate , and ( ii ) establishes an upper bound on the number of iterations of bottom - up evaluation for computing the least fixpoint , in terms of the maximum height of any simple tree for any ground atom . this is the first step in showing that p - programs of this type have a polynomial time data complexity . we will use the next three lemmas ( lemmas [ lem : key][lem : boundfora ] ) in proving this theorem . [ lem : key ] let @xmath484 be any ground atom , and let @xmath342 be a ddt for @xmath253 corresponding to @xmath485 , for some @xmath486 . suppose @xmath342 is @xmath345-non - simple , for some atom @xmath345 . then there is a ddt @xmath487 for @xmath253 with the following properties : \(i ) the certainty of @xmath253 computed by @xmath487 equals that computed by @xmath342 . + ( ii ) the number of violations of simplicity of @xmath487 with respect to @xmath345 is less than that of @xmath342 . * proof. let @xmath342 be the ddt described in the hypothesis of the claim . let @xmath253 be the label of the root @xmath288 of @xmath342 , and assume without loss of generality that @xmath345 is identical to @xmath253 . ( the case when @xmath345 is distinct from @xmath253 is similar . ) let @xmath248 be the last goal node from the root down ( e.g. in the level - order ) , which is distinct from the root and is labeled by @xmath253 . since @xmath342 corresponds to applications of the @xmath1 operator , we have the following . ( ) every branch of @xmath248 must be isomorphic to some branch of @xmath288 which does not contain the node @xmath248 . this can be seen as follows . let @xmath488 be the iteration such that the subtree of @xmath342 rooted at @xmath248 , say @xmath489 , corresponds to @xmath490 . then clearly , every rule applicable in iteration @xmath56 is also applicable in iteration @xmath55 . this means every branch of @xmath489 constructed from a sequence of rule applications is also constructible in iteration @xmath55 and hence there must be a branch of @xmath342 that is isomorphic to such a branch . it follows from the isomorphism that the isomorphic branch of @xmath342 can not contain the node @xmath248 . associate a logical formula with each node of @xmath342 as follows . \(i ) the formula associated with each ( rule ) leaf is * true. + ( ii ) the formula associated with a goal node with rule children @xmath491 and associated formulas @xmath492 , is @xmath493 . + ( iii ) the formula associated with a rule parent with goal children @xmath494 and associated formulas @xmath495 is @xmath496 . let the formula associated with the node @xmath248 be @xmath3 . to simplify the exposition , but at no loss of generality , let us assume that in @xmath342 , every goal node has exactly two rule children . then the formula associated with the root @xmath288 can be expressed as @xmath497 . by ( ) above , we can see that @xmath3 logically implies @xmath498 , @xmath499 . by the structure of a ddt , we can then express @xmath498 as @xmath500 , for some formula @xmath113 . construct a ddt @xmath487 from @xmath342 by deleting the parent of the node @xmath248 , as well as the subtree rooted at @xmath248 . we claim that ( * * ) the formula associated with the root of @xmath487 is equivalent to that associated with the root of @xmath342 . to see this , notice that the formula associated with the root of @xmath342 can now be expressed as @xmath501 . by simple application of propositional identities , it can be seen that this formula is equivalent to @xmath502 . but this is exactly the formula associated with the root of t , which proves ( * * ) . finally , we shall show that @xmath276 , together with any conjunctive mode , satisfy the following absorption laws : @xmath503 . + @xmath504 . the first of these laws follows from the fact that for all modes @xmath362 we consider in this paper , @xmath505 , where @xmath50 is the lattice ordering . the second is the dual of the first . in view of the absorption laws , it can be seen that the certainty for @xmath253 computed by @xmath487 above is identical to that computed by @xmath342 . this proves the lemma , since @xmath487 has at least one fewer violations of simplicity with respect to @xmath253 . [ lem : simple ] let @xmath342 be a ddt for an atom @xmath253 . then there is a simple ddt for @xmath253 such that the certainty of @xmath253 computed by it is identical to that computed by @xmath342 . * proof. follows by an inductive argument using lemma [ lem : key ] . [ lem : boundfora ] let @xmath253 be an atom and @xmath506 be the maximum height of any simple ddt for @xmath253 . then certainty of @xmath253 in @xmath485 is identical to that in @xmath507 , for all @xmath508 . proof. let @xmath342 be the ddt for @xmath253 corresponding to @xmath485 . note that height(@xmath509 . let @xmath111 represent the certainty computed by @xmath342 for @xmath253 , which is @xmath510 . by lemma [ lem : simple ] , there is a simple ddt , say @xmath487 , for @xmath253 , which computes the same certainty for @xmath253 as @xmath342 . clearly , height(@xmath511 . let @xmath112 represent the certainty computed by @xmath487 for @xmath253 , @xmath512 . by the soundness theorem , and monotonicity of @xmath1 , we can write @xmath513 . it follows that @xmath514 . now we can complete the proof of theorem [ term - bound ] . proof of theorem [ term - bound]. let @xmath515 be the maximum height of any simple ddt for any atom . it follows from the above lemmas that the certainty of any atom in @xmath485 is identical to that in @xmath516 , for all @xmath517 , from which the theorem follows . it can be shown that the height of simple ddts is polynomially bounded by the database size . this makes the above result significant . this allows us to prove the following theorem regarding the data complexity of the above class of p - programs . [ complexity ] let @xmath16 be a p - program such that only positive correlation is used as the disjunctive mode for recursive predicates . then its least fixpoint can be computed in time polynomial in the database size . in particular , bottom - up naive evaluation terminates in time polynomial in the size of the database , yielding the least fixpoint . by theorem [ term - bound ] we know that the least fixpoint model of @xmath16 can be computed in at most @xmath458 iterations where @xmath518 is the maximum height of any simple ddt for any ground atom with respect to @xmath16 ( @xmath55 iterations to arrive at the fixpoint , and one extra iteration to verify that a fixpoint has been reached . ) notice that each goal node in a ddt corresponds to a database predicate . let @xmath519 be the maximum arity of any predicate in @xmath16 , and @xmath398 be the number of constants occurring in the program . notice that under the data complexity measure ( definition [ defn : datacomplexity ] ) @xmath519 is a constant . the maximum number of distinct goal nodes that can occur in any branch of a simple ddt is @xmath520 . this implies the height @xmath373 above is clearly a polynomial in the database size @xmath398 . we have thus shown that bottom - up evaluation of the least fixpoint terminates in a polynomial number of iterations . the fact that the amount of work done in each iteration is polynomial in @xmath398 is easy to see . the theorem follows . we remark that our proof of theorem [ complexity ] implies a similar result for van emden s framework . to our knowledge , this is the first polynomial time result for rule - based programming with ( probabilistic ) uncertainty . we should point out that the polynomial time complexity is preserved whenever modes other than positive correlation are associated with non - recursive predicates ( for disjunction ) . more generally , suppose @xmath521 is the set of all recursive predicates and @xmath522 is the set of non - recursive predicates in the kb , which are possibly defined in terms of those in @xmath521 . then any modes can be freely used with the predicates in @xmath522 while keeping the data complexity polynomial . finally , if we know that the data does not contain cycles , we can use any mode even with a recursive predicate and still have a polynomial time data complexity . we also note that the framework of annotation functions used in @xcite enables an infinite family of modes to be used in propagating confidences from rule bodies to heads . the major differences with our work are ( i ) in @xcite a fixed mode " for disjunction is imposed unlike our framework , and ( ii ) they do not study the complexity of query answering , whereas we establish the conditions under which the important advantage of polynomial time data complexity of classical datalog can be retained . more importantly , our work has generated useful insights into how modes ( for disjunction ) affect the data complexity . finally , a note about the use of positive correlation as the disjunctive mode for recursive predicates ( when data might contain cycles ) . the rationale is that different derivations of such recursive atoms could involve some amount of overlap ( the degree of overlap depends on the data ) . now , positive correlation ( for disjunction ) tries to be conservative ( and hence sound ) by assuming the extent of overlap is maximal , so the combined confidence of the different derivations is the least possible ( under @xmath50 ) . thus , it _ does _ make sense even from a practical point of view . we motivated the need for modeling both belief and doubt in a framework for manipulating uncertain facts and rules . we have developed a framework for probabilistic deductive databases , capable of manipulating both belief and doubt , expressed as probability intervals . belief doubt pairs , called confidence levels , give rise to a rich algebraic structure called a trilattice . we developed a probabilistic calculus permitting different modes for combining confidence levels of events . we then developed the framework of p - programs for realizing probabilistic deductive databases . p - programs inherit the ability to parameterize " the modes used for combining confidence levels , from our probabilistic calculus . we have developed a declarative semantics , a fixpoint semantics , and proved their equivalence . we have also provided a sound and complete proof procedure for p - programs . we have shown that under disciplined use of modes , we can retain the important advantage of polynomial time data complexity of classical datalog , in this extended framework . we have also compared our framework with related work with respect to the aspects of termination and intuitive behavior ( of the semantics ) . the parametric nature of modes in p - programs is shown to be a significant advantage with respect to these aspects . also , the analysis of trilattices shows insightful relationships between previous work ( ng and subrahmanian @xcite ) and ours . interesting open issues which merit further research include ( 1 ) semantics of p - programs under various trilattice orders and various modes , including new ones , ( 2 ) query optimization , ( 3 ) handling inconsistency in a framework handling uncertainty , such as the one studied here . the authors would like to thank the anonymous referees for their careful reading and their comments , many of which have resulted in significant improvements to the paper . kifer , m. , & li , a. ( 1988 ) . on the semantics of rule - based expert systems with uncertainty . gyssens , m. , paradaens , j. , & van gucht , d. ( eds ) , _ conf . on database theory_. bruges , belgium : springer - verlag lncs-326 . lakshmanan , l. v. s. , & shiri , n. ( 1997 ) . a parametric approach to deductive databases with uncertainty . . ( a preliminary version appeared in proc . workshop on logic in databases ( lid96 ) , springer - verlag , lncs-1154 , san miniato , italy ) . ng , r. t. , & subrahmanian , v. s. ( 1991 ) . . report umiacs - tr-91 - 49 , cs - tr-2647 . institute for advanced computer studies and department of computer science university of maryland , college park , md 20742 .	we propose a framework for modeling uncertainty where both belief and doubt can be given independent , first - class status . we adopt probability theory as the mathematical formalism for manipulating uncertainty . an agent can express the uncertainty in her knowledge about a piece of information in the form of a _ confidence level _ , consisting of a pair of intervals of probability , one for each of her belief and doubt . the space of confidence levels naturally leads to the notion of a _ trilattice _ , similar in spirit to fitting s bilattices . intuitively , the points in such a trilattice can be ordered according to truth , information , or precision . we develop a framework for _ probabilistic deductive databases _ by associating confidence levels with the facts and rules of a classical deductive database . while the trilattice structure offers a variety of choices for defining the semantics of probabilistic deductive databases , our choice of semantics is based on the truth - ordering , which we find to be closest to the classical framework for deductive databases . in addition to proposing a declarative semantics based on valuations and an equivalent semantics based on fixpoint theory , we also propose a proof procedure and prove it sound and complete . we show that while classical datalog query programs have a polynomial time data complexity , certain query programs in the probabilistic deductive database framework do not even terminate on some input databases . we identify a large natural class of query programs of practical interest in our framework , and show that programs in this class possess polynomial time data complexity , not only do they terminate on every input database , they are guaranteed to do so in a number of steps polynomial in the input database size . [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ]	34,519	444
strong interactions are described by su(3 ) yang - mills field theory , and the fundamental degrees of freedom of quantum chromodynamics ( qcd ) are gluons and quarks . at high temperature this theory is weakly coupled , allowing for the use of perturbative methods . the leading order contributions to the characteristic collective excitations and the equation of state of a quark - gluon plasma were already determined many years ago @xcite . a gauge invariant approach to non - leading corrections was derived more recently in form of the hard thermal loop ( htl ) approximation @xcite and its improved versions @xcite . the theory of qcd predicts the appearance of a phase transition between the quark - gluon dominated high energy region and the hadronic state in the low energy region . the two states are characterized by a dramatic difference in the number of degrees of freedom . perturbative qcd can describe successfully only the asymptotic state at very high momenta , but it fails close to this phase transition where non - perturbative effects become dominant . where the perturbative region begins is still a matter of debate @xcite . the numerical method of lattice qcd can describe both sides of this phase transition . recent developments of this field have yielded dramatic improvements both in the extrapolation to the continuum limit and for the inclusion of dynamical fermions . the existence of this phase transition generated a large experimental effort to create it in the laboratory and investigate it in detail through heavy ion collisions . however , to verify the appearance of a deconfined state in such experiments we need accurate knowledge about its subsequent hadronization . there is a general interest to create phenomenological models for this phase transition which agree with the perturbative results at high energy and with the lattice qcd data at low energies . such models must specify the basic degrees of freedom in the plasma state which will participate in the formation of hadrons during the confinement phase transition . the experience from phase transitions in solid state physics and other fields suggests the introduction of quasi - particles with effective masses generated through the interactions among the basic constituents . if a large part of the interaction can be included into the effective masses , then a quasi - particle description of the interacting matter can be generated in which the quasi - particles move freely or interact only weakly @xcite . such a model would be a preferable starting point for phenomenological hadronization models and for a description of the strongly interacting matter near the phase transition . our goal is to identify the appropriate degrees of freedom through an analysis of lattice qcd data . similar attempts started as soon as good lattice qcd results on the equation of state of strongly interacting matter became available , and they developed in parallel with the improvements of the lattice data . it was observed already quite some time ago that the formula for the free energy density of a massive ideal gas gives a quite satisfactory description of the numerical lattice qcd simulations @xcite . in ref . @xcite temperature dependent screening masses and coupling constants were extracted from su(2 ) lattice data @xcite and compared with perturbative results . in ref . @xcite it was shown that the obtained thermal mass can be indeed used as an effective mass in the equation of state , comparing favorably with su(2 ) lattice qcd data . early ( very poor ) lattice data on the pressure and energy density of pure su(3 ) gauge theory in the temperature region @xmath0 @xcite were interpreted with a constant gluon mass , @xmath1 mev , and a constant bag constant @xcite . newer su(3 ) lattice data of better quality @xcite made a reanalysis of the gluonic equation of state possible which yielded a new expression for the temperature dependent coupling constant in the thermal gluon mass @xcite . a quasi - particle based , thermodynamically consistent analysis of the su(2 ) @xcite and su(3 ) @xcite lattice data was performed in ref . @xcite whose authors investigated in detail the temperature dependence of the thermal gluon mass and of the bag constant . new su(3 ) lattice data with a complete continuum extrapolation appeared in refs . @xcite , and again a phenomenological analysis proved the applicability of an equation of state with massive gluons @xcite . furthermore it was shown @xcite that at @xmath2 the debye screening mass extracted from lattice qcd correlation functions and the thermal gluon mass fitted to the lattice qcd equation of state are consistent with each other , such that at high temperature the perturbative qcd with effective massive degrees of freedom provides a good description of the lattice qcd results @xcite . in ref . @xcite the lattice data fit yielded @xmath3 gluon degrees of freedom , supporting the presence of massive _ transverse _ modes only . a direct investigation of the screening mass in su(3 ) lattice calculations was performed in ref . @xcite on both sides of the phase transition . a somewhat different analysis of pure su(3 ) lattice data was presented in ref . it assumes the existence of massless gluons above a certain minimal momentum ( `` cut - off model '' ) together with glueball - like non - perturbative massive excitations . this model could reproduce the su(3 ) lattice data quite well @xcite . all of the above analyses were performed for the case of pure su(2 ) and su(3 ) gauge theories , because sufficiently high - quality lattice qcd data existed only in these cases ( for su(3 ) the latest results can be found in refs . @xcite ) . however , new lattice data including dynamical fermions appeared recently , with @xmath4 @xcite and @xmath5 @xcite quark flavors . extended investigations are under evaluation for the cases @xmath6 and @xmath7 ( see ref . @xcite ) , but in their present state they can not yet be used to extract temperature dependent thermal masses . these new lattice results can now be used for the extraction of a phenomenological equation of state for the quark - gluon plasma ( qgp ) containing massive gluons and quarks , which to our knowledge has not been published elsewhere . similar work is under way in the group of a. peshier and b. kmpfer @xcite . the understanding of the dynamical generation of effective quark and gluon masses could be very important in many research fields . dynamical mass generation is an essential ingredient for the solution of the infrared catastrophe in hot gauge theories @xcite and the formulation of effective field theoretical approaches to thermal qcd @xcite . it is at the heart of recent new approximation schemes such as the `` screened perturbation theory '' @xcite . screening masses have also successfully been used to regulate the low-@xmath8 behavior in the parton cascade approach to ultrarelativistic heavy ion collisions @xcite . massive quarks are the basic degrees of freedom in hadronization models : the phenomenological model alcor is based on the coalescence of massive quarks and antiquarks into hadrons @xcite ; transport descriptions of the hadronization based on the nambu - jona - lasinio model contain massive quarks as well @xcite . massive quarks and gluons obtained from htl approximation @xcite were already applied to estimate charm production at rhic and lhc energies @xcite . in sec . [ sec2 ] we summarize the present knowledge about the thermal and screening masses in perturbative thermal qcd . in sec . [ sec3 ] we overview the equation of state of quasi - particles , its most important thermodynamical properties and discuss the criterium of thermodynamical consistency . in sec . [ sec4 ] we analyze pure su(3 ) lattice qcd results ; we determine the relevant number of gluonic degrees of freedom , the temperature dependent gluon mass and coupling constant , the temperature dependent bag constant and investigate the screening mass . in sec . [ sec5 ] we repeat this analysis on lattice data with @xmath4 and @xmath5 dynamical fermion species . in sec . [ sec6 ] we generate the equation of state of the qgp in general and investigate the speed of sound in this system . in sec . [ sec7 ] we include hadronic matter into our investigation and describe phenomenologically the phase transition between qgp and hadronic matter on the basis of our new equation of state obtained from lattice qcd data . in sec . [ sec8 ] we discuss our results . there are essentially two ways to define an effective dynamical gluon mass : via the pole of the effective gluon propagator , or via the long - range behavior of the potential between two heavy color sources . in both cases one has to address the question of gauge invariance of the result for the mass , since the defining objects are not themselves gauge invariant . on a perturbative level , it was noticed by klimov @xcite and weldon @xcite that the leading term in a high - temperature expansion of the 1-loop gluon polarization tensor was gauge invariant . it was later shown by heinz @xcite that the same result for the gluon polarization tensor could be obtained from classical color kinetic theory in the linear response approximation . from this expression for the polarization tensor weldon @xcite derived the following dispersion relation for transverse gluons with momenta @xmath9 : @xmath10 one might object that the high temperature limit in which the polarization tensor was obtained is inconsistent with the limit @xmath9 under which eq . ( [ weld1 ] ) was derived . however , rebhan et al . showed recently @xcite that the validity of eq . ( [ weld1 ] ) does not depend on the high temperature limit taken in ref . @xcite ; indeed , near the light cone the transverse 1-loop polarization function is given by the gauge invariant result @xmath11 independent of the magnitude of @xmath12 relative to @xmath13 . since for large momenta @xmath14 the gluon dispersion relation moves arbitrarily close to the light cone , @xmath15 can be interpreted as the thermal gluon mass in the high momentum limit . its inclusion as a gluon mass term in higher orders of perturbation theory removes certain classes of collinear singularities near the light cone , and within the context of hard thermal loop resummation @xcite it leads to the so - called `` improved htl resummation scheme '' @xcite . the `` debye screening mass '' @xmath16 , on the other hand , is related to the behavior of gluonic excitations at small momenta . it can be defined either through the static limit of the gluon polarization tensor @xmath17 for @xmath18 or via the behavior of the potential between two heavy color charges at large distances , @xmath19 @xcite . in the first case a gauge invariant result can be obtained from the leading term of the high temperature expansion for the 1-loop gluon polarization operator @xcite or , equivalently , from the 1-loop result within the htl resummation scheme @xcite for the gluon polarization operator at momenta @xmath20 . in the second case , care must be taken to make the external sources gauge invariant @xcite . in the htl approximation one obtains from the behavior of + the debye mass @xcite @xmath21 this result is related to the 1-loop plasmon frequency , @xmath22 , obtained from the gluon dispersion relation ( pole of the propagator ) for @xmath23 , by @xmath24 . similarly to the gluon propagator , one can study the quark propagator in the 1-loop approximation . in the high temperature or low - momentum limit @xmath25 @xcite or , equivalently , in the htl approximation @xcite one obtains @xmath26 this can be interpreted as the effective quark mass for soft quarks with momenta @xmath27 . for high momenta @xmath28 the fermion dispersion relation again approaches the light cone , and one can make use of the gauge invariant light cone limit of the 1-loop fermion self energy @xcite : @xmath29 again this is an exact 1-loop result , independent of the value of @xmath12 , and @xmath30 can be interpreted as the thermal quark mass for high momentum quarks , @xmath14 . the thermal gluon and quark masses obtained from perturbative qcd are displayed in eqs . ( [ infmg ] ) and ( [ infmq ] ) . all masses depend on the temperature @xmath13 and the strong coupling constant @xmath31 . at higher orders of the loop expansion , the coupling @xmath31 begins to run as a function of @xmath13 , giving rise to a temperature dependent effective coupling constant @xmath32 . in @xmath33 gauge theory at @xmath34 , in the presence of @xmath35 quark flavors the 1-loop expression for the running coupling constant as a function of the momentum transfer @xmath36 is @xmath37 here @xmath38 is the cut - off parameter . it was shown in ref . @xcite that in a thermal system it makes sense to introduce a temperature dependent function @xmath39 by the following parametrization : @xmath40 we expect @xmath41 to be linear in the region where perturbation theory is valid @xcite . in other words , writing @xmath42 we expect @xmath43 to be a constant in the perturbative region , i.e. at high @xmath13 . at low @xmath13 a possible @xmath13-dependence of @xmath44 reflects non - perturbative corrections . in su(3 ) at different @xmath35 we have different critical temperatures @xmath45 and different normalization parameters @xmath46 @xcite : @xmath47 for @xmath4 the critical temperature in su(3 ) was determined as @xmath48 mev @xcite , but the normalization factor @xmath46 is not known by the authors . from eqs . ( [ tla1 ] ) and ( [ tla2 ] ) we will estimate this value by interpolation and we will use @xmath49 . we will try to fit the lattice data with a non - interacting gas of massive quarks and gluons , with masses given by the perturbative expressions ( [ infmg ] ) and ( [ infmq ] ) , but with @xmath50 replaced by a phenomenological running coupling @xmath39 . using eqs . ( [ gfun1 ] ) and ( [ gfun2 ] ) we can then extract a function @xmath43 . in a very early work @xcite in pure su(3 ) gauge theory this function @xmath43 was determined from a similar fit to the heavy quark - antiquark potential ; the fit result was a constant , @xmath51 . however , the authors of ref . @xcite used @xmath52 which differs strongly from the now accepted value . with our normalization ( [ tla1 ] ) , @xmath51 would correspond to @xmath53 . another result was obtained recently from a numerical fit of the equation of state in pure su(3 ) gauge theory @xcite : @xmath54 obviously , this parametrization differs from eq . ( [ gfun2 ] ) . our expectation is to obtain a constant @xmath43 function in that temperature region where perturbative qcd and the quasi - particle picture of the lattice qcd results overlap . a strong temperature dependence of @xmath43 implies non - perturbative effects . we summarize our results on the function @xmath43 for different values of @xmath35 in sects . [ sec4 ] and [ sec5 ] . in this section we introduce the equation of state of a pure gluon plasma and a quark - gluon plasma consisting of massive quasi - particles . this equation of state ( resp . its parameters ) will be determined from the su(3 ) lattice data . the key point is to consider temperature dependent effective masses , @xmath55 , which are dynamically generated . we can introduce the dispersion relation for quasi - particles : @xmath56 here @xmath57 is the momentum of the quasi - particle and @xmath58 is its energy . a pure gluon plasma contains only gluons ( @xmath59 , @xmath60 ) , a qgp consists of quarks and antiquarks also ( @xmath61 , @xmath62 ) . we can define the bose or fermi distribution functions @xmath63 for the quasi - particles as @xmath64^{-1 } = [ \exp ( \omega_i /t ) \pm 1 ] ^{-1 } \ . \label{dist}\ ] ] in addition to the effective masses @xmath55 we also need the effective number of degrees of freedom of the quasi - particles to determine pressure and energy density . the correct value for the number of effective gluonic degrees of freedom is not obvious , since a massless vector boson has two helicity states , but massive vector bosons have three spin states . one possibility is to introduce a temperature dependent effective degeneracy factor @xmath65 for the gluons . we will determine the value of @xmath65 in a pure gluon plasma ( @xmath60 ) , where quark degrees of freedom do not interfere . later we can assume the same value for @xmath65 in a qgp . another possibility is to fix the number of gluonic degrees of freedom ( e.g. at its perturbatively expected value @xmath66 ) as well as the number of quark and antiquark degrees of freedom . in this case we have the freedom to introduce an effective interaction term for those contributions of the strong interaction which can not be absorbed by the presence of effective masses . we parametrize this term by a temperature dependent `` bag constant '' , @xmath67 . this interaction term @xmath67 can be determined in a pure gluon plasma as well as in a quark - gluon plasma . as we will see in the following section , in the qgp phase quarks and anti - quarks will also contribute to @xmath67 , so its physical meaning differs from the bag constant parameter of the mit bag model ; nevertheless we will use the same notation for convenience . let us first consider pure su(3 ) gauge theory where we wish to find out the number of contributing gluonic quasi - particle degrees of freedom , @xmath65 . a strongly interacting gluon plasma contains not only the transverse gluon modes , but at low momenta @xmath68 there exist also longitudinal plasma excitations . it was shown in ref . @xcite that at high momenta @xmath14 the longitudinal modes disappear , in the sense that the residue of the corresponding pole in the propagator becomes exponentially small . since the equation of state is dominated by particles with momenta @xmath69 , we expect at high temperatures @xmath70 ( where @xmath31 becomes small ) the contribution of the longitudinal modes to be negligible . on the other hand , at low temperatures ( where @xmath31 can become of order 1 or larger ) their contribution may be relatively large . therefore we can not a priori neglect them from the analysis of lattice qcd data . we will introduce a temperature dependent effective number of gluon degrees of freedom @xmath65 , generalizing the approach of ref . @xcite where @xmath71 was taken as a constant ( temperature independent ) fit parameter . we expect that in the high temperature limit there are only transverse gluonic quasi - particle modes , yielding @xmath72 . at low temperatures the longitudinal gluons are expected to also contribute , giving @xmath73 . with these assumptions the thermodynamical pressure @xmath74 of pure gluon matter ( which is nothing but the grand canonical thermodynamical potential ) can be written as @xmath75 from the thermodynamic identity @xmath76 ( we work at zero net baryon density , @xmath77 ) one obtains the following relation for the energy density : @xmath78 here @xmath79 summarizes the extra terms stemming from the temperature derivative of the effective mass and effective number of degrees of freedom . on the other hand , a consistent quasi - particle picture demands @xmath80 @xcite . requiring this identity yields a differential equation for @xmath65 : @xmath81 here @xmath82 is an integration constant . if the functions @xmath15 and @xmath65 satisfy this selfconsistency condition , then we have a thermodynamically consistent quasi - particle description . lattice calculations yield numerical values for the functions @xmath74 and @xmath83 . these two sets of data are sufficient to determine for each value of @xmath13 the quantities @xmath15 and @xmath65 such that they satisfy eqs . ( [ dpt]-[det ] ) with @xmath84 . since the lattice results were created in a thermodynamically consistent way we expect that the resulting functions @xmath15 and @xmath65 will satisfy eq . ( [ dt ] ) . we will check this as a test for the consistency of our extraction procedure for @xmath15 and @xmath65 . our numerical results on @xmath15 and @xmath65 will be summarized in sec . [ sec4 ] . we can also investigate the high temperature limit from another point of view @xcite . let us fix the value of the gluonic degrees of freedom at every temperature to @xmath85 , assuming that only transverse gluons contribute to the thermodynamic quantities , and calculate directly any extra contribution to the pressure and the energy density . in this way both pure gluon matter and a quark - gluon plasma can be investigated . we will here consider a qgp with @xmath35 quark flavors and the usual number of quark degrees of freedom : @xmath86 . the effective quark mass @xmath87 will be related to the effective gluon mass @xmath15 through the value of the temperature dependent coupling constant @xmath32 via the perturbative expressions in eqs . ( [ infmg ] ) and ( [ infmq ] ) for the effective masses . this means that we absorb non - perturbative features into a common non - perturbative fit function @xmath32 , without touching the perturbative form of the eqs . ( [ infmg ] ) and ( [ infmq ] ) itself . the deviation of @xmath88 and @xmath89 from the ideal gas values corresponding to the effective masses @xmath55 and the fixed degeneracy factors @xmath90 , @xmath91 and @xmath92 will be parametrized by a temperature dependent function @xmath67 . at high temperature , where we expect a perturbative picture based on free quarks and transverse gluons to be valid , @xmath67 should vanish . at lower temperature , @xmath67 provides another measure for non - perturbative physics which can not be absorbed into effective quark and gluon masses . both quarks and gluons contribute to @xmath67 . we introduce @xmath67 by writing the pressure in the form @xmath93 motivated by the mit bag model . thermodynamic identities yield the following relation for the energy density : @xmath94 here @xmath95 summarizes the extra terms obtained from the temperature derivative of @xmath67 and of the effective mass @xmath55 . similarly as above thermodynamic consistency of our quasi - particle model requires @xmath96 . this yields to the following integral representation for @xmath67 @xcite : @xmath97 again @xmath82 is an integration constant . if this expression is satisfied by the functions @xmath55 and @xmath67 , we have a thermodynamically correct quasi - particle description containing effective gluons and quarks with effective thermal masses . similarly as above , from the numerical values for the functions @xmath98 and @xmath99 obtained from lattice qcd and assuming @xmath100 , we can determine the appropriate values for @xmath55 ( resp . @xmath32 ) and @xmath67 through eqs . ( [ pt ] ) and ( [ et ] ) . as before we expect that the obtained functions @xmath55 and @xmath67 will satisfy eq . ( [ bt ] ) . once again this is a good numerical consistency check for our extraction of the functions @xmath55 , @xmath67 . we analyze the latest lattice data in the continuum limit for pure su(3 ) gauge theory @xcite . the calculations of ref . @xcite show a @xmath101 error on the data for @xmath98 , @xmath99 and @xmath102 . we include this error into our analysis ; it will be indicated in figs . 1 , 2 and 3 by dotted lines above and below the solid lines corresponding to the results extracted from the mean values . we begin by determining the effective number of gluonic quasi - particle degrees of freedom in a gluon plasma , extracting the functions @xmath15 and @xmath65 as explained in sec . [ sec3.1 ] . 1a shows the temperature dependent effective gluon mass , @xmath15 , while in fig . 1b we display the effective number of gluonic degrees of freedom , @xmath65 . at high temperature , @xmath103 , the number of degrees of freedom is rather constant , @xmath104 . please note that the @xmath105 error of the lattice data results in an uncertainty of about 1 gluon degree of freedom . within this uncertainty the result is surprisingly close to the naive expectation that at high temperature only the transverse gluonic modes are present . in the low temperature region , @xmath106 , the function @xmath65 increases , which one may wish to associate with the increasing contribution of the longitudinal modes . however , for @xmath107 , @xmath65 rises very strongly and exceeds even the value of @xmath108 expected for massive gluons with @xmath109 color and @xmath110 helicity states . this indicates the presence of strong non - perturbative effects at low temperature not all of which can be absorbed by the effective gluon mass . in fact , part of the strong rise in @xmath65 is driven by the strong increase of the effective gluon mass which leads to exponential suppression of the thermodynamical contributions to @xmath98 and @xmath99 . we consider this behavior of @xmath65 as physically unreasonable and believe that a parametrization of these non - perturbative effects via an extra interaction term makes more sense . to this end we fix the effective number of gluonic modes to @xmath66 as suggested by the behavior of @xmath65 in fig . 1b and our perturbative prejudice at large @xmath13 . this means that we consider only transverse effective gluons as quasi - particle states and subsume all further interaction effects into a non - perturbative `` bag constant '' @xmath67 . the procedure is described in sec . [ sec3.2 ] . 2a shows the temperature dependent transverse gluon mass @xmath15 resulting from this approach . comparing with fig . 1a one sees that at large values of @xmath13 the effective mass is now somewhat larger , compensating for the choice of a fixed @xmath66 instead of the best fit value @xmath111 in fig . 1b . for smaller temperatures , however , the rise of @xmath15 is much weaker than in fig . 1a , and @xmath15 also shows a much weaker sensitivity to the statistical error of the lattice data on @xmath98 and @xmath99 . this indicates that this type of parametrization provides a more reasonable fit to the lattice data than the one above in terms of @xmath15 and @xmath65 . please note that the effective gluon masses shown in fig . 2a correspond in the region @xmath112 to a coupling constant @xmath113 , in agreement with other extraction procedures : for example , the authors of ref . @xcite obtained from the interquark potential at small distances the value @xmath114 which translates into @xmath115 . 2b shows the function @xmath116 , the interaction term relative to the total energy density . at high temperatures , @xmath117 , the bag constant is very small , only about @xmath118 of the total energy density , although slightly negative . we take the smallness of @xmath67 as confirmation for the validity of a quasi - particle picture at large temperature . near @xmath119 @xmath67 changes sign , increasing for smaller values of @xmath13 , but never exceeding a value of @xmath120 of the total energy density even at @xmath45 . note that this relatively small deviation from an ideal quasi - particle picture gave rise to the dramatic and unphysical behavior of @xmath65 ( and @xmath15 ) in the alternative approach above . clearly , the parametrization via @xmath67 is more economical and leads to a more selfconsistent picture . the dashed line in fig . 2b shows the result of an integration of eq . ( [ bt ] ) , using the extracted @xmath15 from fig . 2a and choosing boundary conditions @xmath121 at @xmath122 . the agreement with the numerically extracted values for @xmath67 is nearly perfect indicating the thermodynamical consistency of the lattice data and our extraction procedure . from the temperature dependent gluon mass , @xmath15 , one can determine the temperature dependent coupling constant , @xmath32 , and extract the function @xmath43 given in eq . ( [ gfun2 ] ) . the result for pure su(3 ) gauge theory is shown in fig . the dotted lines again indicate the @xmath101 systematic error of the lattice data . one can see that at high temperature we obtain a constant value , @xmath123 . at low temperature @xmath43 decreases indicating a larger coupling constant and a larger thermal gluon mass . close to the phase transition the function approaches the value @xmath124 ; the precise value depends , however , on the latent heat at the phase transition point which is still under intensive investigation in lattice qcd calculations . the obtained curve can be fitted rather well with the following function ( dashed line in fig . 3 ) : @xmath125 if we start from this approximate function of @xmath43 for the coupling constant @xmath32 , we obtain an analytically parametrized , approximate thermal gluon mass @xmath15 ( see eqs . ( [ infmg],[gfun1],[gfun2 ] ) ) . once we fix the boundary condition @xmath121 at @xmath126 as determined from the numerical analysis of the lattice data ( see fig . 2b ) , the bag constant @xmath67 can be uniquely reconstructed by integrating eq . ( [ bt ] ) . with @xmath15 and @xmath67 known , @xmath98 and @xmath99 are easily evaluated using eqs . ( [ pt],[et ] ) ( with @xmath127 ) . this means that eq . ( [ kfun ] ) together with the value for @xmath82 provide a complete analytical parametrization of the equation of state . in fig . 4 we compare the original lattice qcd data ( diamonds ) with the thus reconstructed values ( dashed and dash - dotted lines ) . the dash - dotted lines neglect the interaction term @xmath67 and thus represent only the quasi - particle contribution to @xmath128 and @xmath129 . clearly the agreement between the model and the lattice data is at most qualitative in this case . including the bag term , however , ( dashed lines ) we obtain nearly perfect reproduction of the lattice data . fig . 4 demonstrates the usefulness of a thermodynamically consistent quasi - particle picture . the deviation from the stefan - boltzmann values indicates that the massless degrees of freedom can not provide a satisfactory description , while the quasi - particle model plus bag term leads to qualitative improvements . newest lattice calculations including dynamical fermions can be used to determine the equation of state for a realistic quark - gluon plasma including light quarks . the numerical lattice qcd results can be found in refs . @xcite for @xmath4 and in ref . @xcite for @xmath5 . following these articles we interpolated the expected continuum result in both cases . from ref . @xcite for @xmath4 we used the data for a current quark mass @xmath131 ( octagons in figs . 9 and 10 in ref . @xcite and circles in fig . 7 here ) . the statistical error of the lattice data is small , @xmath132 for the pressure and @xmath133 for the energy density . from ref . @xcite for @xmath5 we used the data set for a current quark mass @xmath134 and we considered their extrapolation of the energy density to the chiral limit ( see @xmath135 in fig . 4 of ref . @xcite and in fig . 7 here ) . in this case the statistical errors are even smaller , but the systematic errors resulting from the extrapolations to the chiral and continuum limits are hard to judge . to estimate the possible influence of errors on the lattice data for our extraction procedure we will consider a universal @xmath136 error in both cases ; the corresponding uncertainty will be indicated by dotted lines in figs . 5 and 6 . in the framework of perturbation theory the effects of dynamical quarks on the gluon dynamics are small and can be simply included into the thermal gluon mass and debye screening length . no additional gluonic collective modes arise . however , at small momenta there are additional fermionic collective modes , the `` plasminos '' @xcite . based on our experience in pure gluodynamics they are not expected to contribute to the equation of state at high temperature ; again the residues of their poles in the quark propagator vanish exponentially for moments @xmath14 @xcite . we will therefore assume a fixed number of degrees of freedom , given by the perturbative values at high @xmath13 , @xmath66 , @xmath137 , and include all collective interaction effects into an interaction term @xmath67 as for the purely gluonic system above . the obtained results are qualitatively very similar to the previous ones for the purely gluonic case . 5a shows the obtained results for the effective gluon mass , @xmath138 . since @xmath139 , fig . 5a can be viewed as giving the temperature dependence of the effective coupling constant @xmath32 which according to eq . ( [ infmq ] ) also determine the effective quark mass . from fig . 5 and eq . ( [ infmg ] ) we extract @xmath140 in the region @xmath141 for @xmath5 , somewhat larger than for the purely gluonic case . 5b shows the interaction term relative to the total energy density @xmath116 both for @xmath4 and @xmath5 . similarly to the result in pure su(3 ) gauge theory , the interaction term @xmath67 remains small at @xmath142 , and does not exceed @xmath120 of the energy density in the low temperature region . in the case @xmath4 the lattice data allow us to extract @xmath15 and @xmath67 only in the temperature region @xmath143 , however the characteristics of the obtained curves are close to the results obtained for @xmath5 . 6 displays the obtained functions @xmath43 ( solid lines ) together with the expected errors ( dotted lines ) stemming from an assumed typical @xmath101 uncertainty in the lattice data . for all three cases , @xmath144 0 , 2 and 4 , the functions @xmath43 are very similar ( within the admittedly large error bars ) . within the existing uncertainties we can thus define a universal function @xmath43 ; differences due to variations of @xmath145 and @xmath35 can be largely absorbed into @xmath45 and @xmath146 in eq . ( [ gfun2 ] ) . it would be very interesting to see whether the existence of such a universal function @xmath43 is confirmed by future higher quality lattice data . since the data for purely gluonic su(3 ) span a much larger range than those including dynamical fermions , we will use the analytical form ( [ kfun ] ) extracted from the gluonic case also in our quasi - particle picture for the full qgp . for the zero point of @xmath67 we will again choose the value @xmath147 determined in the purely gluonic su(3 ) case , although ( modulo large numerical uncertainties ) the @xmath148 data seem to favor a somewhat smaller value . for a qgp with @xmath4 quark flavours fig . 7a shows the lattice data ( circles for pressure and energy density ) and the reconstructed values for @xmath99 and @xmath98 , both including the bag constant ( dashed lines ) and neglecting the interaction term ( dash - dotted lines ) . for @xmath5 quark flavours the analogous quantities are displayed in fig . the squares give directly the lattice data from ref . @xcite for the energy density , while the diamonds represent the continuous line for the pressure given in fig . 2 of ref . @xcite . in each case inclusion of the interaction term @xmath67 improves the picture . the agreement is weaker than in the pure su(3 ) case ( see fig . 4 ) , but the pressure is reproduced rather nicely as well as the energy density at high temperature . note that here we used the function @xmath43 from eq . ( [ kfun ] ) . the discrepancy in the region @xmath149 may be connected to the use of massive current quarks in the actual lattice simulations and associated uncertainties in the extrapolation to the chiral limit ( for a discussion see ref . @xcite ) . in view of those the @xmath150 discrepancy between the reconstructed curves and data is surprisingly small , especially if we remember that we used here the parametrization which was obtained from the pure su(3 ) case . in hydrodynamical models of strongly interacting matter the dynamical behavior is mostly determined by the value of the speed of sound : @xmath151 since the energy density @xmath99 was obtained from eq . ( [ entr ] ) we can rewrite this as @xmath152 which is more directly connected to the lattice data . for a noninteracting ( @xmath153 ) gas of massless particles the speed of sound is @xmath154 . in ref . @xcite , using eq . ( [ spe1 ] ) , it was found that even for a non - interacting ( @xmath153 ) massive quark - gluon gas the speed of sound remains very close to @xmath154 . ( in that work thermal masses from earlier htl calculations @xcite and a running coupling constant according to eqs . ( [ gfun1],[gfun2 ] ) with a constant @xmath43 were used . ) if there are , however , strong remaining interactions which can not be absorbed in the masses but require a non - vanishing and possibly strongly @xmath13-dependent @xmath67 , this will generate deviations of @xmath155 from the ideal gas value , especially near @xmath45 @xcite . a strong drop of @xmath155 near @xmath45 will give rise to a `` soft point '' in the equation of state at which the ability of the matter to generate expansion flow is minimal . since in our analysis the interaction term @xmath67 remains small and changes smoothly with @xmath13 , we expect only weak modifications of the speed of sound around @xmath45 . 8 displays the speed of sound @xmath155 for @xmath156 in our quasi - particle picture and confirms this expectation . including the bag term @xmath67 ( dashed lines ) , @xmath157 in the high temperature limit , and @xmath155 decreases to a minimum value @xmath158 around @xmath45 . if we neglect the interaction term @xmath67 and calculate the speed of sound @xmath159 from the expression @xmath160 where @xmath161 and @xmath162 , then ( see dash - dotted line in fig . 8) the speed of sound shows only a very weak temperature dependence , decreasing from @xmath163 at @xmath164 to @xmath165 at @xmath45 . if we instead use eq . ( [ spe1 ] ) also in this case ( as in ref . @xcite ) , where @xmath67 cancels in the numerator , then the resulting speed of sound is very close to the full result ( dashed line ) , because of the small influence of @xmath67 on the energy density in the denominator of eq . ( [ spe1 ] ) . our result agrees with the calculations in ref . @xcite in the high temperature region and differs around @xmath45 because of the non - linear behavior of @xmath43 indicating non - perturbative effects . these results on the speed of sound show that a massive quark - gluon plasma will expand rapidly and that therefore the phase transition should be a fast process without detectable duration effects . even where the pressure already shows large ( @xmath166 ) deviations from the massless ideal gas law , the speed of sound still deviates by less than 10% from the value @xmath167 . in this way our analysis of lattice qcd results favors fast hadronization models @xcite of the deconfined phase . in this section we investigate some phenomenological consequences of the large effective quark and gluon masses . 9 displays the effective masses in gev for @xmath156 . all masses behave similarly , since they are connected to each other by the function @xmath43 in the effective coupling constant @xmath32 . just above the critical temperature the effective masses drop , but then they remain rather constant in the temperature region @xmath168 . the mass values for the case @xmath4 are smaller than for @xmath60 due to the smaller critical temperature @xmath169 mev instead of 260 mev . for @xmath5 they increase again due to the larger contribution from quark loops and the somewhat larger @xmath170 mev . the gluon masses are large in the cases @xmath171 , of order @xmath172 mev in the temperature region @xmath173 ( solid lines ) . in the same temperature region the quark masses are of order @xmath174 mev ( dashed lines ) . large effective masses yield smaller number density at the same energy density . thus the density of these massive quasi - particles will be smaller than in the massless case . 10 demonstrates this by displaying the density ratio of the massive quasi - particles and the massless ones , @xmath175 ( solid lines for gluons and dotted lines for quarks ) as a function of temperature . at large temperature the ratio flattens and eventually approaches 1 , but close to the critical temperature the ratio drops very quickly to values much smaller than 1 . this indicates that the massive quark - gluon plasma is much more dilute than a massless qgp , especially near @xmath45 . now we compare the abundances of quarks and gluons relative to each other . ( [ infmg ] ) and ( [ infmq ] ) imply that in our model the gluon effective mass is always larger than the quark effective mass by the ratio @xmath176 which even increases with the number of quark flavor @xmath35 . each gluon degree of freedom will thus be suppressed compared to each quark degree of freedom . after multiplying the corresponding degeneracy factors we can compute the density ratio for gluons @xmath177 which is shown in fig . 11 as a function of @xmath13 . the horizontal lines show the ratio @xmath178 for the massless qgp . the deviation caused by the effective mass is clearly seen , and a considerable suppression of the relative gluon number occurs near the phase transition . near @xmath45 , for @xmath4 we have @xmath179 which means that only 25% of the particles are gluons while the remaining 75% are quarks and antiquarks . for @xmath5 the gluon suppression is even stronger and near @xmath45 we find only 10% gluons and 90% quarks and anti - quarks . these results show that the massive qgp is a dilute system dominated by massive quarks and antiquarks . gluons are much heavier than quarks and their number densities are suppressed . these properties indicate that lattice qcd results support the formation of a quark - antiquark dominated plasma state just above the critical temperature , which contains mostly massive quarks and antiquarks . this system hadronizes quickly . fast hadronization is also favored by the fact that the effective quark masses are already close to the masses in the `` constituent quark '' picture of hadrons . clusterization of these massive quarks leads directly to multiquark states with similar masses as those of the finally observed hadrons , allowing for immediate hadron formation without having to wait for further energy transfer . lattice data in pure su(3 ) gauge theory and new lattice results with @xmath4 and @xmath5 dynamical fermions suggest that beyond the phase transition temperature strong non - perturbative effects dominate the deconfined state . in order to understand this phenomenon and to facilitate the application of lattice results in dynamical descriptions of the evolution of the deconfined phase and its hadronization we analyzed the lattice data in a phenomenological model assuming the appearance of massive quasi - particles , namely massive quarks and massive transverse gluons . we determined a temperature dependent effective coupling constant @xmath32 , which determines the effective masses , and the interaction term @xmath67 , which summarizes contributions not to be included into the effective masses . we found that such a quasi - particle picture works very well . the interaction term @xmath67 remains small and the extracted dynamical mass is consistent with perturbative qcd results at high temperature while it includes non - perturbative contributions at low temperature . in all cases , @xmath156 , the obtained characteristics are very similar . if we determine @xmath32 in the purely gluonic case and extend it in an appropriate way for @xmath180 , we can reproduce approximately the @xmath130 lattice data . this fact suggest the existence of a universal description of non - perturbative effects in the fermionless and fermionic cases . our analysis leads to a thermodynamically consistent quasi - particle model for the qgp equation of state which is parametrized via an effective coupling constant @xmath32 through eqs . ( [ gfun1],[gfun2],[kfun ] ) and the zero point @xmath82 of the interaction term @xmath67 . further lattice data with improved quality over a wider temperature region are needed to fix @xmath32 and @xmath82 more accurately . the successful reproduction of lattice data by this quasi - particle model indicates the validity of such an approximation . the consequences of the appearance of these quasi - particles are very interesting : near @xmath45 lattice results indicate the existence of a qcd phase which is dominated by massive quarks and anti - quarks , while the even heavier gluons are suppressed . this quark - antiquark plasma can be characterized by a relatively large speed of sound which indicates fast dynamical evolution and the lack of long time delays during hadronization . massive quarks and anti - quarks can form clusters , 2- and 3-body objects , which can be easily associated with the hadronic mass spectra : massive quarks ( @xmath181 mev ) above the critical temperature turn into similarly massive `` constituent quarks '' inside hadrons below @xmath45 in a smooth but rapid hadronization process . discussions with t.s . bir , e. laermann , b. mller , a. schfer , m. thoma , j. zimnyi , and especially with b. kmpfer and a. peshier , are gratefully acknowledged . p.l . is grateful for the warm hospitality of the institute fr theoretische physik at universitt regensburg . this work was supported in part by daad by the national scientific research fund ( hungary ) otka grant nos . f019689 and t016206 ( p.l . ) , and by dfg , bmbf and gsi ( u.h . ) . 99 j. kapusta , nucl . phys . * b148 * ( 1979 ) 461 . v.v . klimov , zh . * 82 * ( 1982 ) 336 ( sov . jetp * 55 * ( 1982 ) 199 ) . weldon , phys . rev . * d26 * ( 1982 ) 1394 ; * d26 * ( 1982 ) 2789 . pisarski , physica * a158 * ( 1989 ) 146 . j. frenkel and j.c . taylor , nucl . phys . * b334 * ( 1990 ) 199 ; + e. braaten and r.d . pisarski , nucl . phys . * b337 * ( 1990 ) 569 . m. le bellac , _ thermal field theory _ ( cambridge university press , cambridge , england , 1996 ) ; + m. h. thoma , hep - ph/9503400 . r. baier , s. peigne , and d. schiff , z. phys . * c62 * ( 1994 ) 337 ; + f. flechsig , a.k . rebhan , and h. schulz , phys . rev . * d52 * ( 1995 ) 2994 ; + u. krmmer , a.k . rebhan , and h. schulz , ann . phys . * 238 * ( 1995 ) 286 . u. krmmer , m. kreuzer , and a.k . rebhan , ann . phys . * 201 * ( 1990 ) 223 ; + f. flechsig and a.k . rebhan , nucl . phys . * b464 * ( 1996 ) 279 . k. kajantie , m. laine , j. peisa , a. rajantie , k. rummukainen , and m. shaposhnikov , phys . . lett . * 79 * , 3130 ( 1997 ) . gorenstein and s.n . yang , phys . rev . * d52 * ( 1995 ) 5206 . f. karsch , z. phys . * c38 * ( 1988 ) 147 ; + f. karsch , nucl . * b9 * ( 1989 ) 357 . f. karsch , m.t . mehr , and h. satz , z. phys . * c37 * ( 1988 ) 617 . j. engels , j. fingberg , k. redlich , h. satz , and m. weber , z. phys . * c42 * ( 1989 ) 341 . v. goloviznin and h. satz , z. phys . * c57 * ( 1993 ) 671 . a. ukawa , nucl . phys . * a498 * ( 1989 ) 227c . bir , p. lvai , and b. mller , phys . rev . * d42 * ( 1990 ) 3078 . j. engels , j. fingberg , f. karsch , d. miller , and m. weber , phys . * b252 * ( 1990 ) 625 . a. peshier , b. kmpfer , o.p . pavlenko , and g. soff , phys . lett . * b337 * ( 1994 ) 235 . g. boyd , j. engels , f. karsch , e. laermann , c. legeland , m. ltgemeier , and b. petersson , phys . * 75 * ( 1995 ) 4169 ; nucl . * b469 * ( 1996 ) 419 . e. laermann , nucl . phys . * a610 * ( 1996 ) 1c . a. peshier , b. kmpfer , o.p . pavlenko , and g. soff , phys . rev . * d54 * ( 1996 ) 2399 . b. grossman , s. gupta , u.m . heller , and f. karsch , nucl * b417 * ( 1994 ) 289 . rischke , m.i . gorenstein , a. schfer , h. stcker , and w. greiner , phys . lett . * b278 * ( 1992 ) 19 ; + d.h . rischke , j. schaffner , m.i . gorenstein , a. schfer , h. stcker , and w. greiner , z. phys . * c56 * ( 1992 ) 325 . c. bernard , t. blum , c. detar , s. gottlieb , k. rummukainen , u.m . heller , j.e . hetrick , d. toussaint , l. krkkinen , r.l . sugar , and m. wingate , phys . rev . * d55 * ( 1997 ) 6861 . s. gottlieb , u.m . heller , a.d . kennedy , s. kim , j.b . kogut , c. liu , r.l . renken , d.k . sinclair , r.l . sugar , d. toussaint , and k.c . wang , phys . rev . * d55 * ( 1997 ) 6852 . j. engels , r. joswig , f. karsch , e. laermann , m. ltgemeier , and b. petersson , phys . * b396 * ( 1997 ) 210 . lattice95 conf . 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( [ kfun ] ) is denoted by the dashed line , and the horizontal line at @xmath184 indicates its asymptotic value . fig . 4 : : : lattice data for pure su(3 ) gauge theory on @xmath185 and @xmath186 ( diamonds ) @xcite , their reconstruction with a free quasi - particle picture with @xmath153 ( dash - dotted lines ) , and including the interaction term @xmath67 ( dashed lines ) . the horizontal line at @xmath187 indicates the stefan - boltzmann value for massless non - interacting particles . fig . 5 : : : the temperature dependent effective gluon mass , @xmath15 , in units of temperature @xmath13 ( a ) and the interaction term @xmath67 normalized by the energy density @xmath99 ( b ) in su(3 ) gauge theory for @xmath35=2 and @xmath35=4 as a function of @xmath182 . : : : the function @xmath43 from the effective coupling constant @xmath32 in su(3 ) gauge theory @xmath156 ( solid lines ) and the influence of @xmath101 systematic error of lattice data on the function @xmath43 . : : : su(3 ) lattice data ( symbols ) and our model reconstruction ( lines ) : @xmath4 @xcite ( a ) and @xmath5 @xcite ( b ) . the reconstructed quantities are denoted by dash - dotted lines in a simple quasi - particle picture and dashed lines if the interaction term @xmath67 is included . horizontal lines indicate the stefan - boltzmann values for massless non - interacting particles , 12.17 and 19.08 , respectively . : : : the speed of sound @xmath155 , calculated for @xmath156 . the dash - dotted lines denote the results @xmath159 from the free quasi - particle model ( @xmath153 ) while the dashed lines with asymptotic value @xmath188 indicate the results including the interaction term @xmath67 . note that the differences between @xmath144 0 , 2 and 4 are hardly visible . fig . 9 : : : the effective masses of quarks ( dashed curves ) and gluons ( solid curves ) in su(3 ) gauge theory with @xmath1440 , 2 and 4 quark flavour . 10 : : : the density ratios between massive ( @xmath189 ) and massless ( @xmath190 ) quarks and gluons in su(3 ) gauge theory for @xmath1440 , 2 and 4 . fig . 11 : : : the relative gluon density @xmath191 for massive , @xmath178 for massless particles in @xmath4 ( solid lines ) and in @xmath5 ( dashed lines ) qgp .	we analyze recent results of su(3 ) lattice qcd calculations with a phenomenological parametrization for the quark - gluon plasma equation of state based on a quasi - particle picture with massive quarks and gluons . at high temperature we obtain a good fit to the lattice data using perturbative thermal quark and gluon masses from an improved htl scheme . at temperatures close to the confinement phase transition the fitted masses increase above the perturbative value , and a non - zero ( but small ) bag constant is required to fit the lattice data . = 7.2pt = 7.2pt 14.5pt	16,542	163
_ * `` ... an infinite series of times , in a dizzily growing , ever spreading network of diverging , converging and parallel times . this web of time the strands of which approach one another , bifurcate , intersect or ignore each other through the centuries embraces every possibility . '' [ borges ] * _ _ `` quid est ergo tempus ? si nemo ex me quaerat , scio ; si quaerenti explicare velim , nescio . '' _ @xcite as augustinus hipponensis present physicists are in the situation where time is an essential physical parameter whose meaning is intuitively clear , but several problems arise when they try to provide a clear definition of time . first of all the definition of time is different in different branches of physics as classical and non - relativistic quantum mechanics ( a fixed background parameter ) , special relativity ( a proper time for each observer , time as fourth coordinate ; set of privileged inertial frames ) or general relativity ( time is a general spacetime coordinate : not at all an absolute time , time non orientability , closed timelike curves)@xcite . more specifically , the problem of time @xcite in physics consists in the fact that a straightforward quantization of the general relativistic evolution equation and constraints generates the wheeler - de witt equation @xmath0 , where @xmath1 is the global hamiltonian of the universe and @xmath2 is its state . obviously , this means that the state of the universe must be static , which clashes with our everyday experience of an evolving world . apart from the wheeler - de witt equation , one can also consider that a time shift of the state of the whole universe must be unobservable from a purely physical consideration : if shifting the state of the whole universe , there is nothing external that can keep track of this shift , so this shift must be unobservable ( it is a trivial application of mach s principle ) . same considerations would apply , of course , also to the spatial degrees of freedom : a global shift in position of the whole state of the universe must similarly be unobservable , so the universe must be in an eigenstate of its momentum operator . in the following we will only consider temporal degrees of freedom . incidentally , this idea of a static universe surprisingly re - proposes , of course rephrased in modern language , ideas stemming from the work of parmenides of elea [ `` the phenomena of movement and change are simply appearances of a static , eternal reality '' ] and then diffused in roman - hellenistic word [ `` tempus item per se non est , sed rebus ab ipsis consequitur sensus , transactum quid sit in aevo , tum quae res instet , quid porro deinde sequatur '' titus lucretius carus , de rerum natura ] . the page and wootters scheme @xcite , which developed some previous ideas @xcite , is based on the fact that there exist states of a system composed by entangled subsystems that are stationary , but one can interpret the component subsystems as evolving . one can then suppose that the global state of the universe as one of this static entangled state , whereas the state of the subsystems ( us , for example ) can evolve . this solves in an extremely elegant way the problem of time . incidentally , page and wootters proposal naturally embodies the philosophy of relationalism @xcite and operationalism , since time is only defined in relation to clocks and to its measurement procedure @xcite . up to now , all these considerations were of theoretical character . here we epitomize an experimental approach @xcite to this problem by providing an emblematic example of page and wootters idea at work , visualizing how time could emerge from a static ( with respect to an abstract external time ) entangled state . even though the total state of a system is static , time is recovered as correlations between a subsystem that acts as a clock and the rest of the system that evolves according to such a clock . we use a system composed of two entangled photons ( a paradigmatic system for several emblematic experiments @xcite ) : the rotation of the polarization of the first acts as a clock for proving an evolution of the polarization of the second . nonetheless , we demonstrate that the joint polarization state of both photons does not evolve . the page - wooters mechanism has been criticised @xcite , and gambini et al . have proposed some new ideas @xcite to overcome these criticisms . out experiment demonstrates also some aspects of these ideas . the experimental setup is schematically depicted in figure 1 . it consists of two blocks , `` preparation '' and `` measurement '' . the preparation block allows producing a family of ququarts biphoton polarization states @xcite of the form : @xmath3 basis ; bs for beam splitter.,scaledwidth=60.0% ] it includes two orthogonally oriented bbo crystals ( @xmath4 ) that , pumped by a @xmath5 mw cw ar laser operating at @xmath6 nm , generate a pair of the basic ququart states via type - i spontaneous parametric down conversion in collinear , frequency degenerate regime around the central wavelength of @xmath7 nm . the basic state amplitudes are controlled with the help of a thompson prism ( v ) , oriented verticaly and the half - wave plate @xmath8 at an angle @xmath9 . two @xmath10 mm quartz plates ( qp ) , that can be rotated along the optical axis , introduce a phase shift @xmath11 between horizontally and vertically polarized type - i entangled @xcite biphotons . non - linear crystals and quartz plates are placed into a temperature - stabilized closed box for achieving stable phase - matching conditions all the time . the preparation block also includes a non - polarizing beam splitter ( bs ) , which is used to split the initial ( collinear ) biphoton field into two spatial modes @xmath12 . for preparing the singlet bell state @xmath13 parameters @xmath14 , were selected and an additional half - wave plate @xmath8 at @xmath15 was introduced in the transmitted arm . the measurement part can be mounted in two different modes `` observer '' and `` super - observer '' . `` observer '' mode ( figure 1 , block a ) serves as proving that one subsystem ( polarization of the upper photon ) evolves with respect to a clock constituted by the other subsystem ( polarization of the lower photon ) . in the `` super - observer '' mode ( ( figure 1 , block b ) ) we show that the global state of the two photons is static with respect to any `` external '' clock . let us analyze the observer mode first . * `` observer '' mode : * each arm of a measurement block contains interference filters ( if ) with central wavelength @xmath7 nm ( fwhm @xmath10 nm ) and polarizing beam splitter ( pbs ) . four avalanche photodiode detectors ( apd ) are placed at the ends and connected to a coincidence count scheme ( cc ) with @xmath16 ns time window . in this mode the polarization of the photon in the lower arm is used as a clock for proving an evolution of the polarization of the photon in the upper arm . in other words , the first polarizing beam splitter pbs acts as a non - demolition measurement that measures ( in the h / v basis ) the polarization of the second photon in the state @xmath17 ( [ eq : singlet ] ) . once it is initialized in the @xmath18 state , the polarization of both photons evolves in the birefringent quarts plates ( a ) at @xmath15 as @xmath19 , where @xmath20 is the materials optical thickness . the clock readout is performed on the lower detectors , which measure whether the photon is @xmath21 or @xmath22 . this clock is extremely primitive , since it has a dial with only two values : either it found with @xmath18 polarization , corresponding to time @xmath23 or with @xmath24 polarization , corresponding to time @xmath25 . in the `` observer '' mode we do not have access to any external clock , the only information we obtain are correlations ( coincidences ) between the detectors . thus , this primitive clock allows extracting the information about the polarization of the upper photon in a defined state , for example @xmath22 , only from registering coincidence counts between the detectors @xmath26 ( that corresponds to the measurement at the moment time @xmath23 ) or 2 - 3 ( that corresponds to time @xmath25 ) . furthermore , one does not have any knowledge about the past time after the clock initialization ( knowledge about this time is equivalent to the presence of an external clock ) , so we have to record all events taking place in selected points of time @xmath23 and @xmath25 . from the experimental point of view it means measurements of coincidences between the detectors 1 - 3 and 2 - 3 for all possible thicknesses of the birefringent plates a @xcite . averaging the number of counts from detectors 1 - 3 and 2 - 3 over the total number of photon pairs from all detectors 1 - 2 - 3 - 4 , gives the probability of finding the upper photon in polarization state @xmath22 @xmath27 and @xmath28 at two points of time . to obtain a more interesting clock , we perform the same conditional probability measurement introducing varying time delays to the clock photon , implemented through quartz plates of variable thickness ( dashed box b in figure [ f : schema ] , block b ) . [ even though he has no access to abstract coordinate time , he can have access to systems that implement known time delays , that he can calibrate separately . ] now , we obtain a sequence of time - dependent values for the conditional probability : @xmath29 and @xmath30 , where @xmath31 is the time delay of the clock photon obtained by inserting the quartz plate b with thickness @xmath32 in the clock photon path . the experimental results are presented in fig . [ f : resultsgp ] , where each colour represents a different delay : the yellow points refer to @xmath33 ; the red points to @xmath34 , etc . [ f : resultsgp ] _ _ our results are in very good agreement with the theory ( dashed line ) @xcite . the reduction in visibility of the sinusoidal time dependence of the probability is caused by the decoherence effect due to the use of a low - resolution clock ( our clock outputs only two possible values ) , a well known effect @xcite . * `` super - observer '' mode : * the former parter of the experiment is addressed to show that one photon `` evolves '' when using the other photon as a clock . now we have to show that the global state of two photons is static with respect to any `` external '' clock . to do so , we need to repeat the whole experiment in `` super - observer '' mode , and show that for arbitrary thicknesses of the birefringent plates which represent an `` external '' time the state is invariant . this setup of the experiment is shown in figure 1(block b ) . here the @xmath35 beam splitter ( bs ) on the output of a balanced interferometer is performing a quantum erasure of the polarization measurement performed by the polarizing beam splitter pbs1 . in the ideal case the beam splitter should coherently transmit a photon that passes in the upper arm and reflects a photon that passes in the lower arm of the interferometer without affecting its polarization . without loss of generality , in our setup we use a conventional beam splitter and post - selection on one of the exits . for temporal stability the interferometer was placed into a closed box and narrow interference filters ( @xmath7 nm , fwhm=@xmath10 nm ) were inserted before the detectors . a set of quartz plates ( a ) with different thicknesses are used to introduce an `` external '' time . for analyzing an output state of transformed ququarts we used the quantum state tomography procedure . necessary projective measurements were realized by polarization filters placed in front of detectors . each filter consists of a sequence of quarter- and half - wave plates and a polarization prism which transmits vertical polarization . two apds linked to a coincidence scheme with 1.5 ns time window were used as single photon detectors . in our experiment we used the protocol of ref . registering the coincidence rate for 16 different projections , that were realized by half and quarter plates and a fixed analyzer , it was possible to reconstruct the arbitrary polarization state of ququarts . the accuracy of quantum tomography is estimated through the fidelity @xmath36 where @xmath37 is the initial ( theoretical ) density matrix and @xmath38 is the reconstructed density matrix after transformation . our results are summarized in table i , demonstrating the stationarity of the global state respect to the evolution . in summary , by running our experiment in two different modes ( `` observer '' and `` super - observer '' mode ) we have experimentally shown how the same energy - entangled hamiltonian eigenstate can be perceived as evolving by the internal observers that test the correlations between a clock subsystem and the rest ( also when considering two - time measurements ) , whereas it is static for the super - observer that tests its global properties . our experiment is a practical implementation of the page and wooters@xcite and gambini et al . @xcite mechanisms but , obviously , it can not discriminate between this and other proposed solutions for the problem of time @xcite , representing only an `` illustration '' of this phenomenon . developments to higher dimensional systems could eventually be envisaged by using atoms @xcite . in closing , we note that the time - dependent graphs of fig . [ f : resultsgp ] have been obtained without any reference to an external time ( or phase ) reference , but only from measurements of correlations between the clock photon and the rest representing an implementation of a ` relational ' measurement of a physical quantity ( time ) relative to an internal quantum reference frame @xcite , i.e. an example of relational metrology . this research was supported by the john templeton foundation ( the opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the john templeton foundation ) , e.v.moreva acknowledges the support from the dynasty foundation and russian foundation for basic research ( project 13 - 02 - 01170-a ) . aurelius augustinus hipponensis quid est ergo tempus ? si nemo ex me quaerat , scio ; si quaerenti explicare velim , nescio confessiones , xi , 14 anderson e 2012 _ ann . phys . _ * 524 * 757 ( _ preprint _ gr - qc/1009.2157v3 sorkin r d 1994 _ int . j. theor . phys . _ * 33 * 523 ; 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on a daily basis , people undergo numerous interactions with objects that barely register on a conscious level . for instance , imagine a person shopping at a grocery store as shown in figure [ fig : main ] . suppose she picks up a can of juice to load it in her shopping cart . the distance of the can is maintained fixed due to the constant length of her arm . when she checks the expiration date on the can , the distance and orientation towards the can is adjusted with respect to her eyes so that she can read the label easily . in the next aisle , she may look at a lcd screen at a certain distance to check the discount list in the store . thus , this example shows that spatial arrangement between objects and humans is subconsciously established in 3d . in other words , even though people do not consciously plan to maintain a particular distance and orientation when interacting with various objects , these interactions usually have some consistent pattern . this suggests the existence of an egocentric object prior in the person s field of view , which implies that a 3d salient object should appear at a predictable location , orientation , depth , size and shape when mapped to an egocentric rgbd image . our main conjecture stems from the recent work on human visual perception @xcite , which shows that _ humans possess a fixed size prior for salient objects_. this finding suggests that a salient object in 3d undergoes a transformation such that people s visual system perceives it with an approximately fixed size . even though , each person s interactions with the objects are biased by a variety of factors such as hand dominance or visual acuity , common trends for interacting with objects certainly exist . in this work , we investigate whether one can discover such consistent patterns by exploiting egocentric object prior from the first - person view in rgbd frames . our problem can be viewed as an inverse object affordance task @xcite . while the goal of a traditional object affordance task is to predict human behavior based on the object locations , we are interested in predicting potential salient object locations based on the human behavior captured by an egocentric rgbd camera . the core challenge here is designing a representation that would encode generic characteristics of visual saliency without explicitly relying on object class templates @xcite or hand skin detection @xcite . specifically , we want to design a representation that captures how a salient object in the 3d world , maps to an egocentric rgbd image . assuming the existence of an egocentric object prior in the first - person view , we hypothesize that a 3d salient object would map to an egocentric rgbd image with a predictable shape , location , size and depth pattern . thus , we propose an egoobject representation that represents each region of interest in an egocentric rgbd video frame by its _ shape _ , _ location _ , _ size _ , and _ depth_. note that using egocentric camera in this context is important because it approximates the person s gaze direction and allows us to see objects from a first - person view , which is an important cue for saliency detection . additionally , depth information is also beneficial because it provides an accurate measure of object s distance to a person . we often interact with objects using our hands ( which have a fixed length ) , which suggests that depth defines an important cue for saliency detection as well . thus assuming the existence of an egocentric object prior , our egoobject representation should allow us to accurately predict pixelwise saliency maps in egocentric rgbd frames . to achieve our goals , we create a new egocentric rgbd saliency dataset . our dataset captures people s interactions with objects during various activities such as shopping , cooking , dining . additionally , due to the use of egocentric - stereo cameras , we can accurately capture depth information of each scene . finally we note that our dataset is annotated for the following three tasks : saliency detection , future saliency prediction , and interaction classification . we show that we can successfully apply our proposed egocentric representation on this dataset and achieve solid results for these three tasks . these results demonstrate that by using our egoobject representation , we can accurately characterize an egocentric object prior in the first - person view rgbd images , which implies that salient objects from the 3d world map to an egocentric rgbd image with predictable characteristics of shape , location , size and depth . we demonstrate that we can learn this egocentric object prior from our dataset and then exploit it for 3d saliency detection in egocentric rgbd images . region proposals . for each of the regions @xmath1 we then generate a feature vector @xmath2 that captures shape , location , size and depth cues and use these features to predict the 3d saliency of region @xmath1 . ] * saliency detection in images . * in the past , there has been much research on the task of saliency detection in 2d images . some of the earlier work employs bottom - up cues , such as color , brightness , and contrast to predict saliency in images @xcite . additionally , several methods demonstrate the importance of shape cues for saliency detection task @xcite . finally , some of the more recent work employ object - proposal methods to aid this task @xcite . unlike the above listed methods that try to predict saliency based on contrast , brightness or color cues , we are more interested in expressing an egocentric object prior based on shape , location , size and depth cues in an egocentric rgbd image . our goal is then to use such prior for 3d saliency detection in the egocentric rgbd images . * egocentric visual data analysis . * in the recent work , several methods employed egocentric ( first - person view ) cameras for the tasks such as video summarization @xcite , video stabilization @xcite , object recognition @xcite , and action and activity recognition @xcite . in comparison to the prior egocentric approaches we propose a novel problem , which can be formulated as an inverse object affordance problem : our goal is to detect 3d saliency in egocentric rgbd images based on human behavior that is captured by egocentric - stereo cameras . additionally , unlike prior approaches , we use * egocentric - stereo * cameras to capture egocentric rgbd data . in the context of saliency detection , the depth information is important because it allows us to accurately capture object s distance to a person . since people often use hands ( which have fixed length ) to interact with objects , depth information defines an important cue for saliency detection in egocentric rgbd environment . unlike other methods , which rely on object detectors @xcite , or hand and skin segmentation @xcite , we propose egoobject representation that is based solely on shape , location , size and depth cues in an egocentric rgbd images . we demonstrate that we can use our representation successfully to predict 3d saliency in egocentric rgbd images . based on our earlier hypothesis , we conjecture that objects from the 3d world map to an egocentric rgbd image with some predictable _ shape _ , _ location _ , _ size _ and _ depth_. we encode such characteristics in a region of interest @xmath3 using an egoobject map , @xmath4 \in \mathds{r}^{n_s\times n_l \times n_b \times n_d \times n_c}$ ] where @xmath5 , @xmath6 , @xmath7 , @xmath8 , and @xmath9 are the number of the feature dimension for shape @xmath10 , location @xmath11 , size @xmath12 , depth @xmath13 , and context @xmath14 , respectively . a shape feature , @xmath15^\mathsf{t}$ ] captures a geometric properties such as area , perimeter , edges , and orientation of @xmath3 . * @xmath16 : perimeter divided by the squared root of the area , the area of a region divided by the area of the bounding box , major and minor axes lengths . * @xmath17 : we employ boundary cues @xcite , which include , sum and average contour strength of boundaries in region @xmath3 and also minimum and maximum ultrametric - contour values that lead to appearance and disappearance of the smaller regions inside @xmath3 @xcite . * @xmath18 : eccentricity and orientation of @xmath3 and also the diameter of a circle with the same area as region @xmath3 . a location feature @xmath19^\mathsf{t}$ ] encode spatial prior of objects imaged in an egocentric view : * @xmath20 : normalized bounding box coordinates and the centroid of a region @xmath3 . * @xmath21 : we also compute horizontal and vertical distances from the centroid of @xmath1 to the center of an image , and also to the mid - points of each border in the image . a size feature @xmath22^\mathsf{t}$ ] encodes the size of the bounding box and area of the region . * @xmath23 : area and perimeter of region @xmath3 . * @xmath24 : area and aspect ratio of the bounding box corresponding to the region @xmath3 . @xmath25^\mathsf{t}$ ] encodes a spatial distribution depth within @xmath3 . * @xmath26 : minimum , average , maximum , depth and also standard deviation of depth in a region @xmath3 . * @xmath27 : @xmath28 spatial depth histograms over the region @xmath3 . * @xmath29 : @xmath30 depth histograms over the region @xmath3 aligned to its major axis . * @xmath31 : @xmath28 spatial _ normalized _ depth histograms over the region @xmath3 . * @xmath32 : @xmath30 _ normalized _ depth histograms over the region @xmath3 aligned to its major axis . in addition , we include a context feature @xmath14 that encodes a spatial relationship between near regions in the egocentric image . given two regions , @xmath33 computes a distance between two features , i.e. , @xmath34^\mathsf{t}.\nonumber\end{aligned}\ ] ] given a target region , @xmath3 , the context feature @xmath35^\mathsf{t}$ ] computes the relationship with @xmath36 neighboring regions , @xmath37 : * @xmath38 : @xmath39 * @xmath40 : @xmath41 * @xmath42 : @xmath43 * @xmath44 : @xmath45 where @xmath46 and @xmath47 are the feature vector constructued by the min - pooling and max - pooling of neighboring regions for each dimension . @xmath48 takes average of neighboring features and @xmath49 is the feature of the top @xmath50 nearest neighbor . * summary . * for every region of interest @xmath3 in an egocentric rgbd frame , we produce a @xmath51 dimensional feature vector denoted by @xmath52 . we note that some of these features have been successfully used previously in tasks other than 3d saliency detection @xcite . additionally , observe that we do not use any object - level feature or hand or skin detectors as is done @xcite . this is because , in this work , we are primarily interested in studying the idea that salient objects from the 3d world are mapped to an egocentric rgbd frame with a consistent shape , location , size and depth patterns . we encode these cues with our egoobject representation and show its effectiveness on egocentric rgbd data in the later sections of the paper . given an rgbd frame as an input to our problem , we first feed rgb channels to an mcg @xcite method , which generates @xmath53 proposal regions . then , for each of these regions @xmath3 , we generate our proposed features @xmath52 and use it as an input to the random forest classifier ( rf ) . using a rf , we aim to learn the function that takes the feature vector @xmath52 corresponding to a particular region @xmath3 as an input , and produces an output for one our proposed tasks for region @xmath3 ( i.e. saliency value or interaction classification ) . we can formally write this function as @xmath54 . we apply the following pipeline for the following three tasks : 3d saliency detection , future saliency prediction , and interaction classification . however , for each of these tasks we define a different output objective @xmath55 and train rf classifier according to that objective separately for each task . below we describe this procedure for each task in more detail . * 3d saliency detection . * we train a random forest _ regressor _ to predict region s @xmath3 intersection over union ( iou ) with a ground truth salient object . to train the rf regressor we sample @xmath56 regions from our dataset , and extract our features from each of these regions . we then assign a corresponding ground truth iou value to each of them and train a rf regressor using @xmath57 trees . our rf learns the mapping @xmath58 $ ] where @xmath55 denotes the ground truth iou value between the @xmath3 and the ground truth salient object . to deal with the imbalance issue , we sample an equal number of examples corresponding to the iou values of @xmath59,[0.25,0.5],[0.5,0.75]$ ] , and @xmath60 $ ] . at testing time , we use mcg @xcite to generate @xmath53 regions of interest . we then apply our trained rf for every region @xmath3 and predict @xmath61 , which denotes the saliency of a region @xmath3 . we note that mcg produces the set of regions that overlap with each other . thus , for the pixels belonging to multiple overlapping regions @xmath62 , we assign a saliency value that corresponds to the maximum predicted value across the overlapping regions ( i.e. @xmath63 ) . we illustrate the basic pipeline of our approach in fig . [ fig : method ] . * future saliency prediction . * for future saliency prediction , given a video frame , we want to predict , which object will be salient ( i.e. used by a person ) after @xmath64 seconds . we hypothesize that the gaze direction is one of the most informative cues that are indicative of person s future behavior . however , gaze signal may be noisy if we consider only a single frame in the video . for instance , this may happen due to the person s attention being focused somewhere else for a split second or due to the shift in the camera . to make our approach more robust to the fluctuations of person s gaze , we incorporate simple temporal features into our system . our goal is to use these temporal cues to normalize the gaze direction captured by an egocentric camera and make it more robust to the small camera shifts . thus , given a frame @xmath65 which encodes time @xmath66 , we also consider frames @xmath67 . we pair up each of these frames @xmath68 with @xmath65 and compute their respective homography matrix @xmath69 . we then use each @xmath69 to recompute the image center @xmath70 in the current frame @xmath65 . for every region @xmath1 we then recompute its distance @xmath71 to the new center @xmath70 for all @xmath72}$ ] and concatenate these new distances to the original features @xmath52 . such gaze normalization scheme ensures robustness to our system in the case of gaze fluctuations . seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] seconds , where the same object is salient . our goal here is to predict an object that will be salient after @xmath66 seconds . ] * interaction classification . * most of the current computer vision systems classify objects by specific object class templates ( cup , phone , etc ) . however , these templates are not very informative and can not be used effectively beyond the tasks of object detection . adding object s function , and the type of interaction related to that object would allow researchers to tackle a wider array of problems overlapping vision and psychology . to predict an interaction type at a given frame , for each frame we select top @xmath73 highest ranked regions @xmath74 according to their predicted saliency score . we then classify each of these regions either as sight or touch . finally , we take the majority label from these @xmath73 classification predictions , and use it to classify an entire frame as sight or touch . \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c \| c ? c & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap & mf & ap + fts @xcite & 7.3 & 0.6 & 8.9 & 1.3 & 10.6 & 1.2 & 5.8 & 0.5 & 4.5 & 1.1 & 17.2 & 2.0 & 20.0 & 2.5 & 5.0 & 0.7 & 9.9 & 1.2 + mcg @xcite & 10.4 & 4.7 & 13.8 & 7.0 & 21.1 & 12.7 & 7.1 & 2.9 & 12.5 & 5.7 & 23.6 & 12.2 & 31.2 & 14.9 & 11.1 & 5.1 & 16.4 & 8.1 + gbmr @xcite & 8.0 & 3.0 & 15.6 & 6.8 & 14.7 & 7.0 & 6.8 & 3.0 & 4.3 & 1.3 & 32.7 & 18.3 & 46.0 & 30.2 & 12.9 & 5.7 & 17.6 & 9.4 + salobj @xcite & 7.2 & 2.7 & 19.9 & 7.4 & 21.3 & 10.0 & 15.4 & 5.1 & 5.8 & 2.2 & 24.1 & 9.2 & 49.3 & 28.3 & 9.0 & 3.4 & 19.0 & 8.5 + gbvs @xcite & 7.2 & 3.0 & 21.3 & 11.4 & 20.0 & 10.6 & 16.1 & 8.8 & 4.3 & 1.4 & 23.1 & 13.8 & 50.9 & * 50.2 & 11.6 & 5.7 & 19.3 & 13.1 + objectness @xcite & 11.5 & 5.6 & * 35.1 & * 24.3 & 39.2 & 29.4 & 11.7 & 6.9 & 4.7 & 1.9 & 27.1 & 17.1 & 47.4 & 42.2 & 13.0 & 6.4 & 23.7 & 16.7 + * ours ( rgb ) & 25.7 & 16.2 & 34.9 & 21.8 & 37.0 & 23.0 & 23.3 & 14.4 & * 28.9 & * 18.5 & 32.0 & 18.7 & * 56.0 & 39.6 & 30.3 & 21.8 & 33.5 & 21.7 + * ours ( rgb - d ) & * 36.9 & * 26.6 & 30.6 & 18.2 & * 55.3 & * 45.4 & * 26.8 & * 19.3 & 18.8 & 10.5 & * 37.9 & * 25.4 & 50.6 & 38.4 & * 40.2 & * 28.5 & * 37.1 & * 26.5 + * * * * * * * * * * * * * * * * * * * * we now present our egocentric rgbd saliency dataset . our dataset records people s interactions with their environment from a first - person view in a variety of settings such as shopping , cooking , dining , etc . we use egocentric - stereo cameras to capture the depth of a scene as well . we note that in the context of our problem , the depth information is particularly useful because it provides an accurate distance from an object to a person . since we hypothesize that a salient object from the 3d world maps to an egocentric rgbd frame with a predictable depth characteristic , we can use depth information as an informative cue for 3d saliency detection task . our dataset has annotations for three different tasks : saliency detection , future saliency prediction , and interaction classification . these annotations enables us to train our models in a supervised fashion and quantitatively evaluate our results . we now describe particular characteristics of our dataset in more detail . * data collection . * we use two stereo gopro hero 3 ( black edition ) cameras with @xmath75 baseline to capture our dataset . all videos are recorded at @xmath76 with @xmath77 . the stereo cameras are calibrated prior to the data collection and synchronized manually with a synchronization token at the beginning of each sequence . * depth computation . * we compute disparity between the stereo pair after stereo rectification . a cost space of stereo matching is generated for each scan line and match each pixel by exploiting dynamic programming in a coarse - to- fine manner . * sequences . * we record @xmath78 video sequences that capture people s interactions with object in a variety of different environments . these sequences include : cooking , supermarket , eating , hotel @xmath79 , hotel @xmath80 , dishwashing , foodmart , and kitchen sequences . * saliency annotations . * we use grabcut software @xcite to annotate salient regions in our dataset . we generate @xmath81 annotated frames for kitchen , cooking , and eating sequences , @xmath82 and @xmath83 annotated frames for supermarket and foodmart sequences respectively , @xmath84 and @xmath85 annotated frames for hotel @xmath79 and hotel @xmath80 sequences respectively , and @xmath86 annotated frames for dishwashing sequence ( for a total of @xmath87 frames with per - pixel salient object annotations ) . in fig . [ fig : data_po ] , we illustrate a few images from our dataset and the depth channels corresponding to these images . to illustrate ground truth labels , we overlay these images with saliency annotations . additionally , in fig . [ fig : data_stats ] , we provide statistics that capture different properties of our dataset such as the location , depth , and size of annotated salient regions from all sequences . each video sequence from our dataset is marked by a different color in this figure . we observe that these statistics suggest that different video sequences in our dataset exhibit different characteristics , and captures a variety of diverse interactions between people and objects . * annotations for future saliency prediction . * in addition , we also label our dataset to predict future saliency in egocentric rgbd image after @xmath64 frames . specifically , we first find the frame pairs that are @xmath64 frames apart , such that the same object is present in both of the frames . we then check that this object is non - salient in the earlier frame and that it is salient in the later frame . finally , we generate per - pixel annotations for these objects in both frames . we do this for the cases where the pair of frames are @xmath88 , and @xmath89 seconds apart . we produce @xmath90 annotated frames for kitchen , @xmath91 for cooking , @xmath92 for eating , @xmath93 for supermarket , @xmath90 for hotel @xmath79 , @xmath94 for hotel @xmath80 , @xmath95 for foodmart , and @xmath90 frames dishwashing sequences . we present some examples of these annotations in fig . [ fig : data_fut ] . * annotations for interaction classification . * to better understand the nature of people s interactions with their environment we also annotate each interaction either as _ sight _ or as _ touch_. in this section , we present the results on our egocentric rgbd saliency dataset for three different tasks , which include 3d saliency detection , future saliency prediction and interaction classification . we show that using our egoobject feature representation , we achieve solid quantitative and qualitative results for each of these tasks . to evaluate our results , we use the following procedure for all three tasks . we first train random forest ( rf ) using the training data from @xmath96 sequences . we then use this trained rf to test it on the sequence that was not used in the training data . such a setup ensures that our classifier is learning a meaningful pattern in the data , and thus , can generalize well on new data instances . we perform this procedure for each of the @xmath78 sequences separately and then use the resulting rf model to test on its corresponding sequence . for the saliency detection and future saliency prediction tasks , our method predicts pixelwise saliency for each frame in the sequence . to evaluate our results we use two different measures : a maximum f - score ( mf ) along the precision - recall curve , and average precision ( ap ) . for the task of interaction classification , we simply classify each interaction either as sight or as touch . thus , to evaluate our performance we use the fraction of correctly classified predictions . we now present the results for each of these tasks in more detail . detecting 3d saliency in an egocentric rgbd setting is a novel and relatively unexplored problem . thus , we compare our method with the most successful saliency detection systems for 2d images . sequences . these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] these visualizations demonstrate that in each of these sequences , our method captures an egocentric object prior that has a distinct shape , location , and size pattern . ] in table [ po_table ] , we present quantitative results for the saliency detection task on our dataset . we observe that our approach outperforms all the other methods by @xmath97 and @xmath98 in mf and ap evaluation metrics respectively . these results indicate that saliency detection methods designed for _ non - egocentric _ images do not generalize well to the _ egocentric _ images . this can be explained by the fact that in most _ non - egocentric _ saliency detection datasets , images are displayed at a pretty standard scale , with little occlusions , and also close to the center of an image . however , in the egocentric environment , salient objects are often occluded , they appear at a small scale and around many other objects , which makes this task more challenging . furthermore , we note that none of these baseline methods use depth information . based on the results , in table [ po_table ] , we observe that adding depth features to our framework provides accuracy gains of @xmath99 and @xmath100 according to mf and ap metrics respectively . finally , we observe that the results of different methods vary quite a bit from sequence to sequence . this confirms that our egocentric rgbd saliency dataset captures various aspects of people s interactions with their environment , which makes it challenging to design a method that would perform equally well in each of these sequences . based on the results , we see that our method achieves best results in @xmath96 and @xmath89 sequences ( out of @xmath78 ) according to mf and ap evaluation metrics respectively , which suggests that exploiting egocentric object prior via shape , location , size , and depth features allows us to predict visual saliency robustly across all sequences . additionally , we present our qualitative results in fig . [ fig : po_preds ] . our saliency heatmaps in this figure suggest that we can accurately capture different types of salient interactions with objects . furthermore , to provide a more interesting visualization of our learned egocentric object priors , we average our predicted saliency heatmaps for each of the @xmath89 selected sequences and visualize them in fig . [ fig : avg_preds ] . we note that these averaged heatmaps have a certain shape , location , and size characteristics , which suggests the existence of an egocentric object prior in egocentric rgbd images . in fig . [ fig : feats ] , we also analyze , which features contribute the most for the saliency detection task . the feature importance is quantified by the mean squared error reduction when splitting the node by that feature in a random forest . in this case , we manually assign each of our @xmath51 features to one of @xmath78 groups . these groups include shape , location , size , depth , shape context , location context , size context and depth context features ( as shown in fig . [ fig : feats ] ) . for each group , we average the importance value of all the features belonging to that group and present it in figure [ fig : feats ] . based on this figure , we observe that shape features contribute the most for saliency detection . additionally , since location features capture an approximate gaze of a person , they are deemed informative as well . furthermore , we observe that size and depth features also provide informative cues for capturing the saliency in an egocentric rgbd image . as expected , the context feature are least important . in this section , we present our results for the task of future saliency prediction . we test our trained rf model under three scenarios : predicting a salient object that will be used after @xmath88 , and @xmath89 seconds respectively . as one would expect , predicting the event further away from the present frame is more challenging , which is reflected by the results in table [ fut_table ] . for this task , we aim to use our egoobject representation to learn the cues captured by egocentric - stereo cameras that are indicative of person s future behavior . we compare our future saliency detector ( fsd ) to the saliency detector ( sd ) from the previous section and show that we can achieve superior results , which implies the existence and consistency of the cues that are indicative of person s future behavior . such cues may include person s gaze direction ( captured by an egocentric camera ) , or person s distance to an object ( captured by the depth channel ) , which are both pretty indicative of what the person may do next . in fig . [ fig : fut_preds ] , we visualize some of our future saliency predictions . based on these results , we observe , that even in a difficult environment such as supermarket , our method can make meaningful predictions . .future saliency results according to max f - score ( mf ) and average precision ( ap ) evaluation metrics . given a frame at time @xmath66 , our future saliency detector ( fsd ) predicts saliency for times @xmath101 , and @xmath102 ( denoted by seconds ) . as our baseline we use a saliency detector ( sd ) from section [ tech_approach ] of this paper . we show that in every case we outperform this baseline according to both metrics . this suggests that using our representation , we can consistently learn some of the egocentric cues such as gaze , or person s distance to an object that are indicative of people s future behavior . [ cols="^,^,^,^,^",options="header " , ] in this section , we report our results on the task of interaction classification . in this case , we only have two labels ( sight and touch ) and so we evaluate the performance as a fraction of correctly classified predictions . we compare our approach with a depth - based baseline , for which we learn an optimal depth threshold for each sequence , then for a given input frame , if a predicted salient region is further than this threshold , our baseline classifies that interaction as _ sight _ , otherwise the baseline classifies it as _ touch_. due to lack of space , we do not present the full results . however , we note that our approach outperforms depth - based baseline in @xmath89 out of @xmath78 categories and achieves @xmath103 higher accuracy on average in comparison to this baseline . we also illustrate some of the qualitative results in fig . [ fig : po_preds ] . these results indicate that we can use our representation to successfully classify people s interactions with objects by sight or touch . in this paper , we introduced a new psychologically inspired approach to a novel 3d saliency detection problem in egocentric rgbd images . we demonstrated that using our psychologically inspired egoobject representation we can achieve good results for the three following tasks : 3d saliency detection , future saliency prediction , and interaction classification . these results suggest that an egocentric object prior exists and that using our representation , we can capture and exploit it for accurate 3d saliency detection on our egocentric rgbd saliency dataset .	on a minute - to - minute basis people undergo numerous fluid interactions with objects that barely register on a conscious level . recent neuroscientific research demonstrates that humans have a fixed size prior for salient objects . this suggests that a salient object in 3d undergoes a consistent transformation such that people s visual system perceives it with an approximately fixed size . this finding indicates that there exists a consistent egocentric object prior that can be characterized by shape , size , depth , and location in the first person view . in this paper , we develop an egoobject representation , which encodes these characteristics by incorporating shape , location , size and depth features from an egocentric rgbd image . we empirically show that this representation can accurately characterize the egocentric object prior by testing it on an egocentric rgbd dataset for three tasks : the 3d saliency detection , future saliency prediction , and interaction classification . this representation is evaluated on our new egocentric rgbd saliency dataset that includes various activities such as cooking , dining , and shopping . by using our egoobject representation , we outperform previously proposed models for saliency detection ( relative @xmath0 improvement for 3d saliency detection task ) on our dataset . additionally , we demonstrate that this representation allows us to predict future salient objects based on the gaze cue and classify people s interactions with objects .	8,862	338

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